Fix text typos

app.py

@@ -31,31 +31,30 @@ st.subheader("Setting optimal prices with Bayesian stats 📈")
 st.header("Let's start with the basics 🏁")
 
 st.markdown("The beginning is usually a good place to start so we'll kick things off there.")
-st.markdown("""The one crucial piece information we need to find the optimal price is
-**how demand behaves over different price points**.  \nIf we can get make a decent guess of what we 
-can expect demand to be for a wide range of prices, we can figure out which price optimizes our target
-(i.e., revenue, profit, ...).""")
+st.markdown("""The one crucial piece information we need in order to find the optimal price is
+**how demand behaves over different price points**.  \nIf we can make a decent guess of what we 
+can expect demand to be for a wide range of prices, we can figure out which price optimizes our 
+target (i.e., revenue, profit, ...).""")
 st.markdown("""For the keen economists amongst you, this is beginning to sound a lot like a
 **demand curve**.""")
 
 st.markdown("""Estimating a demand curve, sounds easy enough right?  \nLet's assume we have 
 demand with constant price elasticity; so a certain percent change in price will cause a
-constant percent change in demand, independent of the price level. This is often seen as a 
-reasonable proxy for demand curves in the wild.""")
-st.markdown("So our data will look something like this:")
+constant percent change in demand, independent of the price level. In economics, this is often used 
+as a proxy for demand curves in the wild.""")
+st.markdown("So our demand data looks something like this:")
 st.image("assets/images/ideal_case_demand.png")
 st.markdown("""Alright now we can get out our trusted regression toolbox and fit a nice curve 
-through the data because we know that our constant-elasticity demand function looks something 
-like this:""")
+through the data because we know that our constant-elasticity demand function has this form:""")
 st.latex(sympy.latex(sympy.Eq(sympy.Function(D)(p), a*p**(-eta), evaluate=False))) 
 st.write("with shape parameter a and price elasticity η")
 st.image("assets/images/ideal_case_demand_fitted.png")
-st.markdown("""Now that we have a reasonable estimate of our demand, we can derive our expected
-profit at different price points because we know the following holds:""")
+st.markdown("""Now that we have a reasonable estimate of our demand function, we can derive our 
+expected profit at different price points because we know the following holds:""")
 st.latex(f"{profit} = {p}*{sympy.Function(D)(p)} - [{var_cost}*{sympy.Function(D)(p)} + {fixed_cost}]")
 st.image("assets/images/ideal_case_profit_curve.png")
-st.markdown("""Finally we can dust off our good old-fashioned high-school math and find the
-price which we expect will optimize profit which is ultimately the goal of all this.""")
+st.markdown("""Finally we can dust off our good old high-school math book and find the
+price which we expect will optimize profit which was ultimately the goal of all this.""")
 st.image("assets/images/ideal_case_optimal_profit.png")
 st.markdown("""Voilà there you have it: we should price this product at 4.24 and we can expect
 a bottom-line profit of 7.34""")
@@ -63,9 +62,9 @@ st.markdown("So can we kick back & relax now?  \nWell, there are a few issues wi
 
 # (2) Dynamic demand curves
 st.header("The demands they are a-changin' 🎸")
-st.markdown("""We've got a first bit of bad news: unfortunately, you can't just estimate a demand 
-curve once and be done with it.  \nWhy? Because demand is influenced by many factors (e.g., market
-trends, competitor actions, human behavior, etc.) that tend to change a lot over time.""")
+st.markdown("""We arrive at our first bit of bad news: unfortunately, you can't just estimate a 
+demand curve once and be done with it.  \nWhy? Because demand is influenced by many factors (e.g., 
+market trends, competitor actions, human behavior, etc.) that tend to change a lot over time.""")
 st.write("Below you can see an (exaggerated) example of what we're talking about:")
 
