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| \title[A Channel-Based Perspective on Conjugate Priors]{A | |
| Channel-Based Perspective on Conjugate Priors\thanks{The research | |
| leading to these results has received funding from the European | |
| Research Council under the European Union's Seventh Framework | |
| Programme (FP7/2007-2013) / ERC grant agreement nr.~320571}} | |
| \author[B. Jacobs]{B\ls A\ls R\ls T\ns | |
| J\ls A\ls C\ls O\ls B\ls S\\ | |
| Institute for Computing and Information Sciences | |
| Radboud University\addressbreak | |
| P.O.Box 9010, 6500 GL Nijmegen, the Netherlands} | |
| \begin{document} | |
| \maketitle | |
| \begin{abstract} | |
| A desired closure property in Bayesian probability is that an updated | |
| posterior distribution be in the same class of distributions --- say | |
| Gaussians --- as the prior distribution. When the updating takes place | |
| via a statistical model, one calls the class of prior distributions | |
| the `conjugate priors' of the model. This paper gives (1)~an abstract | |
| formulation of this notion of conjugate prior, using channels, in a | |
| graphical language, (2)~a simple abstract proof that such conjugate | |
| priors yield Bayesian inversions, and (3)~a logical description of | |
| conjugate priors that highlights the required closure of the priors | |
| under updating. The theory is illustrated with several standard | |
| examples, also covering multiple updating. | |
| \end{abstract} | |
| \section{Introduction}\label{sec:intro} | |
| The main result of this paper, Theorem~\ref{thm:conjugateinversion}, | |
| is mathematically trivial. But it is not entirely trivial to see that | |
| this result is trivial. The effort and contribution of this paper lies | |
| in setting up a framework --- using the abstract language of channels, | |
| Kleisli maps, and string diagrams for probability theory --- to define | |
| the notion of conjugate prior in such a way that there is a trivial | |
| proof of the main statement, saying that conjugate priors yield | |
| Bayesian inversions. This is indeed what conjugate priors are meant to | |
| be. | |
| Conjugate priors form a fundamental topic in Bayesian theory. They | |
| are commonly described via a closure property of a class of prior | |
| distributions, namely as being closed under certain Bayesian updates. | |
| Conjugate priors are especially useful because they do not only | |
| involve a closure \emph{property}, but also a particular | |
| \emph{structure}, namely an explicit function that performs an | |
| analytical computation of posterior distributions via updates of the | |
| parameters. This greatly simplify Bayesian analysis. For instance, the | |
| $\betachan$ distribution is conjugate prior to the Bernoulli (or | |
| `flip') distribution, and also to the binomial distribution: updating | |
| a $\betachan(\alpha,\beta)$ prior via a Bernoulli/binomial statistical | |
| model yields a new $\betachan(\alpha',\beta')$ prior, with adapted | |
| parameters $\alpha',\beta'$ that can be computed explicitly from | |
| $\alpha,\beta$ and the observation at hand. Despite this importance, | |
| the descriptions in the literature of what it means to be a conjugate | |
| prior are remarkably informal. One does find several lists of classes | |
| of distributions, for instance at Wikipedia\footnote{See | |
| \url{https://en.wikipedia.org/wiki/Conjugate_prior} or online lists, | |
| such as | |
| \url{https://www.johndcook.com/CompendiumOfConjugatePriors.pdf}, | |
| consulted at Sept.\ 10, 2018}, together with formulas about how to | |
| re-compute parameters. The topic has a long and rich history in | |
| statistics (see \textit{e.g.}~\cite{Bishop06}), with much emphasis on | |
| exponential families~\cite{DiaconisY79}, but a precise, general | |
| definition is hard to find. | |
| We briefly review some common approaches, without any pretension to be | |
| complete: the definition in~\cite{Alpaydin10} is rather short, based | |
| on an example, and just says: ``We see that the posterior has the same | |
| form as the prior and we call such a prior a \emph{conjugate prior}.'' | |
| Also~\cite{RussellN03} mentions the term `conjugate prior' only in | |
| relation to an example. There is a separate section in~\cite{Bishop06} | |
| about conjugate priors, but no precise definition. Instead, there is | |
| the informal description ``\ldots the posterior distribution has the | |
| same functional form as the prior.'' The most precise definition | |
| (known to the author) is in~\cite[\S5.2]{BernardoS00}, where the | |
| conjugate family with respect to a statistical model, assuming a | |
| `sufficient statistic', is described. It comes close to our | |
| channel-based description, since it explicitly mentions the conjugate | |
| family as a conditional probability distribution with (re-computed) | |
| parameters. The approach is rather concrete however, and the high | |
| level of mathematical abstraction that we seek here is missing | |
| in~\cite{BernardoS00}. | |
| This paper presents a novel systematic perspective for precisely | |
| defining what conjugate priorship means, both via diagrams and via | |
| (probabilistic) logic. It uses the notion of `channel' as starting | |
| point. The basis of this approach lies in category theory, especially | |
| effectus theory~\cite{Jacobs15d,ChoJWW15b}. However, we try to make | |
| this paper accessible to non category theorists, by using the term | |
| `channel' instead of morphism in a Kleisli category of a suitable | |
| monad. Moreover, a graphical language is used for channels that | |
| hopefully makes the approach more intuitive. Thus, the emphasis of the | |
| paper is on \emph{what it means} to have conjugate priors. It does | |
| not offer new perspectives on how to find/obtain them. | |
| The paper is organised as follows. It starts in | |
| Section~\ref{sec:ideas} with a high-level description of the main | |
| ideas, without going into technical details. Preparatory definitions | |
| are provided in Sections~\ref{sec:Kleisli} and~\ref{sec:inversion}, | |
| dealing with channels in probabilistic computation, with a | |
| diagrammatic language for channels, and with Bayesian inversion. | |
| Then, Section~\ref{sec:conjugate} contains the novel channel-based | |
| definition of conjugate priorship; it also illustrates how several | |
| standard examples fit in this new | |
| setting. Section~\ref{sec:conjugatepriorinversion} establishes the | |
| (expected) close relationship between conjugate priors and Bayesian | |
| inversions. Section~\ref{sec:logic} then takes a fresh perspective, by | |
| re-describing the Bayesian-inversion based formulation in more logical | |
| terms, using validity and updating. This re-formulation captures the | |
| intended closure of a class of priors under updating in the most | |
| direct manner. It is used in Section~\ref{sec:multiple} to illustrate | |
| how multiple updates are handled, typically via a `sufficient | |
| statistic'. | |
| \section{Main ideas}\label{sec:ideas} | |
| This section gives an informal description of the main ideas | |
| underlying this paper. It starts with a standard example, and then | |
| proceeds with a step-by-step introduction to the essentials of the | |
| perspective of this paper. | |
| \begin{figure} | |
| \begin{center} | |
| \includegraphics[width=0.3\textwidth]{coin.eps} | |
| \includegraphics[width=0.3\textwidth]{coin-H.eps} | |
| \includegraphics[width=0.3\textwidth]{coin-HTTT.eps} | |
| \end{center} | |
| \caption{Uniform prior, and two posterior probability density | |
| functions on $[0,1]$, after observing head, and after observing | |
| head-tail-tail-tail. These functions correspond respectively to | |
| $\betachan(1,1)$, $\betachan(2,1)$, | |
| $\betachan(2,4)$. Example~\ref{ex:efprobcoin} below explains how | |
| these three plots are obtained, via actual Bayesian updates | |
| (inversions), and not by simply using the $\betachan$ functions.} | |
| \label{fig:coin} | |
| \end{figure} | |
| A well-known example in Bayesian reasoning is inferring the (unknown) | |
| bias of a coin from a sequence of consecutive head/tail observations. | |
| The bias is a number $r \in [0,1]$ in the unit interval, giving the | |
| `Bernoulli' or `flip' probability $r$ for head, and $1-r$ for | |
| tail. Initially we assume a uniform distribution for $r$, as described | |
| by the constant probability density function (pdf) on the left in | |
| Figure~\ref{fig:coin}. After observing one head, this pdf changes to | |
| the second picture. After observing head-tail-tail-tail we get the | |
| third pdf. These pictures are obtained by Bayesian inversion, see | |
| Section~\ref{sec:inversion}. | |
| It is a well-known fact that all the resulting distributions are | |
| instances of the $\betachan(\alpha,\beta)$ family of distributions, | |
| for different parameters $\alpha,\beta$. After each observation, one | |
| can re-compute the entire updated distribution, via Bayesian | |
| inversion, as in Example~\ref{ex:efprobcoin}. But in fact there is a | |
| much more efficient way to obtain the revised distribution, namely by | |
| computing the new parameter values: increment $\alpha$ by one, for | |
| head, and increment $\beta$ by one for tail, see | |
| Examples~\ref{ex:betaflip} and~\ref{ex:betaflipupdate} for | |
| details. The family of distributions $\betachan(\alpha,\beta)$, | |
| indexed by parameters $\alpha,\beta$, is thus suitably closed under | |
| updates with Bernoulli. It is the essence of the statement that | |
| $\betachan$ is conjugate prior to Bernoulli. This will be made precise | |
| later on. | |
| Let $X = (X, \Sigma)$ be a measurable space, where $\Sigma \subseteq | |
| \Pow(X)$ is a $\sigma$-algebra of measurable subsets. We shall write | |
| $\Giry(X)$ for the set of probability distributions on $X$. Elements | |
| $\omega\in\Giry(X)$ are thus countably additive functions | |
| $\omega\colon\Sigma \rightarrow [0,1]$ with $\omega(X) = 1$. | |
| \begin{description} | |
| \item[Idea 1:] A \emph{family} of distributions on $X$, indexed by a | |
| measurable space $P$ of parameters, is a (measurable) function $P | |
| \rightarrow \Giry(X)$. Categorically, such a function is a | |
| \emph{Kleisli} map for $\Giry$, considered as monad on the category | |
| of measurable spaces (see Section~\ref{sec:Kleisli}). These Kleisli | |
| maps are also called \emph{channels}, and will be written simply as | |
| arrows $P \rightarrow X$, or diagrammatically as boxes | |
| \ $\scriptstyle\vcenter{\hbox{\begin{tikzpicture}[font=\tiny] | |
| \node[arrow box, scale=0.5] (c) at (0,0) {$\;$}; | |
| \draw (c) to (0,0.35); | |
| \draw (c) to (0,-0.35); | |
| \node at (0.2,0.3) {$X$}; | |
| \node at (0.2,-0.3) {$P$}; | |
| \end{tikzpicture}}}$ where we imagine that information is flowing upwards. | |
| \end{description} | |
| \noindent The study of families of distributions goes back a long way, | |
| \textit{e.g.} as `experiments'~\cite{Blackwell51}. | |
| Along these lines we shall describe the family of $\betachan$ | |
| distributions as a channel with $P = \R_{>0}\times\R_{>0}$ and $X = | |
| [0,1]$, namely as function: | |
| \begin{equation} | |
| \label{diag:beta} | |
| \vcenter{\xymatrix@C+1pc{ | |
| \R_{>0}\times\R_{>0}\ar[r]^-{\betachan} & \Giry([0,1]) | |
| }} | |
| \end{equation} | |
| \noindent For $(\alpha,\beta) \in \R_{>0}\times\R_{>0}$ there is the | |
| probability distribution $\betachan(\alpha,\beta) \in \Giry([0,1])$ | |
| determined by its value on a measurable subset $M\subseteq [0,1]$, | |
| which is obtained via integration: | |
| \begin{equation} | |
| \label{eqn:beta} | |
| \begin{array}{rcl} | |
| \betachan(\alpha,\beta)(M) | |
| & = & | |
| {\displaystyle\int_{M}} | |
| \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}\intd x, | |
| \end{array} | |
| \end{equation} | |
| \noindent where $B(\alpha,\beta) = \int_{[0,1]} | |
| x^{\alpha-1}(1-x)^{\beta-1} \intd x$ is a normalisation constant. | |
| A conjugate prior relationship involves a family of distributions $P | |
| \rightarrow \Giry(X)$ which is closed wrt.\ updates based on | |
| observations (or: data) from a separate domain $O$. Each `parameter' | |
| element $x\in X$ gives rise to a separate distribution on $O$. This is | |
| what is usually called a \emph{statistical} or \emph{parametric} | |
| model. We shall also describe it as a channel. | |
| \begin{description} | |
| \item[Idea 2:] The observations for a family $P \rightarrow \Giry(X)$ | |
| arise via another ``Kleisli'' map $X \rightarrow \Giry(O)$ | |
| representing the statistical model. Conjugate priorship will be | |
| defined for two such composable channels $P \rightarrow X | |
| \rightarrow O$, where $O$ is the space of observations. | |
| \end{description} | |
| In the above coin example, the space $O$ of observations is the | |
| two-element set $2 = \{0,1\}$ where $0$ is for tail and $1$ for head. | |
| The Bernoulli channel is written as $\flipchan \colon [0,1] | |
| \rightarrow \Giry(2)$. A probability $r\in [0,1]$ determines a | |
| Bernoulli/flip/coin probability distribution $\flipchan(r) \in | |
| \Giry(2)$ on $2$, formally sending the subset $\{1\}$ to $r$ and | |
| $\{0\}$ to $1-r$. | |
| \begin{description} | |
| \item[Idea 3:] A channel $c\colon P\rightarrow X$ is a conjugate prior | |
| to a channel $d\colon X \rightarrow O$ if there is a \emph{parameter | |
| translation function} $h\colon P\times O \rightarrow P$ satisfying | |
| a suitable equation. | |
| \end{description} | |
| \noindent The idea is that $c(p)$ is a prior, for $p\in P$, which gets | |
| updated via the statistical model (channel) $d$, in the light of | |
| observation $y\in O$. The revised, updated distribution is | |
| $c(h(p,y))$. The model $d$ is usually written as a conditional | |
| probability $d(y\mid \theta)$. | |
| In the coin example we have $h\colon \R_{>0}\times\R_{>0} \times 2 | |
| \rightarrow \R_{>0}\times\R_{>0}$ given by $h(\alpha,\beta,1) = | |
| (\alpha+1,\beta)$ and $h(\alpha,\beta,0) = (\alpha,\beta+1)$, see | |
| Example~\ref{ex:betaflip} below for more information. | |
| What has been left unexplained is the `suitable' equation that the | |
| parameter translation function $h\colon P\times O \rightarrow P$ | |
| should satisfy. It is not entirely trivial, because it is an equation | |
| between channels in what is called the Kleisli category $\Kl(\Giry)$ | |
| of the Giry monad $\Giry$. At this stage we need to move to a more | |
| categorical description. The equation, which will appear in | |
| Definition~\ref{def:conjugateprior}, bears similarities with the | |
| notion of Bayesian inversion, which will be introduced in | |
| Section~\ref{sec:inversion}. | |
| \section{Channels and conditional probabilities}\label{sec:Kleisli} | |
| This section will describe conditional probabilities as arrows and | |
| will show how to compose them. Thereby we are entering the world of | |
| category theory. We aim to suppress the underlying categorical | |
| machinery and make this work accessible to readers without such | |
| background. For those with categorical background knowledge: we will | |
| be working in the Kleisli categories of the distribution monad $\Dst$ | |
| for discrete probability, and of the Giry monad $\Giry$ for continuous | |
| probability, see \textit{e.g.}~\cite{Giry82,Panangaden09,Jacobs17a}. | |
| Discrete distributions may be seen as a special case of continuous | |
| distributions, via a suitable inclusion map $\Dst \rightarrow \Giry$. | |
| Hence one could give one account, using $\Giry$ only. However, in | |
| computer science, unlike for instance in statistics, discrete | |
| distributions are so often used that they merit separate treatment. | |
| We thus start with discrete probability. We write a (finite, discrete) | |
| distribution on a set $X$ as a formal convex sum $r_{1}\ket{x_1} + | |
| \cdots + r_{n}\ket{x_n}$ of elements $x_{i}\in X$ and probabilities | |
| $r_{i}\in [0,1]$ with $\sum_{i}r_{i}=1$. The `ket' notation $\ket{-}$ | |
| is syntactic sugar, used to distinguish elements of $x$ from their | |
| occurrence $\ket{x}$ in such formal convex sums\footnote{Sometimes | |
| these distributions $\sum_{i}r_{i}\ket{x_i}$ are called | |
| `multinomial' or `categorical'; the latter terminology is confusing | |
| in the present context.}. A distribution as above can be identified | |
| with a `probability mass' function $\omega \colon X \rightarrow [0,1]$ | |
| which is $r_{i}$ on $x_{i}$ and $0$ elsewhere. We often implicitly | |
| identify distributions with such functions. We shall write $\Dst(X)$ | |
| for the set of distributions on $X$. | |
| We shall focus on functions of the form $c\colon X \rightarrow | |
| \Dst(Y)$. They give, for each element $x\in X$ a distribution $c(x)$ | |
| on $Y$. Hence such functions form an $X$-indexed collection | |
| $\big(c(x)\big)_{x\in X}$ of distributions $c(x)$ on $Y$. They can be | |
| understood as \emph{conditional} probabilities $P(y\mid x) = r$, if | |
| $c(x)$ is of the form $\cdots r\ket{y}\cdots$, with weight $r = | |
| c(x)(y)\in[0,1]$ for $y\in Y$. Thus, by construction, $\sum_{y} | |
| P(y\mid x) = 1$, for each $x\in X$. Moreover, if the sets $X$ and $Y$ | |
| are finite, we can describe $c\colon X \rightarrow \Dst(Y)$ as a | |
| stochastic matrix, with entries $P(y\mid x)$, adding up to one --- per | |
| row or column, depending on the chosen orientation of the matrix. | |
| We shall often write functions $X \rightarrow \Dst(Y)$ simply as | |
| arrows $X \rightarrow Y$, call them `channels', and write them as | |
| `boxes' in diagrams. This arrow notation is justified, because there | |
| is a natural way to compose channels, as we shall see shortly. But | |
| first we describe \emph{state transformation}, also called | |
| \emph{prediction}. Given a channel $c\colon X \rightarrow \Dst(Y)$ and | |
| a state $\omega\in\Dst(X)$, we can form a new state, written as $c \gg | |
| \omega$, on $Y$. It is defined as: | |
| \begin{equation} | |
| \label{eqn:discstatransf} | |
| \begin{array}{rcl} | |
| c \gg \omega | |
| & \coloneqq & | |
| {\displaystyle\sum_y} \big(\sum_{x} \omega(x)\cdot c(x)(y)\big)\bigket{y}. | |
| \end{array} | |
| \end{equation} | |
| \noindent The outer sum $\sum_{y}$ is a formal convex sum, whereas the | |
| inner sum $\sum_{x}$ is an actual sum in the unit interval $[0,1]$. | |
| Using state transformation $\gg$ it is easy to define composition of | |
| channels: given functions $c\colon X \rightarrow \Dst(Y)$ and $d\colon | |
| Y \rightarrow \Dst(Z)$, we use the ordinary composition symbol | |
| $\after$ to form a composite channel $d \after c \colon X \rightarrow | |
| \Dst(Z)$, where: | |
| \begin{equation} | |
| \label{eqn:discretecomposition} | |
| \begin{array}{rcccl} | |
| (d \after c)(x) | |
| & \coloneqq & | |
| d \gg c(x) | |
| & = & | |
| {\displaystyle\sum_{z\in Z}} \big(\sum_{y} c(x)(y)\cdot d(y)(z)\big) | |
| \bigket{z}. | |
| \end{array} | |
| \end{equation} | |
| \noindent Essentially, this is matrix composition for stochastic | |
| matrices. Channel composition $\after$ is associative, and also has a | |
| neutral element, namely the identity channel $\eta \colon X | |
| \rightarrow X$ given by the `Dirac' function $\eta(x) = 1\ket{x}$. It | |
| is not hard to see that $(d \after c) \gg \omega = d \gg (c \gg | |
| \omega)$. | |
| \medskip | |
| We turn to channels in continuous probability. As already mentioned in | |
| Section~\ref{sec:ideas}, we write $\Giry(X)$ for the set of | |
| probability distributions $\omega\colon \Sigma_{X} \rightarrow [0,1]$, | |
| where $X = (X,\Sigma_{X})$ is a measurable space. These probability | |
| distributions are (also) called states. The set $\Giry(X)$ carries a | |
| $\sigma$-algebra itself, but that does not play an important role | |
| here. Each element $x\in X$ yields a probability measure | |
| $\eta(x)\in\Giry(X)$, with $\eta(x)(M) = \indic{M}(x)$, which is $1$ | |
| if $x\in M$ and $0$ otherwise. This map $\indic{M} \colon X | |
| \rightarrow [0,1]$ is called the indicator function for the subset | |
| $M\in\Sigma_{X}$. | |
| For a state/measure $\omega\in\Giry(X)$ and a measurable function | |
| $f\colon X \rightarrow \R_{\geq 0}$ we write $\int f\intd \omega$ for | |
| the Lebesgue integral, if it exists. We follow the notation | |
| of~\cite{Jacobs13a} and refer there for details, or alternatively, | |
| to~\cite{Panangaden09}. We recall that an integral $\int_{M} f\intd | |
| \omega$ over a measurable subset $M\subseteq X$ of the domain of $f$ | |
| is defined as $\int \indic{M}\cdot f \intd\omega$, and that $\int | |
| \indic{M} \intd\omega = \omega(M)$. Moreover, $\int f \intd\eta(x) = | |
| f(x)$. | |
| For a measurable function $g\colon X\rightarrow Y$ between measurable | |
| spaces $X,Y$ there is the `push forward' function $\Giry(g) \colon | |
| \Giry(X) \rightarrow \Giry(Y)$, given by $\Giry(g)(\omega)(N) = | |
| \omega\big(g^{-1}(N)\big)$. It satisfies: | |
| \begin{equation} | |
| \label{eqn:invmeasure} | |
| \begin{array}{rcl} | |
| \displaystyle\int f \intd \Giry(g)(\omega) | |
| & = & | |
| \int f \after g \intd \omega. | |
| \end{array} | |
| \end{equation} | |
| \noindent Often, the measurable space $X$ is a subset $X\subseteq\R$ | |
| of the real numbers and a probability distribution $\omega$ on $X$ is | |
| given by a probability density function (pdf), that is, by a | |
| measurable function $f\colon X \rightarrow \R_{\geq 0}$ with $\int_{X} | |
| f(x) \intd x = 1$. Such a pdf $f$ gives rise to a state $\omega | |
| \in\Giry(X)$, namely: | |
| \begin{equation} | |
| \label{eqn:statefrompdf} | |
| \begin{array}{rcl} | |
| \omega(M) | |
| & = & | |
| \displaystyle \int_{M} f(x) \intd x. | |
| \end{array} | |
| \end{equation} | |
| \noindent We then write $\omega = \int f$. | |
| In this continuous context a channel is a measurable function $c\colon | |
| X \rightarrow \Giry(Y)$, for measurable spaces $X,Y$. Like in the | |
| discrete case, it gives an $X$-indexed collection | |
| $(\big(c(x)\big)_{x\in X}$ of probability distributions on $Y$. | |
| The channel $c$ can transform a state $\omega\in\Giry(X)$ on $X$ | |
| into a state $c \gg \omega \in\Giry(Y)$ on $Y$, given on a measurable | |
| subset $N\subseteq Y$ as: | |
| \begin{equation} | |
| \label{eqn:statetransformation} | |
| \begin{array}{rcl} | |
| \big(c \gg \omega\big)(N) | |
| & = & | |
| \displaystyle\int c(-)(N) \intd \omega. | |
| \end{array} | |
| \end{equation} | |
| \noindent For another channel $d\colon Y \rightarrow \Giry(Z)$ there | |
| is a composite channel $d \after c \colon X \rightarrow \Giry(Z)$, via | |
| integration: | |
| \begin{equation} | |
| \label{eqn:continuouscomposition} | |
| \begin{array}{rcccl} | |
| \big(d \after c\big)(x)(K) | |
| & \coloneqq & | |
| \big(d \gg c(x)\big)(K) | |
| & = & | |
| \displaystyle\int d(-)(K) \intd c(x) | |
| \end{array} | |
| \end{equation} | |
| In many situations a channel $c\colon X \rightarrow \Giry(Y)$ is given | |
| by an indexed probability density function (pdf) $u\colon X \times Y | |
| \rightarrow \R_{\geq 0}$, with $\int u(x,y)\intd y = 1$ for each $x\in | |
| X$. The associated channel $c$ is: | |
| \begin{equation} | |
| \label{eqn:channelfrompdf} | |
| \begin{array}{rcl} | |
| c(x)(N) | |
| & = & | |
| \displaystyle \int_{N} u(x,y) \intd y. | |
| \end{array} | |
| \end{equation} | |
| \noindent In that case we simply write $c = \int u$ and call $c$ a | |
| pdf-channel. We have already seen such a description of the | |
| $\betachan$ distribution as a pdf-channel in~\eqref{eqn:beta}. | |
| (In these pdf-channels $X \rightarrow Y$ we use a collection of pdf's | |
| $u(x,-)$ which are all dominated by the Lebesgue measure. This | |
| domination happens via the relationship $\ll$ of absolute continuity, | |
| using the Radon-Nikodym Theorem, see | |
| \textit{e.g.}~\cite{Panangaden09}.) | |
| Various additional computation rules for integrals are given in the | |
| Appendix. | |
| \section{Bayesian inversion in string diagrams}\label{sec:inversion} | |
| In this paper we make superficial use of string diagrams to | |
| graphically represent sequential and parallel composition of channels, | |
| mainly in order to provide an intuitive visual overview. We refer | |
| to~\cite{Selinger11} for mathematical details, and mention here only | |
| the essentials. | |
| A channel $X \rightarrow Y$, for instance of the sort discussed in | |
| the previous section, can be written as a box \ $\scriptstyle\vcenter{\hbox{\begin{tikzpicture}[font=\tiny] | |
| \node[arrow box, scale=0.5] (c) at (0,0) {$\;$}; | |
| \draw (c) to (0,0.35); | |
| \draw (c) to (0,-0.35); | |
| \node at (0.2,0.3) {$Y$}; | |
| \node at (0.2,-0.3) {$X$}; | |
| \end{tikzpicture}}}$ with information flowing upwards, from the | |
| wire labeled with $X$ to the wire labeled with $Y$. Composition of | |
| channels, as in~\eqref{eqn:discretecomposition} | |
| or~\eqref{eqn:continuouscomposition}, simply involves connecting | |
| wires (of the same type). The identity channel is just a wire. | |
| We use a triangle notation \ \raisebox{.3em}{$\scriptstyle\vcenter{\hbox{\begin{tikzpicture}[font=\tiny] | |
| \node[state, scale=0.5] (omega) at (0,0) {$\;$}; | |
| \draw (omega) to (0,0.2); | |
| \node at (0.15,0.15) {$X$}; | |
| \end{tikzpicture}}}$} for a state on $X$. It is special case of | |
| a channel, namely of the form $1 \rightarrow X$ with trivial singleton | |
| domain $1$. | |
| In the present (probabilistic) setting we allow copying of wires, | |
| written diagrammatically as $\copier$. We briefly describe such copy | |
| channels for discrete and continuous probability: | |
| \[ \xymatrix@R-1.8pc{ | |
| X\ar[r]^-{\copier} & \Dst(X\times X) | |
| & & | |
| X\ar[r]^-{\copier} & \Giry(X\times X) | |
| \\ | |
| x\ar@{|->}[r] & 1\ket{x,x} | |
| & & | |
| x\ar@{|->}[r] & \big(M\times N \mapsto \indic{M\cap N}(x)\big) | |
| } \] | |
| After such a copy we can use parallel channels. We briefly describe | |
| how this works, first in the discrete case. For channels $c\colon X | |
| \rightarrow \Dst(Y)$ and $d\colon A \rightarrow \Dst(B)$ we have a | |
| channel $c\otimes d \colon X\times A \rightarrow \Dst(Y\times B)$ | |