 with open("assets/images/dynamic_demand.gif", "rb") as file_:
@@ -86,24 +85,23 @@ st.markdown("""So far, we have assumed that we get (and keep getting) data on de
 different price points.  \n
 Not only is this assumption **unrealistic**, it is also very **undesirable**""")
 st.markdown("""Why? Because getting demand data on a wide spectrum of price points implies that
-we are spending a significant amount of time setting prices at levels that are either too high or
-too low!  \n
+we are spending a significant amount of time setting prices that are either too high or too low!  \n
 Which is ironically exactly the opposite of what we set out to achieve.""")
-st.markdown("In practice, our demand will rather look something like this:")
+st.markdown("In practice, our demand observations will rather look something like this:")
 st.image("assets/images/realistic_demand.png")
-st.markdown("""As we can see, we have tried three price points in the past (€7.5, €10 and €11) for 
-which we have collected demand data.""")
+st.markdown("""As we can see, we have tried three price points in the past (€7.5, €10 and €11) and
+collected demand data.""")
 st.markdown("""On a side note: keep in mind that we still assume the same latent demand curve and
 optimal price point of €4.24  \n
-So (for the sake of the example) we have been massively overpricing our product in the past""")
+So (for the sake of the example) we have been massively overpricing our product in the past.""")
 st.image("assets/images/realistic_demand_latent_curve.png")
-st.markdown("""This constrained data brings along a major challenge in estimating the demand curve 
+st.markdown("""This limited data brings along a major challenge in estimating the demand curve 
 though.  \n
 Intuitively, it makes sense that we can make a reasonable estimate of expected demand at €8 or €9,
 given the observed demand at €7.5 and €10.  \nBut can we extrapolate further to €2 or €20 with the 
-same reasonable confidence?""")
+same reasonable confidence? Probably not.""")
 st.markdown("""This is a nice example of a very well-known problem in statistics called the 
-**\"exploration-exploitation\" trade-off**  \n
+**\"exploration-exploitation trade-off\"**  \n
 👉 **Exploration**: We want to explore the demand for a diverse enough range of price points
 so that we can accurately estimate our demand curve.  \n
 👉 **Exploitation**: We want to exploit all the knowledge we have gained through exploring and
@@ -161,17 +159,17 @@ st.markdown("""And eventually we arrive at a new price: €5.25! Which is indeed
 to the actual optimal price of €4.24""")
 st.markdown("""Now that we have our first updated price point, why stop there? Let's simulate 10 
 demand data points at this price point from out latent demand curve and check whether Thompson 
-sampling will edge us even closer to that optimal €4.24 point""")
+sampling will edge us even closer to that optimal €4.24 point.""")
 st.image("assets/images/updated_prices_demand.png")
-st.markdown("""We know the drill by down.  \n
+st.markdown("""We know the drill by now.  \n
 Let's recalculate our posteriors with this extra information.""")
 st.image(["assets/images/posterior_demand_2.png", "assets/images/posterior_profit_2.png"])
 st.markdown("""We immediately notice that the demand (and profit) posteriors are much less spread
-apart this time around which implies that we are more confident in our predictions""")
-st.markdown("Now, we can sample just one curve from the distribution")
+apart this time around which implies that we are more confident in our predictions.""")
+st.markdown("Now, we can sample just one curve from the distribution.")
 st.image(["assets/images/posterior_demand_sample_2.png", "assets/images/posterior_profit_sample_2.png"])
 st.markdown("""And finally we arrive at a price point of €4.44 which is eerily close to
-the actual optimum of €4.24.""")
+the actual optimum of €4.24""")
 
 # (5) Thompson sampling demo
 st.header("Demo time 🎮")
@@ -187,7 +185,7 @@ elasticity = st.slider(
     key="latent_elasticity",
     min_value=0.05,
     max_value=0.95,
-    value=0.15,
+    value=0.25,
     step=0.05,
 )
 while demo_button:
@@ -221,9 +219,11 @@ st.markdown("""If you do want to learn more, we recommend these links:
 [2](https://thegradient.pub/gaussian-process-not-quite-for-dummies/),
 [3](https://sidravi1.github.io/blog/2018/05/15/latent-gp-and-binomial-likelihood)""")
 
-st.subheader("""👉 Price optimization is much more complex than just finding the point that maximizes a simple profit function. What about inventory constraints, complex cost structures, ...?""")
-st.markdown("""It sure is but the nice thing about our setup is that it consists of three 
-components that you can change pretty much independently from each other.  \n
+st.subheader("""👉 Price optimization is much more complex than just optimizing a simple profit function?""")
+st.markdown("""It sure is. In reality, there are many added complexities that come into play, such
+as inventory/capacity constraints, complex cost structures, ...""")
+st.markdown("""The nice thing about our setup is that it consists of three components that you can 
+change pretty much independently from each other.  \n
 This means that you can make the price optimization pillar arbitrarily custom/complex. As long as
 it takes in a demand function and spits out a price.""")
 st.image("assets/images/flywheel_2.png")