| given by: | |
| \[ \begin{array}{rcl} | |
| (c\otimes d)(x,a) | |
| & = & | |
| \displaystyle \sum_{y,b} c(x)(y)\cdot d(a)(b)\bigket{y,b}. | |
| \end{array} \] | |
| Similarly, in the continuous case, for channels $c\colon X | |
| \rightarrow \Giry(Y)$ and $d\colon A \rightarrow \Giry(B)$ we | |
| get $c\otimes d \colon X\times A \rightarrow \Giry(Y\times B)$ given | |
| by: | |
| \[ \begin{array}{rcl} | |
| (c\otimes d)(x,a)(M\times N) | |
| & = & | |
| c(x)(M)\cdot d(a)(N). | |
| \end{array} \] | |
| \noindent Recall that the product measure on $Y\times B$ is generated | |
| by measurable rectangles of the form $M\times N$, for $M\in\Sigma_{Y}$ | |
| and $N\in\Sigma_{B}$. | |
| We shall use a tuple $\tuple{c,d}$ as convenient abbreviation for | |
| $(c\otimes d) \after \copier$. Diagrammatically, parallel channels are | |
| written as adjacent boxes. | |
| We can now formulate what Bayesian inversion is. The definition is | |
| couched in purely diagrammatic language, but is applied only to | |
| probabilistic interpretations in this paper. | |
| \begin{definition} | |
| \label{def:inversion} | |
| The \emph{Bayesian inversion} of a channel $c\colon X \rightarrow Y$ | |
| with respect to a state $\omega$ of type $X$, if it exists, is a | |
| channel in the opposite direction, written as $c^{\dag}_{\omega} | |
| \colon Y \rightarrow X$, such that the following equation holds. | |
| \begin{equation} | |
| \label{diag:inversion} | |
| \vcenter{\hbox{\begin{tikzpicture}[font=\small] | |
| \node[state] (omega) at (0,0) {$\omega$}; | |
| \node[copier] (copier) at (0,0.3) {}; | |
| \node[arrow box] (c) at (0.5,0.95) {$c$}; | |
| \coordinate (X) at (-0.5,1.5); | |
| \coordinate (Y) at (0.5,1.5); | |
| \draw (omega) to (copier); | |
| \draw (copier) to[out=150,in=-90] (X); | |
| \draw (copier) to[out=15,in=-90] (c); | |
| \draw (c) to (Y); | |
| \end{tikzpicture}}} | |
| \qquad=\qquad | |
| \vcenter{\hbox{\begin{tikzpicture}[font=\small] | |
| \node[state] (omega) at (0,-0.55) {$\omega$}; | |
| \node[copier] (copier) at (0,0.3) {}; | |
| \node[arrow box] (d) at (-0.5,0.95) {$c^{\dag}_{\omega}$}; | |
| \coordinate (X) at (-0.5,1.5); | |
| \coordinate (Y) at (0.5,1.5); | |
| \node[arrow box] (c) at (0,-0.15) {$c$}; | |
| \draw (omega) to (c); | |
| \draw (c) to (copier); | |
| \draw (copier) to[out=165,in=-90] (d); | |
| \draw (copier) to[out=30,in=-90] (Y); | |
| \draw (d) to (X); | |
| \end{tikzpicture}}} | |
| \end{equation} | |
| \end{definition} | |
| The dagger notation $c^{\dag}_{\omega}$ is copied | |
| from~\cite{ClercDDG17}. There the state $\omega$ is left implicit, via | |
| a restriction to a certain comma category of kernels. In that setting | |
| the operation $(-)^{\dag}$ is functorial, and forms a dagger category | |
| (see \textit{e.g.}~\cite{AbramskyC09,Selinger07} for definitions). In | |
| particular, it preserves composition and identities of channels. | |
| Equation~\eqref{diag:inversion} can also be written as: | |
| $\tuple{\idmap, c} \gg \omega = \tuple{c^{\dag}_{\omega}, \idmap} \gg | |
| (c \gg \omega)$. Alternatively, in the discrete case, with variables | |
| explicit, it says: $c(x)(y)\cdot \omega(x) = c^{\dag}_{\omega}(y)(x) | |
| \cdot (c \gg \omega)(y)$. This comes close to the `adjointness' | |
| formulations that are typical for daggers. | |
| Bayesian inversion gives a channel-based description of Bayesian | |
| (belief) updates. We briefly illustrate this for the coin example from | |
| Section~\ref{sec:ideas}, using the $\EfProb$ language~\cite{ChoJ17b}. | |
| \begin{example} | |
| \label{ex:efprobcoin} | |
| In Section~\ref{sec:ideas} we have seen the channel $\flipchan \colon | |
| [0,1] \rightarrow 2$ that sends a probability $r\in [0,1]$ to the coin | |
| state $\flipchan(r) = r\ket{1} + (1-r)\ket{0}$ with bias $r$. The | |
| Bayesian inversion $2 \rightarrow [0,1]$ of this channel performs a | |
| belief update, after a head/tail observation. Without going into | |
| details we briefly illustrate how this works in the $\EfProb$ language | |
| via the following code fragment. The first line describes a channel | |
| \pythoninline{Flip} of type $[0,1] \rightarrow 2$, where $[0,1]$ is | |
| represented as \pythoninline{R(0,1)} and $2 = \{0,1\}$ as | |
| \pythoninline{bool\_dom}. The expression \pythoninline{flip(r)} | |
| captures a coin with bias \pythoninline{r}. | |
| \begin{python} | |
| >>> Flip = chan_fromklmap(lambda r: flip(r), R(0,1), bool_dom) | |
| >>> prior = uniform_state(R(0,1)) | |
| >>> w1 = Flip.inversion(prior)(True) | |
| >>> w2 = Flip.inversion(w1)(False) | |
| >>> w3 = Flip.inversion(w2)(False) | |
| >>> w4 = Flip.inversion(w3)(False) | |
| \end{python} | |
| \noindent The (continuous) states \pythoninline{w1} -- | |
| \pythoninline{w4} are obtained as successive updates of the uniform | |
| state \pythoninline{prior}, after successive observations | |
| \pythoninline{True}-\pythoninline{False}-\pythoninline{False}-\pythoninline{False}, | |
| for head-tail-tail-tail. The three probability density functions in | |
| Figure~\ref{fig:coin} are obtained by plotting the prior state, and | |
| also the two states \pythoninline{w1} and \pythoninline{w4}. | |
| \end{example} | |
| It is relatively easy to define Bayesian inversion in discrete | |
| probability theory: for a channel $c\colon X \rightarrow \Dst(Y)$ and | |
| a state/distribution $\omega\in\Dst(X)$ one can define a channel | |
| $c^{\dag}_{\omega} \colon Y \rightarrow \Dst(X)$ as: | |
| \begin{equation} | |
| \label{eqn:discreteinversion} | |
| \begin{array}{rcccl} | |
| c^{\dag}_{\omega}(y)(x) | |
| & = & | |
| \displaystyle\frac{\omega(x)\cdot c(x)(y)}{(c \gg \omega)(y)} | |
| & = & | |
| \displaystyle\frac{\omega(x)\cdot c(x)(y)}{\sum_{z}\omega(z)\cdot c(z)(y)}, | |
| \end{array} | |
| \end{equation} | |
| \noindent assuming that the denominator is non-zero. This corresponds | |
| to the familiar formula $P(B\mid A) = \nicefrac{P(A,B)}{P(A)}$ for | |
| conditional probability. The state $c^{\dag}_{\omega}(y)$ can | |
| alternatively be defined via updating the state $\omega$ with the | |
| point predicate $\{y\}$, transformed via $c$ into a predicate $c \ll | |
| \indic{\{y\}}$ on $X$, see Section~\ref{sec:logic} | |
| (and~\cite{JacobsZ16}) for details. | |
| \begin{Auxproof} | |
| Just to be sure: | |
| \[ \begin{array}{rcl} | |
| \big(\tuple{c^{\dag}_{\omega}, \idmap} \gg (c \gg \omega)\big)(x,y) | |
| & = & | |
| c^{\dag}_{\omega}(y)(x)\cdot (c \gg \omega)(y) | |
| \\ | |
| & = & | |
| \displaystyle\frac{\omega(x)\cdot c(x)(y)}{c_{*}(\omega)(y)} \cdot | |
| (c \gg \omega)(y) | |
| \\ | |
| & = & | |
| \omega(x)\cdot c(x)(y) | |
| \\ | |
| & = & | |
| (\tuple{\idmap,c} \gg \omega)(x,y). | |
| \end{array} \] | |
| \noindent And: | |
| \[ \begin{array}{rcccl} | |
| \omega|_{c \ll \indic{\{y\}}}(x) | |
| & = & | |
| \displaystyle\frac{\omega(x) \cdot (c \ll \indic{\{y\}})(x)}{\omega | |
| \models c \ll \indix{\{y\}}} | |
| & = & | |
| \displaystyle\frac{\omega(x)\cdot c(x)(y)}{\sum_{x}\omega(x)\cdot c(x)(y)}, | |
| \end{array} \] | |
| \end{Auxproof} | |
| The situation is much more difficult in continuous probability theory, | |
| since Bayesian inversions may not exist~\cite{AckermanFR11,Stoyanov14} | |
| or may be determined only up to measure zero. But when restricted to | |
| \textit{e.g.}~standard Borel spaces, as in~\cite{ClercDDG17}, | |
| existence is ensured, see also~\cite{Faden85,CulbertsonS14}. Another | |
| common solution is to assume that we have a pdf-channel: there is a | |
| map $u\colon X\times Y \rightarrow \R_{\geq 0}$ that defines a channel | |
| $c \colon X \rightarrow \Giry(Y)$, like in~\eqref{eqn:channelfrompdf}, | |
| as $c(x)(N) = \int_{N} u(x,y)\intd y$. Then, for a distribution | |
| $\omega\in\Giry(X)$ we can take as Bayesian inversion: | |
| \begin{equation} | |
| \label{eqn:continuousinversion} | |
| \begin{array}{rcl} | |
| c^{\dag}_{\omega}(y)(M) | |
| & = & | |
| \displaystyle\frac{\int_{M} u(-,y)\intd \omega}{\int_{X} u(-,y)\intd \omega} | |
| \\[+1em] | |
| & = & | |
| \displaystyle\frac{\int_{M} f(x)\cdot u(x,y) \intd x} | |
| {\int_{X} f(x) \cdot u(x,y) \intd x} | |
| \qquad \mbox{when } \omega = \int f(x)\intd x. | |
| \end{array} | |
| \end{equation} | |
| \noindent We prove that this definition satisfies the | |
| inversion Equation~\eqref{diag:inversion}, using the calculation | |
| rules from the Appendix. | |
| \[ \begin{array}{rcl} | |
| \big(\tuple{c^{\dag}_{\omega}, \idmap} \gg (c \gg \omega)\big)(M\times N) | |
| & \smash{\stackrel{\eqref{eqn:statetransformation}}{=}} & | |
| \displaystyle \int \tuple{c^{\dag}_{\omega}, \idmap}(-)(M\times N) | |
| \intd (c \gg \omega) | |
| \\ | |
| & \smash{\stackrel{(\ref{eqn:pdfintegration}, | |
| \ref{eqn:pdfstatetransformation})}{=}} & | |
| \displaystyle\int \big(\int f(x) \cdot u(x,y) \intd x\big) \cdot | |
| \tuple{c^{\dag}_{\omega}, \idmap}(y)(M\times N) \intd y | |
| \\ | |
| & \smash{\stackrel{\eqref{eqn:graphequation}}{=}} & | |
| \displaystyle\int \big(\int f(x) \cdot u(x,y) \intd x\big) \cdot | |
| c^{\dag}_{\omega}(y)(M) \cdot \indic{N}(y) \intd y | |
| \\ | |
| & \smash{\stackrel{\eqref{eqn:continuousinversion}}{=}} & | |
| \displaystyle\int_{N} \big(\int f(x) \cdot u(x,y) \intd x\big) \cdot | |
| \frac{\int_{M} f(x)\cdot u(x,y) \intd x} | |
| {\int f(x)\cdot u(x,y) \intd x} \intd y | |
| \\[+0.8em] | |
| & = & | |
| \displaystyle\int_{N} \int_{M} f(x) \cdot u(x,y) \intd x \intd y | |
| \\ | |
| & \smash{\stackrel{\eqref{eqn:pdfgraphstatetransformation}}{=}} & | |
| (\tuple{\idmap, c} \gg \omega)(M\times N). | |
| \end{array} \] | |
| \section{Conjugate priors}\label{sec:conjugate} | |
| We now come to the core of this paper. As described in the | |
| introduction, the informal definition says that a class of | |
| distributions is conjugate prior to a statistical model if the | |
| associated posteriors are \emph{in the same class} of distributions. | |
| The posteriors can be computed via Bayesian | |
| inversion~\eqref{eqn:continuousinversion} of the statistical model. | |
| This definition of `conjugate prior' is a bit vague, since it loosely | |
| talks about `classes of distributions', without further | |
| specification. As described in `Idea 1' in Section~\ref{sec:ideas}, we | |
| interpret `class of states on $X$' as channel $P\rightarrow X$, where | |
| $P$ is the type of parameters of the class. | |
| We have already seen this channel-based description for the class | |
| $\betachan$ distributions, in~\eqref{diag:beta}, as channel $\betachan | |
| \colon \R_{>0} \times \R_{>0} \rightarrow [0,1]$. This works more | |
| generally, for instance for Gaussian (normal) distributions | |
| $\normchan(\mu, \sigma)$, where $\mu$ is the mean parameter and | |
| $\sigma$ is the standard deviation parameter, giving a channel of the | |
| form: | |
| \begin{equation} | |
| \label{diag:normal} | |
| \vcenter{\xymatrix@C+1pc{ | |
| \R\times\R_{>0}\ar[r]^-{\normchan} & \Giry(\R) | |
| }} | |
| \end{equation} | |
| \noindent It is determined by its value on a measurable subset | |
| $M\subseteq \R$ as the standard integral: | |
| \begin{equation} | |
| \label{eqn:normal} | |
| \begin{array}{rcl} | |
| \normchan(\mu,\sigma)(M) | |
| & = & | |
| \displaystyle\int_{M} | |
| \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}\intd x | |
| \end{array} | |
| \end{equation} | |
| Given a channel $c\colon P \rightarrow X$, we shall look at states | |
| $c(p)$, for parameters $p\in P$, as priors. The statistical model, for | |
| which these $c(p)$'s will be described as conjugate priors, goes from | |
| $X$ to some other object $O$ of `observations'. Thus our starting | |
| point is a pair of (composable) channels the form: | |
| \begin{equation} | |
| \label{diag:conjugatechannels} | |
| \vcenter{\xymatrix{ | |
| P\ar[r]^-{c} & X\ar[r]^-{d} & O | |
| }} | |
| \qquad\qquad\mbox{or, as diagram,}\qquad\qquad | |
| \vcenter{\hbox{\begin{tikzpicture}[font=\small] | |
| \node[arrow box] (c) at (0.0,0.5) {$c$}; | |
| \node[arrow box] (d) at (0.0,1.2) {$d$}; | |
| \coordinate (P) at (0.0,0.0); | |
| \coordinate (Y) at (0.0,1.7); | |
| \draw (P) to (c); | |
| \draw (c) to (d); | |
| \draw (d) to (Y); | |
| \end{tikzpicture}}} | |
| \end{equation} | |
| \noindent Such a pair of composable channels may be seen as a 2-stage | |
| hierarchical Bayesian model. In that context the parameters $P$ are | |
| sometimes called `hyperparameters', see | |
| \textit{e.g.}~\cite{BernardoS00}. There, esp.\ in Defn~5.6 of | |
| conjugate priorship one can also distinguish two channels, written as | |
| $p(\theta\mid\tau)$ and $p(x\mid\theta)$, corresponding respectively | |
| to our channels $c$ and $d$. The $\tau$ form the hyperparameters. | |
| In this setting we come to our main definition that formulates the | |
| notion of conjugate prior in an abstract manner, avoiding classes of | |
| distributions. It contains the crucial equation that was missing in | |
| the informal description in Section~\ref{sec:ideas}. | |
| All our examples of (conjugate prior) channels are maps in the Kleisli | |
| category of the Giry monad, but the formulation applies more | |
| generally. In fact, abstraction purifies the situation and shows the | |
| essentials. The definition below speaks of `deterministic' channels, | |
| between brackets. This part will be explained later on, in the | |
| beginning of Section~\ref{sec:conjugatepriorinversion}. It can be | |
| ignored for now. | |
| \begin{definition} | |
| \label{def:conjugateprior} | |
| In the situation~\eqref{diag:conjugatechannels} we call channel $c$ a | |
| \emph{conjugate prior} to channel $d$ if there is a (deterministic) | |
| channel $h\colon P\times O \rightarrow P$ for which the following | |
| equation holds: | |
| \begin{equation} | |
| \label{eqn:conjugateprior} | |
| \vcenter{\hbox{\begin{tikzpicture}[font=\small] | |
| \node[arrow box] (c) at (0.0,-0.2) {$c$}; | |
| \node[copier] (copier) at (0,0.3) {}; | |
| \node[arrow box] (d) at (0.5,0.95) {$d$}; | |
| \coordinate (X) at (-0.5,1.5); | |
| \coordinate (Y) at (0.5,1.5); | |
| \draw (c) to (copier); | |
| \draw (c) to (0.0,-0.7); | |
| \draw (copier) to[out=150,in=-90] (X); | |
| \draw (copier) to[out=15,in=-90] (d); | |
| \draw (d) to (Y); | |
| \end{tikzpicture}}} | |
| \quad | |
| = | |
| \quad | |
| \vcenter{\hbox{\begin{tikzpicture}[font=\small] | |
| \node[copier] (copier1) at (0,0.3) {}; | |
| \node[copier] (copier2) at (0.25,2.0) {}; | |
| \coordinate (X) at (-0.5,3.8); | |
| \coordinate (Y) at (0.5,3.8); | |
| \node[arrow box] (c1) at (0.25,0.8) {$c$}; | |
| \node[arrow box] (d) at (0.25,1.5) {$d$}; | |
| \node[arrow box] (h) at (-0.5,2.6) {$\;\;h\;\;$}; | |
| \node[arrow box] (c2) at (-0.5,3.3) {$c$}; | |
| \draw (copier1) to (0.0,0.0); | |
| \draw (copier1) to[out=150,in=-90] (h.240); | |
| \draw (copier1) to[out=30,in=-90] (c1); | |
| \draw (c1) to (d); | |
| \draw (d) to (copier2); | |
| \draw (copier2) to[out=165,in=-90] (h.305); | |
| \draw (h) to (c2); | |
| \draw (c2) to (X); | |
| \draw (copier2) to[out=30,in=-90] (Y); | |
| \end{tikzpicture}}} | |
| \end{equation} | |
| \noindent Equivalently, in equational form: | |
| \[ \begin{array}{rcl} | |
| \tuple{\idmap, d} \after c | |
| & = & | |
| ((c\after h)\otimes\idmap) \after \tuple{\idmap, \copier \after d \after c}. | |
| \end{array} \] | |
| \end{definition} | |
| The idea is that the map $h\colon P\times O \rightarrow P$ translates | |
| parameters, with an observation from $O$ as additional | |
| argument. Informally, one gets a posterior state $c(h(p,y))$ from the | |
| prior state $c(p)$, given the observation $y\in O$. The power of this | |
| `analytic' approach is that it involves simple re-computation of | |
| parameters, instead of more complicated updating of entire | |
| states. This will be illustrated in several standard examples below. | |
| The above Equation~\eqref{eqn:conjugateprior} is formulated in an | |
| abstract manner --- which is its main strength. We will derive an | |
| alternative formulation of Equation~\eqref{eqn:conjugateprior} for | |
| pdf-channels. It greatly simplifies the calculations in examples. | |
| \begin{lemma} | |
| \label{lem:conjugatepriorpdf} | |
| Consider composable channels $\smash{P \stackrel{c}{\rightarrow} X | |
| \stackrel{d}{\rightarrow} O}$, as in~\eqref{diag:conjugatechannels}, | |
| for the Giry monad $\Giry$, where $c\colon P\rightarrow \Giry(X)$ and | |
| $d\colon X \rightarrow \Giry(O)$ are given by pdf's $u\colon P\times X | |
| \rightarrow \R_{\geq 0}$ and $v \colon X\times O\rightarrow \R_{\geq | |
| 0}$, as pdf-channels $c = \int u$ and $d = \int v$. Let $c$ be | |
| conjugate prior to $d$, via a measurable function $h\colon P\times O | |
| \rightarrow P$. | |
| Equation~\eqref{eqn:conjugateprior} then amounts to, for an element | |
| $p\in P$ and for measurable subsets $M\subseteq X$ and $N\subseteq O$, | |
| \begin{equation} | |
| \label{eqn:conjugatepriorint} | |
| \begin{array}{rcl} | |
| \lefteqn{\displaystyle\int_{N} \int_{M} u(p,x) \cdot v(x,y) \intd x\intd y} | |
| \\[+0.8em] | |
| & = & | |
| \displaystyle \int_{N} \Big(\int u(p,x)\cdot v(x,y)\intd x\Big) \cdot | |
| \Big(\int_{M} u(h(p,y),x) \intd x\Big) \intd y. | |
| \end{array} | |
| \end{equation} | |
| \noindent In order to prove this equation, it suffices to prove that | |
| the two functions under the outer integral $\int_{N}$ are equal, that | |
| is, it suffices to prove for each $y\in O$, | |
| \begin{equation} | |
| \label{eqn:conjugatepriorfun} | |
| \begin{array}{rcl} | |
| \displaystyle\int_{M} u(p,x) \cdot v(x,y) \intd x | |
| & = & | |
| \displaystyle \Big(\int u(p,x)\cdot v(x,y)\intd x\Big) \cdot | |
| \Big(\int_{M} u(h(p,y),x) \intd x\Big). | |
| \end{array} | |
| \end{equation} | |
| This formulation will be used in the examples below. | |
| \end{lemma} | |
| \begin{myproof} | |
| We extensively use the equations for integration from | |
| Section~\ref{sec:Kleisli} and from the Appendix, in order to | |
| prove~\eqref{eqn:conjugatepriorint}. The left-hand-side of | |
| Equation~\eqref{eqn:conjugateprior} gives the left-hand-side | |
| of~\eqref{eqn:conjugatepriorint}: | |
| \[ \begin{array}{rcccl} | |
| \big(\tuple{\idmap, d} \after c\big)(p)(M\times N) | |
| & \smash{\stackrel{\eqref{eqn:continuouscomposition}}{=}} & | |
| \big(\tuple{\idmap, d} \gg c(p)\big)(M\times N) | |
| & \smash{\stackrel{\eqref{eqn:pdfgraphstatetransformation}}{=}} & | |
| \displaystyle \int_{N} \int_{M} u(p,x) \cdot v(x,y) \intd x \intd y. | |
| \end{array} \] | |
| \noindent Unravelling the right-hand-side of~\eqref{eqn:conjugateprior} | |
| is a bit more work: | |
| \[ \begin{array}{rcl} | |
| \lefteqn{\big((c \after h)\otimes\idmap) \after | |
| \tuple{\idmap, \copier \after d \after c}\big)(p)(M\times N)} | |
| \\ | |
| & \smash{\stackrel{\eqref{eqn:continuouscomposition}}{=}} & | |
| \displaystyle \int (c \after h)\otimes\idmap)(-)(M\times N) | |
| \intd \tuple{\idmap, \copier \after d \after c}(p) | |
| \\ | |
| & \smash{\stackrel{\eqref{eqn:graphequation}}{=}} & | |
| \displaystyle \int ((c \after h)\otimes\idmap)(-)(M\times N) | |
| \intd \big(\eta(p) \otimes | |
| (\copier \after d \after c)(p)\big) | |
| \\ | |
| & \smash{\stackrel{\eqref{eqn:productpdfintegration}}{=}} & | |
| \displaystyle \int \int ((c \after h)\otimes\idmap)(-,-)(M\times N) | |
| \intd \eta(p) \intd (\copier \after d \after c)(p)\big) | |
| \\ | |
| & = & | |
| \displaystyle \int ((c \after h)\otimes\idmap)(p,-)(M\times N) | |
| \intd \Giry(\copier)(d \gg c(p)) | |
| \\ | |
| & \smash{\stackrel{\eqref{eqn:invmeasure}}{=}} & | |
| \displaystyle \int ((c \after h)\otimes\idmap)(p,\copier(-))(M\times N) | |
| \intd (d \gg c(p)) | |
| \\ | |
| & \smash{\stackrel{(\ref{eqn:pdfintegration}, | |
| \ref{eqn:pdfseqcomposition})}{=}} & | |
| \displaystyle \int \Big(\int u(p,x)\cdot v(x,y)\intd x\Big) \cdot | |
| ((c \after h)\otimes\idmap)(p,y,y)(M\times N) \intd y | |
| \\ | |
| & = & | |
| \displaystyle \int \Big(\int u(p,x)\cdot v(x,y)\intd x\Big) \cdot | |
| c(h(p,y))(M) \cdot \indic{N}(y) \intd y | |
| \\ | |
| & = & | |
| \displaystyle \int_{N} \Big(\int u(p,x)\cdot v(x,y)\intd x\Big) \cdot | |
| \Big(\int_{M} u(h(p,y),x) \intd x\Big) \intd y. | |
| \end{array} \] | |
| \noindent By combining this outcome with the earlier one we get the | |
| desired equation~\eqref{eqn:conjugatepriorint}. \QED | |
| \end{myproof} | |
| One can reorganise Equation~\eqref{eqn:conjugatepriorfun} as | |
| a normalisation fraction: | |
| \begin{equation} | |
| \label{eqn:conjugatepriorfunnorm} | |
| \begin{array}{rcl} | |
| \displaystyle\int_{M} u(h(p,y),x) \intd x | |
| & = & | |
| \displaystyle \frac{\int_{M} u(p,x) \cdot v(x,y) \intd x} | |
| {\int u(p,x)\cdot v(x,y)\intd x}. | |
| \end{array} | |
| \end{equation} | |
| \noindent It now strongly resembles | |
| Equation~\eqref{eqn:continuousinversion} for Bayesian inversion. This | |
| connection will be established more generally in | |
| Theorem~\ref{thm:conjugateinversion}. Essentially, the above | |
| normalisation fraction~\eqref{eqn:conjugatepriorfunnorm} occurs | |
| in~\cite[Defn.~5.6]{BernardoS00}. Later, in Section~\ref{sec:logic} we | |
| will see that~\eqref{eqn:conjugatepriorfunnorm} can also be analysed | |
| in terms of updating a state with a random variable. | |
| We are now ready to review some standard examples. The first one | |
| describes the structure underlying the coin example in | |
| Section~\ref{sec:ideas}. | |
| \begin{example} | |
| \label{ex:betaflip} | |
| It is well-known that the beta distributions are conjugate prior to | |
| the Bernoulli `flip' likelihood function. We shall re-formulate this | |
| fact following the pattern of Definition~\ref{def:conjugateprior}, | |
| with two composable channels, as in~\eqref{diag:conjugatechannels}, | |
| namely: | |
| \[\xymatrix@C+1pc{ | |
| \NNO_{>0}\times\NNO_{>0}\ar[r]^-{\betachan} & | |
| [0,1]\ar[r]^-{\flipchan} & 2 | |
| \rlap{\qquad where $2=\{0,1\}$.} | |
| } \qquad \] | |
| \noindent The $\betachan$ channel is as in~\eqref{diag:beta}, but now | |
| restricted to the non-negative natural numbers $\NNO_{>0}$. We recall | |
| that the normalisation constant $B(\alpha,\beta)$ is $\int_{[0,1]} | |
| x^{\alpha-1}(1-x)^{\beta-1} \intd x$. | |
| The $\flipchan$ channel sends a probability $r\in[0,1]$ to the | |
| $\text{Bernoulli}(r)$ distribution, which can also be written as a | |
| discrete distribution $\flipchan(r) = r\ket{1} + (1-r)\ket{0}$. More | |
| formally, as a Kleisli map $[0,1]\rightarrow\Giry(2)$ it is, for a | |
| subset $N\subseteq 2$, | |
| \[ \begin{array}{rcccccl} | |
| \flipchan(r)(N) | |
| & = & | |
| \displaystyle \int_{N} r^{i}\cdot (1-r)^{1-i} \intd i | |
| & = & | |
| \displaystyle\sum_{i\in N} r^{i}\cdot (1-r)^{1-i} | |
| & = & | |
| \left\{{\renewcommand{\arraystretch}{1.0}\begin{array}{ll} | |
| 0 & \mbox{if } N = \emptyset \\ | |
| r & \mbox{if } N = \{1\} \\ | |
| 1-r \;\; & \mbox{if } N = \{0\} \\ | |
| 1 & \mbox{if } N = \{0,1\}. | |
| \end{array}}\right. | |
| \end{array} \] | |
| \noindent The $i$ in $\intd i$ refers here to the counting measure. | |
| In order to show that $\betachan$ is a conjugate prior of $\flipchan$ | |
| we have to produce a parameter translation function $h\colon | |
| \NNO_{>0}\times\NNO_{>0}\times 2 \rightarrow | |
| \NNO_{>0}\times\NNO_{>0}$. It is defined by distinguishing the | |
| elements in $2 = \{0,1\}$ | |
| \begin{equation} | |
| \label{eqn:betaflipfun} | |
| \begin{array}{rclcrcl} | |
| h(\alpha, \beta, 1) | |
| & = & | |
| (\alpha+1, \beta) | |
| & \qquad\mbox{and}\qquad & | |
| h(\alpha, \beta, 0) | |
| & = & | |
| (\alpha, \beta+1). | |
| \end{array} | |
| \end{equation} | |
| \noindent Thus, in one formula, $h(\alpha,\beta,i) = (\alpha+i, | |
| \beta+(1-i))$. | |
| We prove Equation~\eqref{eqn:conjugatepriorfun} for $c = \betachan = | |
| \int u$ and $d = \flipchan = \int v$. We start from its | |
| right-hand-side, for an arbitrary $i\in 2$, | |
| \[ \begin{array}{rcl} | |
| \lefteqn{\displaystyle \Big(\int u(\alpha,\beta,x)\cdot v(x,i)\intd x\Big) | |
| \cdot \Big(\int_{M} u(h(\alpha,\beta,i),x) \intd x\Big)} | |
| \\ | |
| & = & | |
| \displaystyle\Big(\int \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)} \cdot | |
| x^{i}\cdot (1-x)^{1-i} \intd x\Big) \cdot | |
| \Big(\int_{M} \frac{x^{\alpha+i-1}(1-x)^{\beta+(1-i)-1}} | |
| {B(\alpha+i,\beta+(1-i))} \intd x\Big) | |
| \\[+1em] | |
| & = & | |
| \displaystyle\Big(\frac{\int x^{\alpha+i-1}(1-x)^{\beta+(1-i)-1} \intd x} | |
| {B(\alpha,\beta)}\Big) \cdot | |
| \Big(\int_{M} \frac{x^{\alpha-1}(1-x)^{\beta-1}} | |
| {B(\alpha+i,\beta+(1-i))} \cdot x^{i} \cdot (1-x)^{1-i} \intd x\Big) | |
| \\[+1em] | |
| & = & | |
| \displaystyle\Big(\frac{B(\alpha+i,\beta+(1-i))}{B(\alpha,\beta)}\Big) \cdot | |
| \Big(\int_{M} \frac{x^{\alpha-1}(1-x)^{\beta-1}} | |
| {B(\alpha+i,\beta+(1-i))} \cdot x^{i} \cdot (1-x)^{1-i} \intd x\Big) | |
| \\[+1em] | |
| & = & | |
| \displaystyle \int_{M}\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)} | |
| \cdot x^{i} \cdot (1-x)^{1-i} \intd x | |
| \\[+1em] | |
| & = & | |
| \displaystyle \int_{M} u(\alpha,\beta,x)\cdot v(x, i) \intd x. | |
| \end{array} \] | |
| \noindent The latter expression is the left-hand-side | |
| of~\eqref{eqn:conjugatepriorfun}. We see that the essence of the | |
| verification of the conjugate prior equation is the shifting of | |
| functions and normalisation factors. This is a general pattern. | |
| \auxproof{ | |
| We have to prove Equation~\eqref{eqn:conjugateprior}. Its | |
| left-hand-side is, as Kleisli map $\NNO_{>0}\times\NNO_{>0} | |
| \rightarrow \Giry([0,1]\times 2)$, applied to measurable subsets | |
| $M\subseteq [0,1]$ and $N\subseteq 2$, | |
| \[ \begin{array}{rcl} | |
| \big(\tuple{\idmap, \flipchan} \after \betachan\big)(\alpha,\beta)(M\times N) | |
| & = & | |
| \big(\tuple{\idmap, \flipchan} \gg \betachan(\alpha,\beta)\big)(M\times N) | |
| \\ | |
| & \smash{\stackrel{\eqref{eqn:pdfgraphstatetransformation}}{=}} & | |
| \displaystyle \int_{M\times N} \frac{x^{\alpha-1}(1-x)^{\beta-1}} | |
| {B(\alpha,\beta)} \cdot x^{i}\cdot (1-x)^{1-i} \intd (x,i) | |
| \\ | |
| & = & | |
| \displaystyle \int_{M\times N} \frac{x^{\alpha-1+i}(1-x)^{\beta-i}} | |
| {B(\alpha,\beta)} \intd (x,i). | |
| \end{array} \] | |
| \noindent With a bit more effort we show that the right-hand-side | |
| of~\eqref{eqn:conjugateprior} has the same outcome: | |
| \[ \begin{array}{rcl} | |
| \lefteqn{\big(((\betachan \after h)\otimes\idmap) \after | |
| \tuple{\idmap, \copychan \after \flipchan \after \betachan}\big) | |
| (\alpha, \beta)(M\times N)} | |
| \\ | |
| & \smash{\stackrel{\eqref{eqn:graphequation}}{=}} & | |
| \displaystyle \int ((\betachan \after h)\otimes\idmap)(-)(M\times N) | |
| \intd \big(\eta(\alpha,\beta) \otimes | |
| (\copychan \after \flipchan \after \betachan)(\alpha,\beta)\big) | |
| \\ | |
| & = & | |
| \displaystyle \int ((\betachan \after h)\otimes\idmap)(\alpha, \beta, -) | |
| (M\times N) \intd | |
| \Giry(\copychan)\big((\flipchan \after \betachan)(\alpha,\beta)\big) | |
| \\ | |
| & \smash{\stackrel{\eqref{??}}{=}} & | |
| \displaystyle \int ((\betachan \after h)\otimes\idmap)(\alpha, \beta, | |
| \copychan(-))(M\times N) \intd | |
| \big(\flipchan \after \betachan)(\alpha,\beta)\big) | |
| \\ | |
| & \smash{\stackrel{(\ref{eqn:pdfintegration}, | |
| \ref{eqn:pdfseqcomposition})}{=}} & | |
| \displaystyle \int \big(\int \frac{x^{\alpha-1+i}(1-x)^{\beta-i}} | |
| {B(\alpha,\beta)}\intd x\big) \cdot | |
| ((\betachan \after h)\otimes\idmap)(\alpha, \beta, | |
| \copychan(i))(M\times N) \intd i | |
| \\ | |
| & = & | |
| \displaystyle \int \frac{\int x^{\alpha-1+i}(1-x)^{\beta-i} \intd x} | |
| {B(\alpha,\beta)} \cdot | |
| \betachan(h(\alpha, \beta, i))(M) \cdot \indic{N}(i) \intd i | |
| \\ | |
| & = & | |
| \displaystyle \int_{N} \frac{B(\alpha+i, \beta+1-i)}{B(\alpha,\beta)} \cdot | |
| \betachan(\alpha+i, \beta+1-i)(M) \intd i | |
| \\ | |
| & = & | |
| \displaystyle \int_{N} \frac{B(\alpha+i, \beta+1-i)}{B(\alpha,\beta)} \cdot | |
| \big(\int_{M} \frac{x^{\alpha+i-1}\cdot (1-x)^{\beta-i}}{B(\alpha+i, \beta+1-i)} | |
| \intd x\big) \intd i | |
| \\ | |
| & = & | |
| \displaystyle \int_{M\times N} \frac{x^{\alpha-1+i}(1-x)^{\beta-i}} | |
| {B(\alpha,\beta)} \intd (x,i). | |
| \end{array} \] | |
| } | |
| \end{example} | |
| \begin{example} | |
| \label{ex:betabinom} | |
| In a similar way one verifies that the $\betachan$ channel is a | |
| conjugate prior to the binomial channel. For the latter we fix a | |
| natural number $n>0$, and consider the two channels: | |
| \[ \xymatrix@-1pc{ | |
| \NNO_{>0}\times\NNO_{>0}\ar[rr]^-{\betachan} & & | |
| [0,1]\ar[rrr]^-{\binomchan_{n}} & & & \{0,1,\ldots,n\} | |
| } \] | |
| \noindent The binomial channel $\binomchan_{n}$ is defined for $r \in | |
| [0,1]$ and $M\subseteq \{0,1,\ldots,n\}$ as: | |
| \[ \begin{array}{rcccl} | |
| \binomchan_{n}(r)(M) | |
| & = & | |
| {\displaystyle \int_{M}} \binom{n}{i}\cdot r^{i}\cdot (1-r)^{n-i} | |
| \intd i | |
| & = & | |
| {\displaystyle \sum_{i\in M}}\, \binom{n}{i}\cdot r^{i}\cdot (1-r)^{n-i}. | |
| \end{array} \] | |
| \noindent The conjugate prior property requires in this situation a | |
| parameter translation function $h\colon \NNO_{>0}\times\NNO_{>0} | |
| \times \{0,1,\ldots,n\}\rightarrow \NNO_{>0}\times\NNO_{>0}$, which is | |
| given by: | |
| \[ \begin{array}{rcl} | |
| h(\alpha, \beta, i) | |
| & = & | |
| (\alpha+i, \beta+n-i). | |
| \end{array} \] | |
| \noindent The proof of Equation~\eqref{eqn:conjugatepriorfun} is much | |
| like in Example~\ref{ex:betaflip}, with $1-i$ replaced by $n-i$, and | |
| an additional binomial term $\binom{n}{i}$ that is shifted from one | |
| integral to another. | |
| \begin{Auxproof} | |
| \[ \begin{array}{rcl} | |
| \lefteqn{\displaystyle\Big(\int \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)} | |
| \cdot \binom{n}{i}\cdot x^{i} \cdot (1-x)^{n-i} \intd x\Big) \cdot | |
| \Big(\int_{M} \frac{x^{\alpha+i-1}(1-x)^{\beta+(n-i)-1}} | |
| {B(\alpha+i,\beta+(n-i))} \intd x\Big)} | |
| \\[+1em] | |
| & = & | |
| \displaystyle\Big(\frac{\int x^{\alpha+i-1}(1-x)^{\beta+(n-i)-1} \intd x} | |
| {B(\alpha,\beta)}\Big) \cdot \binom{n}{i} \cdot | |
| \Big(\int_{M} \frac{x^{\alpha-1}(1-x)^{\beta-1}} | |
| {B(\alpha+i,\beta+(n-i))} \cdot x^{i} \cdot (1-x)^{n-i} \intd x\Big) | |
| \\ | |
| & = & | |
| \displaystyle\Big(\frac{B(\alpha+i,\beta+(n-i))}{B(\alpha,\beta)}\Big) \cdot | |
| \Big(\int_{M} \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha+i,\beta+(n-i))} \cdot | |
| \binom{n}{i}\cdot x^{i} \cdot (1-x)^{n-i} \intd x\Big) | |
| \\ | |
| & = & | |
| \displaystyle \int_{M}\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)} | |
| \cdot \binom{n}{i} \cdot x^{i} \cdot (1-x)^{n-i} \intd x. | |
| \end{array} \] | |
| \end{Auxproof} | |
| \end{example} | |
| Here is another well-known conjugate prior relationship, namely | |
| between Dirichlet and `multinomial' distributions. The latter | |
| are simply called discrete distributions in the present context. | |
| \begin{example} | |
| \label{ex:dirmon} | |
| Here we shall identify a number $n\in\NNO$ with the $n$-element set | |
| $\{0,1,\ldots,n-1\}$. We then write $\Dst_{*}(n)$ for the set of | |
| $n$-tuples $(x_{0}, \ldots, x_{n-1})\in (\R_{>0})^{n}$ with | |
| $\sum_{i}x_{i} = 1$. | |
| For a fixed $n>0$, let $O = \{y_{0}, \ldots, | |
| y_{n-1}\}$ be a set of `observations'. We consider the following two | |
| channels. | |
| \[ \xymatrix@-1pc{ | |
| (\NNO_{>0})^{n}\ar[rr]^-{\dirchan_n} & & \Dst_{*}(n)\ar[rr]^-{\multchan} & & O | |
| } \] | |
| \noindent The multinomial channel is defined as $\multchan(x_{0}, | |
| \ldots, x_{n-1}) = x_{0}\ket{y_{0}} + \cdots + x_{n-1}\ket{y_{n-1}}$. | |
| It can be described as a pdf-channel, via the function $v(\vec{x},y) | |
| \coloneqq x_{i} \mbox{ if }y=y_{i}$. Then, for $N\subseteq O = | |
| \{y_{0}, \ldots, y_{n-1}\}$, | |
| \[ \begin{array}{rcccl} | |
| \multchan(\vec{x})(N) | |
| & = & | |
| \displaystyle\int_{N} v(\vec{x}, y) \intd y | |
| & = & | |
| \sum\set{x_{i}}{y_{i}\in N}. | |
| \end{array} \] | |
| The Dirichlet channel $\dirchan_n$ is more complicated: for an | |
| $n$-tuple $\vec{\alpha} = (\alpha_{0}, \ldots, \alpha_{n-1})$ it is | |
| given via pdf's $d_n$, in: | |
| \[ \begin{array}{rclcrcl} | |
| \dirchan_{n}(\vec{\alpha}) | |
| & = & | |
| \displaystyle\int d_{n}(\vec{\alpha}) | |
| & \qquad\mbox{where}\qquad & | |
| d_{n}(\vec{\alpha})(x_{0}, \ldots, x_{n-1}) | |
| & = & | |
| {\displaystyle\frac{\Gamma(\sum_{i}\alpha_{i})}{\prod_{i}\Gamma(\alpha_{i})}} | |
| \cdot \prod_{i} x_{i}^{\alpha_{i}-1}, | |
| \end{array} \] | |
| \noindent for $(x_{0}, \ldots, x_{n-1}) \in \Dst_{*}(n)$. The | |
| operation $\Gamma$ is the `Gamma' function, which is defined on | |
| natural numbers $k > 1$ as $\Gamma(k) = (k-1)!$. Hence $\Gamma$ can be | |
| defined recursively as $\Gamma(1) = 1$ and $\Gamma(k+1) = | |
| k\cdot\Gamma(k)$. The above fraction is a normalisation factor since | |
| one has $\frac{\prod_{i} | |
| \Gamma(\alpha_{i})}{\Gamma(\sum_{i}\alpha_{i})} = \int\prod_{i} | |
| x_{i}^{\alpha_{i}-1} \intd \vec{x}$, see | |
| \textit{e.g.}~\cite{Bishop06}. From this one can derive: $\int | |
| x_{i}\cdot d_{n}(\vec{\alpha})(\vec{x}) \intd \vec{x} = | |
| \frac{\alpha_{i}}{\sum_{j}\alpha_j}$. | |
| The parameter translation function $h\colon (\NNO_{>0})^{n} \times O | |
| \rightarrow (\NNO_{>0})^{n}$ is: | |
| \[ \begin{array}{rcl} | |
| h(\alpha_{0}, \ldots, \alpha_{n-1}, y) | |
| & = & | |
| (\alpha_{0}, \ldots, \alpha_{i}+1, \ldots, \alpha_{n-1}) | |
| \quad \mbox{if } y=y_{i}. | |
| \end{array} \] | |
| \noindent We check Equation~\eqref{eqn:conjugatepriorfun}, for $M\subseteq | |
| \Dst_{*}(n)$ and observation $y_{i}\in O$, | |
| \[ \begin{array}{rcl} | |
| \lefteqn{\displaystyle | |
| \Big(\int d_{n}(\vec{\alpha})(\vec{x})\cdot v(\vec{x},y_{i})\intd\vec{x}\Big) | |
| \cdot \Big(\int_{M} d_{n}(h(\vec{\alpha},y_{i}))(\vec{x}) \intd \vec{x}\Big)} | |
| \\[+0.4em] | |
| & = & | |
| \frac{\alpha_i}{\sum_{j}\alpha_{j}} \cdot | |
| {\displaystyle\int_{M}} | |
| \frac{\Gamma(1 + \sum_{j}\alpha_{j})} | |
| {\Gamma(\alpha_{i}+1)\cdot \prod_{j\neq i}\Gamma(\alpha_{j})} \cdot | |
| x_{i}^{\alpha_i} \cdot \prod_{j\neq i} x_{j}^{\alpha_{j}-1} \intd \vec{x} | |
| \\[+0.8em] | |
| & = & | |
| {\displaystyle\Big(\int_{M}} | |
| \frac{\Gamma(\sum_{j}\alpha_{j})} | |
| {\prod_{j}\Gamma(\alpha_{j})} \cdot | |
| x_{i} \cdot \prod_{j} x_{j}^{\alpha_{j}-1} \intd \vec{x} | |
| \\[+0.8em] | |
| & = & | |
| \displaystyle\int_{M} d_{n}(\vec{\alpha})(\vec{x})\cdot | |
| v(\vec{x},y_{i})\intd\vec{x}. | |
| \end{array} \] | |
| \end{example} | |
| We include one more example, illustrating that normal channels are | |
| conjugate priors to themselves. This fact is also well-known. The | |
| point is to illustrate once again how that works in the current | |
| setting. | |
| \begin{example} | |
| \label{ex:normnorm} | |
| Consider the following two normal channels. | |
| \[ \xymatrix@-0.5pc{ | |
| \R\times\R_{>0}\ar[rr]^-{\normchan} & & | |
| \R\ar[rrr]^-{\normchan(-,\nu)} & & & \R_{>0} | |
| } \] | |
| \noindent The channel $\normchan$ is described explicitly | |
| in~\eqref{diag:normal}. Notice that we use it twice here, the second | |
| time with a fixed standard deviation $\nu$, for `noise'. This second | |
| channel is typically used for observation, like in Kalman filtering, | |
| for which a fixed noise level can be assumed. We claim that the first | |
| normal channel $\normchan$ is a conjugate prior to the second channel | |
| $\normchan(-,\nu)$, via the parameter translation function $h\colon | |
| \R\times\R_{>0}\times\R_{>0}\rightarrow \R\times\R_{>0}$ given by: | |
| \[ \begin{array}{rcl} | |
| h(\mu, \sigma, y) | |
| & = & | |
| \displaystyle | |
| (\, \frac{\mu\cdot\nu^{2} + y\cdot\sigma^{2}}{\nu^{2}+\sigma^{2}}, | |
| \frac{\nu\cdot\sigma}{\sqrt{\nu^{2}+\sigma^{2}}} \,) | |
| \end{array} \] | |
| \noindent We prove Equation~\eqref{eqn:conjugatepriorfun}, again | |
| starting from the right. | |
| \[ \begin{array}{rcl} | |
| \lefteqn{\Big({\displaystyle \int} | |
| \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}} \cdot | |
| \frac{1}{\sqrt{2\pi}\nu}e^{-\frac{(y-x)^{2}}{2\nu^{2}}}\intd x\Big) \cdot | |
| \Big({\displaystyle \int_{M}} | |
| \frac{\sqrt{\nu^{2}+\sigma^{2}}}{\sqrt{2\pi}\nu\sigma} | |
| e^{-\frac{(\nu^{2}+\sigma^{2}) (x-\frac{\mu\cdot\nu^{2} + y\cdot\sigma^{2}} | |
| {\nu^{2}+\sigma^{2}})^{2}}{2\nu^{2}\sigma^{2}}}\intd x\Big)} | |
| \\[+1em] | |
| & \smash{\stackrel{(*)}{=}} & | |
| \Big({\displaystyle \int}\frac{1}{\sqrt{2\pi}\sigma\nu} | |
| e^{-\frac{\nu^{2}(x-\mu)^{2} + \sigma^{2}(y-x)^{2}}{2\sigma^{2}\nu^{2}}}\intd x\Big) | |
| \\ | |
| & & \qquad \cdot\; \Big({\displaystyle \int_{M}} | |
| \frac{\sqrt{\nu^{2}+\sigma^{2}}}{\sqrt{2\pi}\nu\sigma} | |
| e^{-\frac{\nu^{2}(x-\mu)^{2} + \sigma^{2}(y-x)^{2} - \nu^{2}\mu^{2} - | |
| \sigma^{2}y^{2} + \frac{(\mu\cdot\nu^{2} + y\cdot\sigma^{2})^{2}} | |
| {\nu^{2}+\sigma^{2}}}{2\nu^{2}\sigma^{2}}}\intd x\Big) | |
| \\ | |
| & = & | |
| \Big({\displaystyle \int}\frac{\sqrt{\nu^{2}+\sigma^{2}}}{\sqrt{2\pi}\nu\sigma} | |
| e^{-\frac{\nu^{2}(x-\mu)^{2} + \sigma^{2}(y-x)^{2} - \nu^{2}\mu^{2} - | |
| \sigma^{2}y^{2} + \frac{(\mu\cdot\nu^{2} + y\cdot\sigma^{2})^{2}} | |
| {\nu^{2}+\sigma^{2}}}{2\sigma^{2}\nu^{2}}}\intd x\Big) | |
| \\ | |
| & & \qquad \cdot\; \Big({\displaystyle \int_{M}} | |
| \frac{1}{\sqrt{2\pi}\sigma\nu} | |
| e^{-\frac{\nu^{2}(x-\mu)^{2} + \sigma^{2}(y-x)^{2}}{2\nu^{2}\sigma^{2}}}\intd x\Big) | |
| \\ | |
| & \smash{\stackrel{(*)}{=}} & | |
| \Big({\displaystyle \int} | |
| \frac{\sqrt{\nu^{2}+\sigma^{2}}}{\sqrt{2\pi}\nu\sigma} | |
| e^{-\frac{(\nu^{2}+\sigma^{2}) (x-\frac{\mu\cdot\nu^{2} + y\cdot\sigma^{2}} | |
| {\nu^{2}+\sigma^{2}})^{2}}{2\nu^{2}\sigma^{2}}}\intd x\Big) \cdot | |
| \Big({\displaystyle \int_{M}} | |
| \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}} \cdot | |
| \frac{1}{\sqrt{2\pi}\nu}e^{-\frac{(y-x)^{2}}{2\nu^{2}}}\intd x\Big) | |
| \\[+1em] | |
| & = & | |
| {\displaystyle \int_{M}} | |
| \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}} \cdot | |
| \frac{1}{\sqrt{2\pi}\nu}e^{-\frac{(y-x)^{2}}{2\nu^{2}}}\intd x. | |
| \end{array} \] | |
| \noindent The last equation holds because the first integral in the | |
| previous line equals one, since, in general, the integral over a pdf | |
| is one. The two marked equations $\smash{\stackrel{(*)}{=}}$ are | |
| justified by: | |
| \[ \begin{array}{rcl} | |
| \lefteqn{(\nu^{2}+\sigma^{2})\big(x-\frac{\mu\cdot\nu^{2} + | |
| y\cdot\sigma^{2}}{\nu^{2}+\sigma^{2}}\big)^{2}} | |
| \\ | |
| & = & | |
| (\nu^{2}+\sigma^{2})x^{2}- | |
| 2(\mu\cdot\nu^{2} + y\cdot\sigma^{2})x | |
| + \frac{(\mu\cdot\nu^{2} + y\cdot\sigma^{2})^{2}} | |
| {\nu^{2}+\sigma^{2}} | |
| \\ | |
| & = & | |
| \nu^{2}(x^{2}-2\mu x + \mu^{2}) + \sigma^{2}(y^{2}-2yx + x^{2}) | |
| - \nu^{2}\mu^{2} - \sigma^{2}y^{2} | |
| + \frac{(\mu\cdot\nu^{2} + y\cdot\sigma^{2})^{2}} | |
| {\nu^{2}+\sigma^{2}} | |
| \\ | |
| & = & | |
| \nu^{2}(x-\mu)^{2} + \sigma^{2}(y-x)^{2} | |
| - \nu^{2}\mu^{2} - \sigma^{2}y^{2} | |
| + \frac{(\mu\cdot\nu^{2} + y\cdot\sigma^{2})^{2}} | |
| {\nu^{2}+\sigma^{2}} | |
| \end{array} \] | |
| \end{example} | |
| \section{Conjugate priors form Bayesian inversions}\label{sec:conjugatepriorinversion} | |
| This section connects the main two notions of this paper, by showing | |
| that conjugate priors give rise to Bayesian inversion. The argument is | |
| a very simple example of diagrammatic reasoning. Before we come to | |
| it, we have to clarify an issue that was left open earlier, regarding | |
| `deterministic' channels, see Definition~\ref{def:conjugateprior}. | |
| \begin{definition} | |
| \label{def:deterministic} | |
| A channel $c$ is called \emph{deterministic} if it commutes with | |
| copiers, that is, if it satisfies the equation on the left below. | |
| \[ \vcenter{\hbox{\begin{tikzpicture}[font=\small] | |
| \node[copier] (c) at (0,0.6) {}; | |
| \node[arrow box] (f) at (0,0.1) {$c$}; | |
| \draw (c) to[out=15,in=-90] (0.4,1.0); | |
| \draw (c) to[out=165,in=-90] (-0.4,1.0); | |
| \draw (f) to (c); | |
| \draw (f) to (0,-0.4); | |
| \end{tikzpicture}}} | |
| \quad=\quad | |
| \vcenter{\hbox{\begin{tikzpicture}[font=\small] | |
| \node[copier] (c) at (0,0.0) {}; | |
| \node[arrow box] (f1) at (-0.4,0.6) {$c$}; | |
| \node[arrow box] (f2) at (0.4,0.6) {$c$}; | |
| \draw (c) to[out=165,in=-90] (f1); | |
| \draw (c) to[out=15,in=-90] (f2); | |
| \draw (f1) to (-0.4,1.1); | |
| \draw (f2) to (0.4,1.1); | |
| \draw (c) to (0,-0.3); | |
| \end{tikzpicture}}} | |
| \hspace*{10em} | |
| \vcenter{\hbox{\begin{tikzpicture}[font=\small] | |
| \node[copier] (c) at (0,0.4) {}; | |
| \node[state] (omega) at (0,0.1) {$\omega$}; | |
| \draw (c) to[out=15,in=-90] (0.4,0.8); | |
| \draw (c) to[out=165,in=-90] (-0.4,0.8); | |
| \draw (omega) to (c); | |
| \end{tikzpicture}}} | |
| \quad=\quad | |
| \vcenter{\hbox{\begin{tikzpicture}[font=\small] | |
| \node[state] (omega1) at (-0.5,0.0) {$\omega$}; | |
| \node[state] (omega2) at (0.5,0.0) {$\omega$}; | |
| \draw (omega1) to (-0.5,0.4); | |
| \draw (omega2) to (0.5,0.4); | |
| \end{tikzpicture}}} | |
| \] | |
| \noindent As a special case, a state $\omega$ is called deterministic | |
| if it satisfies the equation on the right, above. | |
| \end{definition} | |
| The state description is a special case of the channel description | |
| since a state on $X$ is a channel $1\rightarrow X$ and copying on the | |
| trivial (final) object $1$ does nothing, up to isomorphism. | |
| Few channels (or states) are deterministic. In deterministic and | |
| continuous computation, the ordinary functions $f\colon X \rightarrow | |
| Y$ are deterministic, when considered as a channel $\eta \after f$. | |
| We check this explicitly for point states, since this is what we need | |
| later on. | |
| \begin{example} | |
| \label{ex:deterministicstate} | |
| Let $x$ be an element of a measurable space $X$. The associated point | |
| state $\eta(x) \in \Giry(X)$ is deterministic, where $\eta(x)(M) = | |
| \indic{M}(x)$. We check the equation on the right in | |
| Definition~\ref{def:deterministic}: | |
| \[ \begin{array}{rcl} | |
| \big(\copier \after \eta(x)\big)(M\times N) | |
| & = & | |
| \eta(x,x)(M\times N) | |
| \hspace*{\arraycolsep}=\hspace*{\arraycolsep} | |
| \indic{M\times N}(x,x) | |
| \hspace*{\arraycolsep}=\hspace*{\arraycolsep} | |
| \indic{M}(x) \cdot \indic{N}(x) | |
| \\ | |
| & = & | |
| \eta(x)(M)\cdot \eta(x)(N) | |
| \hspace*{\arraycolsep}=\hspace*{\arraycolsep} | |
| \big(\eta(x)\otimes\eta(x)\big)(M\times N). | |
| \end{array} \] | |
| \end{example} | |
| We now come to the main result. | |
| \begin{theorem} | |
| \label{thm:conjugateinversion} | |
| Let $\smash{P \stackrel{c}{\rightarrow} X \stackrel{d}{\rightarrow} | |
| O}$ be channels, where $c$ is conjugate prior to $d$, say via | |
| $h\colon P \times O \rightarrow P$. Then for each deterministic | |
| (copyable) state $p$, the map $c \after h(p, -) \colon O \rightarrow | |
| X$ is a Bayesian inversion of $d$, wrt.\ the transformed state $c \gg | |
| p$. | |
| \end{theorem} | |
| \begin{myproof} | |
| We have to prove Equation~\eqref{diag:inversion}, for channel $d$ and | |
| state $c \gg p$, with the channel $c \after h(p, -)$ playing the role | |
| of Bayesian inversion $d^{\dag}_{c \gg p}$. This is easiest to see | |
| graphically, using that the state $p$ is deterministic and thus | |
| commutes with copiers $\copier$, see the equation on the right in | |
| Definition~\ref{def:deterministic}. | |
| \[ \vcenter{\hbox{\begin{tikzpicture}[font=\small] | |
| \node[state] (p) at (0,-0.7) {$p$}; | |
| \node[arrow box] (c) at (0.0,-0.2) {$c$}; | |
| \node[copier] (copier) at (0,0.3) {}; | |
| \node[arrow box] (d) at (0.5,0.95) {$d$}; | |
| \coordinate (X) at (-0.5,1.5); | |
| \coordinate (Y) at (0.5,1.5); | |
| \draw (c) to (copier); | |
| \draw (c) to (p); | |
| \draw (copier) to[out=150,in=-90] (X); | |
| \draw (copier) to[out=15,in=-90] (d); | |
| \draw (d) to (Y); | |
| \end{tikzpicture}}} | |
| \quad | |
| \smash{\stackrel{\eqref{eqn:conjugateprior}}{=}} | |
| \quad | |
| \vcenter{\hbox{\begin{tikzpicture}[font=\small] | |
| \node[state] (p) at (0,0.0) {$p$}; | |
| \node[copier] (copier1) at (0,0.3) {}; | |
| \node[copier] (copier2) at (0.25,2.0) {}; | |
| \coordinate (X) at (-0.5,3.8); | |
| \coordinate (Y) at (0.5,3.8); | |
| \node[arrow box] (c1) at (0.25,0.8) {$c$}; | |
| \node[arrow box] (d) at (0.25,1.5) {$d$}; | |
| \node[arrow box] (h) at (-0.5,2.6) {$\;\;h\;\;$}; | |
| \node[arrow box] (c2) at (-0.5,3.3) {$c$}; | |
| \draw (copier1) to (p); | |
| \draw (copier1) to[out=150,in=-90] (h.240); | |
| \draw (copier1) to[out=30,in=-90] (c1); | |
| \draw (c1) to (d); | |
| \draw (d) to (copier2); | |
| \draw (copier2) to[out=165,in=-90] (h.305); | |
| \draw (h) to (c2); | |
| \draw (c2) to (X); | |
| \draw (copier2) to[out=30,in=-90] (Y); | |
| \end{tikzpicture}}} | |
| \quad | |
| = | |
| \quad | |
| \vcenter{\hbox{\begin{tikzpicture}[font=\small] | |
| \node[state] (p1) at (-0.65,2.1) {$p$}; | |
| \node[state] (p2) at (0.25,-0.15) {$p$}; | |
| \node[copier] (copier2) at (0.25,1.5) {}; | |
| \coordinate (X) at (-0.5,3.8); | |
| \coordinate (Y) at (0.5,3.8); | |
| \coordinate (Z) at (0.0,2.0); | |
| \node[arrow box] (c1) at (0.25,0.3) {$c$}; | |
| \node[arrow box] (d) at (0.25,1.0) {$d$}; | |
| \node[arrow box] (h) at (-0.5,2.6) {$\;\;h\;\;$}; | |
| \node[arrow box] (c2) at (-0.5,3.3) {$c$}; | |
| \draw (p1) to[out=90,in=-90] (h.240); | |
| \draw (p2) to[out=90,in=-90] (c1); | |
| \draw (c1) to (d); | |
| \draw (d) to (copier2); | |
| \draw (copier2) to[out=165,in=-90] (Z); | |
| \draw (Z) to[out=90,in=-90] (h.305); | |
| \draw (h) to (c2); | |
| \draw (c2) to (X); | |
| \draw (copier2) to[out=30,in=-90] (Y); | |
| \end{tikzpicture}}} \] | |
| \noindent This is it. \QED | |
| \end{myproof} | |
| When we specialise to Giry-channels we get an `if-and-only-if' | |
| statement, since there we can reason elementwise. | |
| \begin{corollary} | |
| \label{cor:Giry} | |
| Let $\smash{P\xrightarrow{c} X\xrightarrow{d} O}$ be two channels in | |
| $\Kl(\Giry)$, and let $h \colon P\times O \rightarrow P$ be a | |
| measurable function. The following two points are equivalent: | |
| \begin{itemize} | |
| \item[(i)] $c$ is a conjugate prior to $d$, via $h$; | |
| \item[(ii)] $c(h(p,-)) \colon O \rightarrow \Giry(X)$ is a Bayesian | |
| inverse for channel $d$ with state $c(p)$, \textit{i.e.}~is | |
| $d^{\dag}_{c(p)}$, for each parameter $p\in P$. \QED | |
| \end{itemize} | |
| \end{corollary} | |
| \section{A logical perspective on conjugate priors}\label{sec:logic} | |
| This section takes a logically oriented, look at conjugate priors, | |
| describing them in terms of updates of a prior state with a random | |
| variable (or predicate). This new perspective is interesting for two | |
| reasons: | |
| \begin{itemize} | |
| \item it formalises the intuition behind conjugate priors in a | |
| precise manner, see | |
| \textit{e.g.}\ Equations~\eqref{eqn:betaflipupdateeqn} | |
| and~\eqref{eqn:betabinomupdateeqn} below, where the characteristic | |
| closure property for a class of distributions is expressed via | |
| occurrences of these distributions on both sides of an equation; | |
| \item it will be useful in the next section to capture multiple | |
| observations via an update with a conjunction of multiple random | |
| variables. | |
| \end{itemize} | |
| \noindent But first we need to introduce some new terminology. We | |
| shall do so separately for discrete and continuous probability, | |
| although both can be described as instances of the same category | |
| theoretic notions, using effectus theory~\cite{Jacobs15d,Jacobs17a}. | |
| \subsection{Discrete updating} | |
| A \emph{random variable} on a set $X$ is a function $r\colon X | |
| \rightarrow \R$. It is a called a \emph{predicate} if it restricts to | |
| $X\rightarrow [0,1]$. Simple examples of predicates are indicator | |
| functions $\indic{E} \colon X \rightarrow [0,1]$, for a subset/event | |
| $E\subseteq X$, given by $\indic{E}(x) = 1$ if $x\in E$ and | |
| $\indic{E}(x) = 0$ if $x\not\in E$. Indicator functions $\indic{\{x\}} | |
| \colon X \rightarrow [0,1]$ for a singleton subset are sometimes | |
| called point predicates. For two random variables $r,s\colon X | |
| \rightarrow \R$ we write $r\andthen s\colon X \rightarrow \R$ for the | |
| new variable obtained by pointwise multiplication: $(r\andthen s)(x) = | |
| r(x) \cdot s(x)$. | |
| For a random variable $r$ and a discrete | |
| probability distribution (or state) $\omega\in\Dst(X)$ we define the | |
| \emph{validity} $\omega\models r$ as the expected value: | |
| \begin{equation} | |
| \label{eqn:discvalidity} | |
| \begin{array}{rcl} | |
| \omega\models r | |
| & \;\coloneqq\; & | |
| \displaystyle\sum_{x\in X} \omega(x)\cdot r(x). | |
| \end{array} | |
| \end{equation} | |
| \noindent Notice that this is a finite sum, since by definition the | |
| support of $\omega$ is finite. | |
| If we have a channel $c\colon X \rightarrow \Dst(Y)$ and a random | |
| variable $r\colon Y\rightarrow \R$ on its codomain $Y$, then we can | |
| transform it --- or pull it back --- into a random variable on its | |
| domain $X$. We write this pulled back random variable as $c \ll r | |
| \colon X \rightarrow \R$. It is defined as: | |
| \begin{equation} | |
| \label{eqn:discrandvartransform} | |
| \begin{array}{rcccl} | |
| \big(c \ll r\big)(x) | |
| & \coloneqq & | |
| c(x) \models r | |
| & = & | |
| \displaystyle\sum_{y\in Y} c(x)(y)\cdot r(y). | |
| \end{array} | |
| \end{equation} | |
| \noindent This operation $\ll$ interacts nicely with composition | |
| $\after$ of channels, in the sense that $(d\after c) \ll r = c \ll (d | |
| \ll r)$. Moreover, the validity $\omega \models c \ll r$ is the same | |
| as the validity $c\gg \omega\models r$, where $\gg$ is state | |
| transformation, see~\eqref{eqn:discstatransf}. | |
| If a validity $\omega\models r$ is non-zero, then we can define the | |
| \emph{updated} or \emph{conditioned} state $\omega|_{r} \in \Dst(X)$ | |
| via: | |
| \begin{equation} | |
| \label{eqn:discconditioning} | |
| \begin{array}{rclcrcl} | |
| \big(\omega|_{r}\big)(x) | |
| & \coloneqq & | |
| \displaystyle\frac{\omega(x)\cdot r(x)}{\omega\models r} | |
| & \qquad\mbox{that is}\qquad & | |
| \omega|_{r} | |
| & = & | |
| \displaystyle\sum_{x\in X} | |
| \frac{\omega(x)\cdot r(x)}{\omega\models r}\bigket{x}. | |
| \end{array} | |
| \end{equation} | |
| \noindent The first formulation describes the updated distribution | |
| $\omega|_{r}$ as a probability mass function, whereas the second one | |
| uses a formal convex sum. | |
| It is not hard to see that successive updates commute and can be | |
| reduced to a single update via $\andthen$, as in: | |
| \begin{equation} | |
| \label{eqn:discconditioningand} | |
| \begin{array}{rcccccl} | |
| \big(\omega|_{r}\big)|_{s} | |
| & = & | |
| \omega|_{r\andthen s} | |
| & = & | |
| \omega|_{s\andthen r} | |
| & = & | |
| \big(\omega|_{s}\big)|_{r}. | |
| \end{array} | |
| \end{equation} | |
| \noindent One can multiply a random variable $r\colon X \rightarrow | |
| \R$ with a scalar $a\in\R$, pointwise, giving a new random variable | |
| $a\cdot r \colon X \rightarrow \R$. When $a\neq 0$ it disappears from | |
| updating: | |
| \begin{equation} | |
| \label{eqn:discconditioningscal} | |
| \begin{array}{rcl} | |
| \omega|_{a\cdot r} | |
| & = & | |
| \omega|_{r}. | |
| \end{array} | |
| \end{equation} | |
| \begin{proposition} | |
| \label{prop:conjpriorpointupdate} | |
| Assume that composable channels $\smash{P \stackrel{c}{\rightarrow} X | |
| \stackrel{d}{\rightarrow} O}$ for the discrete distribution monad | |
| $\Dst$ are given, where $c$ is conjugate prior to $d$, say via | |
| $h\colon P \times O \rightarrow P$. The distribution for the updated | |
| parameter $h(p,y)$ is then an update of the distribution for the | |
| original parameter $p$, with the pulled-back point predicate for the | |
| observation $y$, as in: | |
| \[ \begin{array}{rcl} | |
| c\big(h(p,y)\big) | |
| & = & | |
| c(p)\big|_{d \ll \indic{\{y\}}}. | |
| \end{array} \] | |
| \end{proposition} | |
| \begin{myproof} | |
| We first notice that the pulled-back singleton predicate $d \ll | |
| \indic{\{y\}} \colon X \rightarrow \R$ is: | |
| \[ \begin{array}{rcccl} | |
| (d \ll \indic{\{y\}})(x) | |
| & \smash{\stackrel{\eqref{eqn:discrandvartransform}}{=}} & | |
| \displaystyle\sum_{z\in Y} d(x)(z) \cdot \indic{\{y\}}(z) | |
| & = & | |
| d(x)(y). | |
| \end{array} \] | |
| \noindent Theorem~\ref{thm:conjugateinversion} tells us that | |
| $c\big(h(p,y)\big)$ is obtained via the Bayesian inversion of $d$, so | |
| that: | |
| \[ \begin{array}[b]{rcl} | |
| c\big(h(p,y)\big)(x) | |
| & = & | |
| d^{\dag}_{c(p)}(y)(x) | |
| \\[+0.4em] | |
| & \smash{\stackrel{\eqref{eqn:discreteinversion}}{=}} & | |
| \displaystyle\frac{c(p)(x)\cdot d(x)(y)}{\sum_{z} c(p)(z)\cdot d(z)(y)} | |
| \\[+1em] | |
| & = & | |
| \displaystyle\frac{c(p)(x)\cdot (d \ll \indic{\{y\}})(x)} | |
| {\sum_{z} c(p)(z)\cdot (d \ll \indic{\{y\}})(z)} | |
| \qquad \mbox{as just noted} | |
| \\[+1em] | |
| & = & | |
| \displaystyle\frac{c(p)(x)\cdot (d \ll \indic{\{y\}})(x)} | |
| {c(p) \models d \ll \indic{\{y\}}} | |
| \\ | |
| & \smash{\stackrel{\eqref{eqn:discconditioning}}{=}} & | |
| c(p)\big|_{d \ll \indic{\{y\}}}. | |
| \end{array} \eqno{\QEDbox} \] | |
| \end{myproof} | |
| In fact, what we are using here is that the Bayesian inversion | |
| $c^{\dag}_{\omega}$ defined in~\eqref{eqn:discreteinversion} is an | |
| update: $c^{\dag}_{\omega} = \omega|_{c \ll \indic{\{y\}}}$. | |
| \subsection{Continuous updating} | |
| We now present the analogous story for continuous probability. A | |
| \emph{random variable} on a measurable space $X$ is a measurable | |
| function $X\rightarrow \R$. It is called a \emph{predicate} if it | |
| restricts to $X\rightarrow [0,1]$. These random variables (and | |
| predicates) are closed under $\andthen$ and scalar multiplication, | |
| defined via pointwise multiplication. In the continuous case one | |
| typically has no point predicates. | |
| Given a measure/state $\omega\in\Giry(X)$ and a random variable | |
| $r\colon X \rightarrow \R$ we define the validity $\omega\models r$ | |
| again as expected value: | |
| \begin{equation} | |
| \label{eqn:contvalidity} | |
| \begin{array}{rcl} | |
| \omega\models r | |
| & \;\coloneqq\; & | |
| \displaystyle\int r \intd\omega. | |
| \end{array} | |
| \end{equation} | |
| \noindent This allows us to define transformation of a random | |
| variable, backwards along a channel: for a channel $c\colon | |
| X\rightarrow \Giry(Y)$ and a random variable $r\colon Y\rightarrow \R$ | |
| we write $c \ll r \colon X \rightarrow \R$ for the pulled-back random | |
| variable defined by: | |
| \begin{equation} | |
| \label{eqn:contrandvartransform} | |
| \begin{array}{rcccl} | |
| \big(c \ll r\big)(x) | |
| & \coloneqq & | |
| c(x) \models r | |
| & = & | |
| \displaystyle\int r \intd\, c(x). | |
| \end{array} | |
| \end{equation} | |
| \noindent The update $\omega|_{r}\in\Giry(X)$ of a state | |
| $\omega\in\Giry(X)$ with a random variable $r\colon X \rightarrow \R$ | |
| is defined on a measurable subset $M\subseteq X$ as: | |
| \begin{equation} | |
| \label{eqn:contconditioning} | |
| \begin{array}{rcccccl} | |
| \big(\omega|_{r}\big)(M) | |
| & \coloneqq & | |
| \displaystyle\frac{\int_{M} r \intd\omega}{\int r \intd\omega} | |
| & = & | |
| \displaystyle\frac{\int_{M} r \intd\omega}{\omega\models r} | |
| & = & | |
| \displaystyle\int_{M}\frac{r}{\omega\models r} \intd\omega. | |
| \end{array} | |
| \end{equation} | |
| \noindent If $\omega = \int f$ for a pdf $f$, this becomes: | |
| \begin{equation} | |
| \label{eqn:contconditioningpdf} | |
| \begin{array}{rcccccl} | |
| \big(\omega|_{r}\big)(M) | |
| & \coloneqq & | |
| \displaystyle\frac{\int_{M} f(x) \cdot r(x) \intd x} | |
| {\int f(x) \cdot r(x) \intd x} | |
| & = & | |
| \displaystyle\frac{\int_{M} f(x) \cdot r(x) \intd x}{\omega\models r} | |
| & = & | |
| \displaystyle\int_{M} \frac{f(x) \cdot r(x)}{\omega\models r} \intd x. | |
| \end{array} | |
| \end{equation} | |
| \noindent The latter formulation shows that the pdf of $\omega|_{r}$ | |
| is the function $x \mapsto \frac{f(x) \cdot r(x)}{\omega\models r}$. | |
| Updating in the continuous case also satisfies the multiple-update and | |
| scalar properties~\eqref{eqn:discconditioningand} and | |
| ~\eqref{eqn:discconditioningscal}. | |
| Again we redescribe conjugate priors in terms of updating. | |
| \begin{proposition} | |
| \label{prop:contpriorpointupdate} | |
| Let $\smash{P \stackrel{c}{\rightarrow} X \stackrel{d}{\rightarrow} | |
| O}$ be channels for the Giry monad $\Giry$, where $c$ and $d$ are | |
| pdf-channels $c = \int u$ and $d = \int v$, for $u\colon P\times X | |
| \rightarrow \R_{\geq 0}$ and $v\colon X\times O \rightarrow \R_{\geq | |
| 0}$. Assume that $c$ be conjugate prior to $d$ via $h\colon P | |
| \times O \rightarrow P$. Then: | |
| \[ \begin{array}{rcl} | |
| c\big(h(p,y)\big) | |
| & = & | |
| c(p)\big|_{v(-,y)}, | |
| \end{array} \] | |
| \noindent where $v(-,y) \colon O \rightarrow \R$ is used as random | |
| variable on $O$. | |
| \end{proposition} | |
| \begin{myproof} | |
| Theorem~\ref{thm:conjugateinversion} gives the first step in: | |
| \[ \begin{array}[b]{rcl} | |
| c\big(h(p,y)\big)(M) | |
| & = & | |
| d^{\dag}_{c(p)}(y)(M) | |
| \\[+0.4em] | |
| & \smash{\stackrel{\eqref{eqn:continuousinversion}}{=}} & | |
| \displaystyle\frac{\int_{M} u(p,x)\cdot v(x,y) \intd x} | |
| {\int u(p,x) \cdot v(x,y) \intd x} | |
| \\[+1em] | |
| & = & | |
| \displaystyle\frac{\int_{M} u(p,x)\cdot v(x,y) \intd x} | |
| {c(p) \models v(-,y)} | |
| \\ | |
| & \smash{\stackrel{\eqref{eqn:contconditioningpdf}}{=}} & | |
| c(p)\big|_{v(-,y)}(M). | |
| \end{array} \eqno{\QEDbox} \] | |
| \end{myproof} | |
| The previous two propositions deal with two \emph{discrete} channels | |
| $c,d$ (for $\Dst$) or with two \emph{continuous} channels (for | |
| $\Giry$). But the update approach also works for mixed channels, | |
| technically because $\Dst$ is a submonad of $\Giry$. We shall not | |
| elaborate these details but give illustrations instead. | |
| \begin{example} | |
| \label{ex:betaflipupdate} | |
| We shall have another look at the $\betachan - \flipchan$ conjugate | |
| prior situation from Example~\ref{ex:betaflip}. We claim that the | |
| essence of these channels being conjugate prior, via the parameter | |
| translation function~\eqref{eqn:betaflipfun}, can be expressed via the | |
| following two state update equations: | |
| \begin{equation} | |
| \label{eqn:betaflipupdateeqn} | |
| \begin{array}{rcl} | |
| \betachan(\alpha,\beta)\big|_{\flipchan \ll \indic{\{1\}}} | |
| & = & | |
| \betachan(\alpha+1,\beta) | |
| \\ | |
| \betachan(\alpha,\beta)\big|_{\flipchan \ll \indic{\{0\}}} | |
| & = & | |
| \betachan(\alpha,\beta+1). | |
| \end{array} | |
| \end{equation} | |
| \noindent These equations follow from what we have proven above. But | |
| we choose to re-prove them here in order to illustrate how updating | |
| works concretely. First note that for a parameter $x\in [0,1]$ we have | |
| predicate values $\big(\flipchan \ll \indic{\{1\}}\big)(x) = x$ and | |
| $\big(\flipchan \ll \indic{\{0\}}\big)(x) = 1-x$. Then: | |
| \[ \begin{array}{rcl} | |
| \betachan(\alpha,\beta) \models \flipchan \ll \indic{\{1\}} | |
| \hspace*{\arraycolsep}\smash{\stackrel{\eqref{eqn:beta}}{=}}\hspace*{\arraycolsep} | |
| \displaystyle\int x\cdot \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}\intd x | |
| & = & | |
| \displaystyle\frac{\int x^{\alpha}(1-x)^{\beta-1}\intd x}{B(\alpha,\beta)} | |
| \\[+0.8em] | |
| & = & | |
| \displaystyle\frac{B(\alpha+1,\beta)}{B(\alpha,\beta)}. | |
| \end{array} \] | |
| \noindent Thus, using~\eqref{eqn:contconditioningpdf}, we obtain | |
| the first equation in~\eqref{eqn:betaflipupdateeqn}: | |
| \[ \begin{array}{rcl} | |
| \betachan(\alpha,\beta)\big|_{\flipchan \ll \indic{\{1\}}}(M) | |
| & = & | |
| \displaystyle\frac{B(\alpha,\beta)}{B(\alpha+1,\beta)} \cdot | |
| \int_{M} x\cdot \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}\intd x | |
| \\[+0.8em] | |
| & = & | |
| \displaystyle\int_{M} \frac{x^{\alpha}(1-x)^{\beta-1}}{B(\alpha+1,\beta)}\intd x | |
| \\ | |
| & = & | |
| \betachan(\alpha+1,\beta)(M). | |
| \end{array} \] | |
| In a similar way we can capture the $\betachan - \binomchan$ conjugate | |
| priorship from Example~\ref{ex:betabinom} as update equation: | |
| \begin{equation} | |
| \label{eqn:betabinomupdateeqn} | |
| \begin{array}{rcl} | |
| \betachan(\alpha,\beta)\big|_{\binomchan_{n} \ll \indic{\{i\}}} | |
| & = & | |
| \betachan(\alpha+i, \beta+n-i). | |
| \end{array} | |
| \end{equation} | |
| \noindent This equation, and also~\eqref{eqn:betaflipupdateeqn}, | |
| hightlight the original ideal behind conjugate priors, expressed | |
| informally in many places in the literature as: we have a class of | |
| distributions --- $\betachan$ in this case --- which is closed under | |
| updates in a particular statistiscal model --- $\flipchan$ or | |
| $\binomchan$ in these cases. | |
| \end{example} | |
| These update formulations~\eqref{eqn:betabinomupdateeqn} | |
| and~\eqref{eqn:betaflipupdateeqn} may be useful when trying to find a | |
| parameter translation function: one can start calculating the state | |
| update on the left-hand-side, using | |
| formulas~\eqref{eqn:discconditioning} | |
| and~\eqref{eqn:contconditioning}, hoping that a distribution of the | |
| same form appears (but with different parameters). | |
| \section{Multiple updates}\label{sec:multiple} | |
| So far we have dealt with the situation where there is a single | |
| observation $y\in O$ that leads to an update of a prior | |
| distribution. In this final section we briefly look at how to handle | |
| multiple observations $y_{1}, \ldots, y_{m}$. This is what typically | |
| happens in practice; it will lead to the notion of \emph{sufficient | |
| statistic}. | |
| A good starting point is the $\betachan - \flipchan$ relationship from | |
| Example~\ref{ex:betaflip} and~\ref{ex:betaflipupdate}, especially in | |
| its snappy update form~\eqref{eqn:betaflipupdateeqn}. Suppose we have | |
| multiple head/tail observations $y_{1}, \ldots, y_{m} \in 2 = \{0,1\}$ | |
| which we wish to incorporate into a prior distribution | |
| $\betachan(\alpha,\beta)$. Following | |
| Equation~\eqref{eqn:betaflipupdateeqn} we use multiple updates, on the | |
| left below, which can be rewritten as a single update, on the | |
| right-hand-side of the equation via conjunction $\andthen$, | |
| using~\eqref{eqn:discconditioningand}: | |
| \[ \begin{array}{rcl} | |
| \lefteqn{\betachan(\alpha,\beta)\big|_{\flipchan \ll \indic{\{y_1\}}} | |
| \big|_{\flipchan \ll \indic{\{y_2\}}} \;\cdots\; \big|_{\flipchan \ll \indic{\{y_m\}}}} | |
| \\ | |
| & = & | |
| \betachan(\alpha,\beta)\big|_{(\flipchan \ll \indic{\{y_1\}}) \,\andthen\, | |
| (\flipchan \ll \indic{\{y_2\}}) \,\andthen\; \cdots \;\andthen\, | |
| (\flipchan \ll \indic{\{y_m\}})}. | |
| \end{array} \] | |
| \noindent The $m$-ary conjunction predicate in the latter expression | |
| amounts to $q(x) = x^{n_1}(1-x)^{n_0}$ where $n_{1} = \sum_{i} y_{i}$ | |
| is the number of $1$'s among the observation $y_i$ and $n_{0} = | |
| \sum_{i} (1-y_{i})$ is the number of $0$'s, see | |
| Example~\ref{ex:betaflipupdate}. Of course the outcome is | |
| $\betachan(\alpha+n_{1}, \beta+n_{0})$. The question that is relevant | |
| in this setting is: can a random variable $p(y_{1}, \ldots, y_{m})$ | |
| with many parameters somehow be simplified, like in $q$ above. This is | |
| where the notion of sufficient statistic arises, see | |
| \textit{e.g.}~\cite{Koopman36,Bishop06}. | |
| \begin{definition} | |
| \label{def:sufstat} | |
| Let $p\colon X\times O^{m} \rightarrow \R$ be a random variable, with | |
| $1+m$ inputs. A \emph{sufficient statistic} for $p$ is a triple of | |
| functions | |
| \[ \xymatrix{ | |
| O^{m}\ar[r]^-{s} & \R | |
| \qquad\quad | |
| O^{m}\ar[r]^-{t} & Z | |
| \qquad\quad | |
| X\times Z\ar[r]^-{q} & \R | |
| } \] | |
| \noindent so that $p$ can be written as: | |
| \begin{equation} | |
| \label{eqn:sufstat} | |
| \begin{array}{rcl} | |
| p(x, y_{1}, \ldots, y_{m}) | |
| & = & | |
| s(y_{1}, \ldots, y_{m}) \cdot q\big(x, t(y_{1}, \ldots, y_{m})\big). | |
| \end{array} | |
| \end{equation} | |
| \end{definition} | |
| In the above $\betachan$ example we would like to simplify the big | |
| conjunction random variable: | |
| \[ \begin{array}{rcl} | |
| p(x, y_{1}, \ldots, y_{m}) | |
| & = & | |
| \Big((\flipchan \ll \indic{\{y_1\}}) \,\andthen\; \cdots \;\andthen\, | |
| (\flipchan \ll \indic{\{y_m\}})\Big)(x). | |
| \end{array} \] | |
| \noindent We can take $Z = \NNO\times\NNO$ with $t(y_{1}, \ldots, | |
| y_{m}) = (n_{1}, n_{0})$, where $n_1$ and $n_{0}$ are the number of | |
| $1$'s and $0$'s in the $y_i$. Then $q(x, n, n') = x^{n}(1-x)^{n'}$. | |
| The function $s$ is trivial and sends everything to $1$. | |
| A sufficient statistic thus summarises, esp.\ via the function $t$, | |
| the essential aspects of a list of observations, in order to simplify | |
| the update. In the coin example, these essential aspects are the | |
| numbers of $1$'s and $0$'s (that is, of heads and tails). In these | |
| situations the conjunction predicate --- like $p$ above --- is usally | |
| called a \emph{likelihood}. | |
| The big advantage of writing a random variable $p$ in the form | |
| of~\eqref{eqn:sufstat} is that updating with $p$ can be | |
| simplified. Let $\omega$ be a distribution on $X$, either discrete or | |
| continuous. Then, writing $\vec{y} = (y_{1}, \ldots, y_{m})$ we get: | |
| \[ \begin{array}{rcccl} | |
| \omega|_{p(-,\vec{y})} | |
| & = & | |
| \omega|_{s(\vec{y})\cdot q(-,t(\vec{y}))} | |
| & = & | |
| \omega|_{q(-,t(\vec{y}))}. | |
| \end{array} \] | |
| \noindent The factor $s(\vec{y})$ drops out because it works | |
| like a scalar, see~\eqref{eqn:discconditioningscal}. | |
| We conclude this section with a standard example of a sufficient | |
| statistic (see \textit{e.g.}~\cite{Bishop06}), for a conjunction | |
| expression arising from multiple updates. | |
| \begin{example} | |
| \label{ex:normnormstat} | |
| Recall the $\normchan - \normchan$ conjugate priorship from | |
| Example~\ref{ex:normnorm}. The first channel there has the form | |
| $\normchan = \int u$, for $u(\mu,\sigma,x) = | |
| \nicefrac{1}{\sqrt{2\pi}\sigma}\cdot | |
| e^{-\nicefrac{(x-\mu)^{2}}{2\sigma^2}}$. The second channel is | |
| $\normchan(-,\nu) = \int v$, for a fixed `noise' factor $\nu$, where | |
| $v(x,y) = \nicefrac{1}{\sqrt{2\pi}\nu}\cdot | |
| e^{-\nicefrac{(y-x)^{2}}{2\nu^2}}$. Let's assume that we have | |
| observations $y_{1}, \ldots, y_{m} \in \R_{> 0}$ which we like to use | |
| to iteratively update the prior distribution $\normchan(\mu,\sigma)$. | |
| Following Proposition~\ref{prop:contpriorpointupdate} we | |
| can describe these updates as: | |
| \[ \begin{array}{rcl} | |
| \normchan(\mu,\sigma)\big|_{v(-,y_1)} \;\cdots\; \big|_{v(-,y_m)} | |
| & = & | |
| \normchan(\mu,\sigma)\big|_{v(-,y_1) \,\andthen \;\cdots\; \andthen\,v(-,y_m)}. | |
| \end{array} \] | |
| \noindent Thus we are interested in finding a sufficient statistics | |
| for the predicate: | |
| \[ \begin{array}{rcl} | |
| p(x,y_{1}, \ldots, y_{m}) | |
| & \coloneqq & | |
| \Big(v(-,y_1) \,\andthen \;\cdots\; \andthen\,v(-,y_m)\Big)(x) | |
| \\ | |
| & = & | |
| v(x,y_{1}) \cdot \;\cdots\; \cdot v(x,y_{m}) | |
| \\ | |
| & = & | |
| \frac{1}{\sqrt{2\pi}\nu}\cdot e^{-\frac{(y_{1}-x)^{2}}{2\nu^{2}}} \;\cdots\; | |
| \frac{1}{\sqrt{2\pi}\nu}\cdot e^{-\frac{(y_{m}-x)^{2}}{2\nu^{2}}} | |
| \\ | |
| & = & | |
| \frac{1}{(\sqrt{2\pi}\nu)^{m}}\cdot e^{-\frac{\sum_{i}(y_{i}-x)^{2}}{2\nu^2}} | |
| \\ | |
| & = & | |
| \frac{1}{(\sqrt{2\pi}\nu)^{m}}\cdot e^{-\frac{\sum_{i}y_{i}^{2}}{2\nu^2}} | |
| \cdot e^{\frac{2(\sum_{i}y_{i})x-mx^{2}}{2\nu^2}} | |
| \\ | |
| & = & | |
| s(y_{1}, \ldots, y_{m}) \cdot q\big(x, t(y_{1}, \ldots, y_{m})\big), | |
| \end{array} \] | |
| \noindent for functions $s,t,q$ given by: | |
| \[ \begin{array}{rclcrclcrcl} | |
| s(y_{1}, \ldots, y_{m}) | |
| & = & | |
| \frac{1}{(\sqrt{2\pi}\nu)^{m}}\cdot e^{-\frac{\sum_{i}y_{i}^{2}}{2\nu^2}} | |
| & \quad & | |
| t(y_{1}, \ldots, y_{m}) | |
| & = & | |
| \sum_{i}x_{i} | |
| & \quad & | |
| q(x, z) | |
| & = & | |
| e^{\frac{2zx-mx^{2}}{2\nu^2}}. | |
| \end{array} \] | |
| \end{example} | |
| \section{Conclusions}\label{sec:conclusions} | |
| This paper contains a novel view on conjugate priors, using the | |
| concept of channel in a systematic manner. It has introduced a precise | |
| definition for conjugate priorship, using a pair of composable | |
| channels $P\rightarrow X\rightarrow O$ and a parameter translation | |
| function $P\times O \rightarrow P$, satisfying a non-trivial equation, | |
| see Definition~\ref{def:conjugateprior}. It has been shown that this | |
| equation holds for several standard conjugate prior examples. There | |
| are many more examples, that have not been checked here. One can be | |
| confident that the same equation holds for those unchecked examples | |
| too, since it has been shown here that conjugate priors amount to | |
| Bayesian inversions. This inversion property is the essential | |
| characteristic for conjugate priors. It has been re-formulated in | |
| logical terms, so that the closure property of a class of priors under | |
| updating is highlighted. | |
| \appendix | |
| \section{Calculation laws for Giry-Kleisli maps with | |
| pdf's}\label{sec:calculation} | |
| We assume that for a probability distribution (state) | |
| $\omega\in\Giry(X)$ and a measurable function $f\colon X \rightarrow | |
| \R_{\geq 0}$ the integral $\int f \intd \omega \in [0,\infty]$ can be | |
| defined as a limit of integrals over simple functions that approximate | |
| $f$. We shall follow the description of~\cite{Jacobs13a}, to which we | |
| refer for details\footnote{In~\cite{Jacobs13a} integration $\int | |
| f\intd \omega$ is defined only for $[0,1]$-valued functions $f$, but | |
| that does not matter for the relevant equations, except that | |
| integrals may not exist for $\R_{\geq 0}$-valued functions (or have | |
| value $\infty$). These integrals are determined by their valued | |
| $\int \indic{M}\intd \omega = \omega(M)$ on indicator functions | |
| $\indic{M}$ for measurable subsets, via continuous and linear | |
| extensions, see also~\cite{JacobsW15a}.}. This integration | |
| satisfies the Fubini property, which can be formulated, for states | |
| $\omega\in\Giry(X)$, $\rho\in\Giry(Y)$ and measurable function | |
| $h\colon X\times Y \rightarrow \R_{\geq 0}$, as: | |
| \begin{equation} | |
| \label{eqn:productpdfintegration} | |
| \begin{array}{rcl} | |
| \displaystyle\int h \intd (\omega\otimes\rho) | |
| & = & | |
| \displaystyle\int \int h \intd \omega \intd\rho. | |
| \end{array} | |
| \end{equation} | |
| \noindent The product state $\omega\otimes\rho \in \Giry(X\times Y)$ | |
| is defined by $(\omega\otimes\rho)(M\times N) = | |
| \omega(M)\cdot\rho(N)$. | |
| \begin{Auxproof} | |
| It suffices to consider the special cases where $g = \indic{M}$ and | |
| $h = \indic{M\times N}$, in: | |
| \[ \begin{array}{rcl} | |
| \displaystyle \int \indic{M} \intd \omega | |
| & = & | |
| \omega(\indic{M}) | |
| \\ | |
| & = & | |
| \displaystyle \int_{M} f(x)\intd x | |
| \\ | |
| & = & | |
| \displaystyle\int f(x) \cdot \indic{M}(x) \intd x | |
| \\ | |
| & = & | |
| \displaystyle\int f(x) \cdot g(x) \intd x | |
| \\ | |
| \displaystyle \int \indic{M\times N} \intd (\omega\otimes\rho) | |
| & = & | |
| (\omega\otimes\rho)(M\times N) | |
| \\ | |
| & = & | |
| \omega(M) \cdot \rho(N) | |
| \\ | |
| & = & | |
| \big(\displaystyle\int_{M} f(x) \intd x\big)\cdot | |
| \big(\displaystyle \int \indic{N} \intd \rho\big) | |
| \\ | |
| & = & | |
| \displaystyle \int \big(\displaystyle\int_{M} f(x) \intd x\big)\cdot \indic{N} | |
| \intd \rho | |
| \\ | |
| & = & | |
| \displaystyle \int \int \indic{M}(x)\cdot f(x)\cdot \indic{N} \intd x | |
| \intd \rho | |
| \\ | |
| & = & | |
| \displaystyle \int \int \indic{M\times N}(x,-)\cdot f(x) \intd x \intd \rho | |
| \\ | |
| & = & | |
| \displaystyle \int \int h(x,-)\cdot f(x) \intd x \intd \rho. | |
| \end{array} \] | |
| \end{Auxproof} | |
| \subsection{States via pdf's}\label{subsec:stateviapdf} | |
| For a subset $X\subseteq \R$, a measurable function $f\colon X | |
| \rightarrow \R_{\geq 0}$ is called a probability density function | |
| (pdf) for a state $\omega\in\Giry(X)$ if $\omega(M) = \int_{M} f(x) | |
| \intd x$ for each measurable subset $M\subseteq X$. In that case we | |
| simply write $\omega = \int f(x) \intd x$, or even $\omega = \int | |
| f$. If $\omega$ is given by such a pdf $f$, integration with state | |
| $\omega$ can be described as: | |
| \begin{equation} | |
| \label{eqn:pdfintegration} | |
| \begin{array}{rcl} | |
| \displaystyle\int g \intd \omega | |
| & = & | |
| \displaystyle\int f(x)\cdot g(x) \intd x. | |
| \end{array} | |
| \end{equation} | |
| \subsection{Channels via pdf's}\label{subsec:channelviapdf} | |
| Let channel $c\colon X \rightarrow \Giry(Y)$ be given as $c = \int u$ | |
| by pdf $u\colon X\times Y \rightarrow \R_{\geq 0}$ as $c(x)(N) = | |
| \int_{N} u(x,y) \intd y$, for each $x\in X$ and measurable $N\subseteq | |
| Y$, like in~\eqref{eqn:channelfrompdf}. If $\omega = \int f$ is a | |
| state on $X$, then state transformation $c \gg \omega \in \Giry(Y)$ | |
| is given by: | |
| \begin{equation} | |
| \label{eqn:pdfstatetransformation} | |
| \hspace*{-0.8em}\begin{array}{rcl} | |
| (c \gg \omega)(N) | |
| \hspace*{\arraycolsep}\smash{\stackrel{\eqref{eqn:statetransformation}}{=}}\hspace*{\arraycolsep} | |
| \displaystyle \int c(-)(N) \intd \omega | |
| & \smash{\stackrel{\eqref{eqn:pdfintegration}}{=}} & | |
| \displaystyle \int f(x) \cdot c(x)(N) \intd x | |
| \\ | |
| & = & | |
| \displaystyle \int_{N} \int f(x) \cdot u(x,y) \intd x \intd y. | |
| \end{array} | |
| \end{equation} | |
| \noindent Hence the pdf of the transformed state $c \gg \omega$ is $y | |
| \mapsto \int f(x) \cdot u(x,y) \intd x$. | |
| Given a channel $d \colon Y \rightarrow \Giry(Z)$, say with $d = \int | |
| v$, then sequential channel composition $d \after c$ is given, for | |
| $x\in X$ and $K\subseteq Z$, by: | |
| \begin{equation} | |
| \label{eqn:pdfseqcomposition} | |
| \begin{array}{rcl} | |
| (d \after c)(x)(K) | |
| \hspace*{\arraycolsep}\smash{\stackrel{\eqref{eqn:continuouscomposition}}{=}}\hspace*{\arraycolsep} | |
| \displaystyle \int d(-)(K) \intd c(x) | |
| & \smash{\stackrel{\eqref{eqn:pdfintegration}}{=}} & | |
| \displaystyle \int u(x,y) \cdot d(y)(K) \intd y | |
| \\ | |
| & = & | |
| \displaystyle \int_{K} \int u(x,y) \cdot v(y,z) \intd y \intd z | |
| \end{array} | |
| \end{equation} | |
| \noindent We see that the pdf of the channel $d \after c$ is $(x,z) | |
| \mapsto \int u(x,y) \cdot v(y,z) \intd y$. | |
| For a channel $e = \int w \colon A \rightarrow \Giry(B)$ we get | |
| a parallel composition channel $c\otimes e \colon X\times A | |
| \rightarrow \Giry(Y\times B)$ given by: | |
| \begin{equation} | |
| \label{eqn:pdfparcomposition} | |
| \begin{array}{rcl} | |
| (c \otimes e)(x,a)(M\times N) | |
| & = & | |
| c(x)(M) \otimes e(a)(N) | |
| \\ | |
| & = & | |
| \displaystyle\big(\int_{M} u(x,y)\intd y\big)\cdot | |
| \big(\int_{N} w(a,b) \intd b\big) | |
| \\[0.8em] | |
| & = & | |
| \displaystyle \int_{M\times N} u(x,y)\cdot w(a,b) \intd (y,b). | |
| \end{array} | |
| \end{equation} | |
| \noindent Hence the pdf of the channel $c\otimes d$ is $(x,a,y,b) | |
| \mapsto u(x,y)\cdot w(a,b)$. | |
| \subsection{Graph channels and pdf's}\label{subsec:graphandpdf} | |
| For a channel $c\colon X \rightarrow \Giry(Y)$ we can form `graph' | |
| channels $\tuple{\idmap,c} = (\idmap\otimes c) \after \copier \colon X | |
| \rightarrow \Giry(X\times Y)$ and $\tuple{c,\idmap} = (c\otimes\idmap) | |
| \after \copier \colon X \rightarrow \Giry(Y\times X)$. For $x\in X$ we | |
| have: | |
| \begin{equation} | |
| \label{eqn:graphequation} | |
| \begin{array}{rclcrcl} | |
| \tuple{\idmap,c}(x) | |
| & = & | |
| \eta(x)\otimes c(x) | |
| & \qquad\mbox{and}\qquad & | |
| \tuple{c,\idmap}(x) | |
| & = & | |
| c(x) \otimes \eta(x). | |
| \end{array} | |
| \end{equation} | |
| \begin{Auxproof} | |
| \[ \begin{array}{rcl} | |
| \tuple{\idmap, c}(x)(M\times N) | |
| & = & | |
| \big(\st_{2} \after (\idmap\times c) \after \Delta\big)(x)(M\times N) | |
| \\ | |
| & = & | |
| \big(\dst \after (\eta\times c) \after \Delta\big)(x)(M\times N) | |
| \\ | |
| & = & | |
| \dst\big((\eta\times c)(x,x)\big)(M\times N) | |
| \\ | |
| & = & | |
| \dst\big(\eta(x), c(x)\big)(M\times N) | |
| \\ | |
| & = & | |
| \eta(x)(M) \cdot c(x)(N) | |
| \\ | |
| & = & | |
| \big(\eta(x) \otimes c(x)\big)(M\times N). | |
| \end{array} \] | |
| \end{Auxproof} | |
| \noindent If $c = \int u$ and $\omega = \int f$ is a state on $X$, then: | |
| \begin{equation} | |
| \label{eqn:pdfgraphstatetransformation} | |
| \begin{array}{rcl} | |
| (\tuple{\idmap,c} \gg \omega)(M\times N) | |
| & \smash{\stackrel{\eqref{eqn:pdfintegration}}{=}} & | |
| \displaystyle \int f(x) \cdot \tuple{\idmap,c}(x)(M\times N) \intd x | |
| \\ | |
| & \smash{\stackrel{\eqref{eqn:graphequation}}{=}} & | |
| \displaystyle \int f(x) \cdot \eta(x)(M) \cdot c(x)(N) \intd x | |
| \\ | |
| & = & | |
| \displaystyle \int_{N} \int_{M} f(x) \cdot u(x,y) \intd x \intd y. | |
| \end{array} | |
| \end{equation} | |
| \noindent We also consider the situation where $d\colon X\times Y | |
| \rightarrow \Giry(Z)$ is of the form $d = \int v$, with composition: | |
| \begin{equation} | |
| \label{eqn:pdfgraphcomposition} | |
| \begin{array}{rcl} | |
| \big(d \after \tuple{\idmap,c}\big)(x)(K) | |
| & \smash{\stackrel{\eqref{eqn:graphequation}}{=}} & | |
| \displaystyle \int d(-)(K) \intd (\eta(x)\otimes c(x)) | |
| \\ | |
| & \smash{\stackrel{\eqref{eqn:productpdfintegration}}{=}} & | |
| \displaystyle \int d(-)(K) \intd \eta(x) \intd c(x) | |
| \\ | |
| & = & | |
| \displaystyle \int d(x,-)(K) \intd c(x) | |
| \\ | |
| & \smash{\stackrel{\eqref{eqn:pdfintegration}}{=}} & | |
| \displaystyle \int u(x,y) \cdot d(x,y)(K) \intd y | |
| \\ | |
| & = & | |
| \displaystyle \int_{K} \int u(x,y) \cdot v(x,y,z) \intd y \intd z. | |
| \end{array} | |
| \end{equation} | |
| \noindent Hence the pdf of the channel $d \after \tuple{\idmap,c}$ | |
| is $(x,z) \mapsto \int u(x,y) \cdot v(x,y,z) \intd y$. | |
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| \end{document} | |