Add demo explanation

app.py

@@ -187,12 +187,20 @@ st.markdown("""We immediately notice that the demand (and profit) posteriors are
 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
+st.markdown("""And finally we arrive at a price point of €4.04 which is eerily close to
 the actual optimum of €4.24""")
 
 # (5) Thompson sampling demo
 st.header("Demo time 🎮")
 st.markdown("Now that we have covered the theory, you can go ahead and try it our for yourself!")
+st.markdown("""You will notice a few things:  \n
+👉 As you increase price elasticity, the demand becomes more sensitive to price changes and thus the
+profit-optimizing price becomes lower (& vice versa).  \n
+👉 As you decrease price elasticity, our demand observations at €7.5, €10 and €11 become
+increasingly larger and increasingly more variable (as their variance is a constant fraction of the
+absolute value). This causes our demand posterior to become increasingly wider and thus Thompson
+sampling will lead to more exploration.
+""")
 
 thompson_sampler = ThompsonSampler()
 demo_button = st.checkbox(

helpers/thompson_sampling.py

@@ -70,11 +70,12 @@ class ThompsonSampler:
         plt.violinplot(price_distr, vert=False, showextrema=False)
 
         for price in self.price_samples:
-            plt.plot(price, 1, marker='o', markersize = 5, color='grey')
+            plt.plot(price, 1, marker='o', markersize = 5, color='grey', label="Price sample")
 
         plt.axhline(1, color='black')
-        plt.axvline(self.latent_price, 0, color='red')
+        plt.axvline(self.latent_price, 0, color='red', label="Latent optimal price")
 
+        _plot_legend()
         return fig
 
     def create_posteriors_plot(self, highlighted_sample=None):
@@ -84,13 +85,21 @@ class ThompsonSampler:
         plt.xlim(0,20)
         plt.ylim(0,10)
 
-        plt.scatter(self.price_observations, self.demand_observations)
-        plt.plot(self.possible_prices, self.latent_demand, color="red")
+        plt.scatter(self.price_observations, self.demand_observations, label="Demand observations")
+        plt.plot(self.possible_prices, self.latent_demand, color="red", label="Latent demand")
 
         for posterior_sample in self.posterior_samples:
-            plt.plot(self.possible_prices, posterior_sample, color="grey", alpha=0.15)
+            plt.plot(
+                self.possible_prices, posterior_sample,
+                color="grey", alpha=0.15, label="Posterior demand"
+            )
         if highlighted_sample is not None:
-            plt.plot(self.possible_prices, highlighted_sample, color="black")
+            plt.plot(
+                self.possible_prices, highlighted_sample,
+                color="black", label="Thompson sampled demand"
+            )
+
+        _plot_legend()
         return fig
 
     def calc_latent_demand(self):
@@ -128,3 +137,9 @@ class ThompsonSampler:
         if st.session_state.latent_elasticity != self.latent_elasticity:
             self.update_elasticity()
         self.pick_posterior()
+
+def _plot_legend():
+    handles, labels = plt.gca().get_legend_handles_labels()
+    labels, ids = np.unique(labels, return_index=True)
+    handles = [handles[i] for i in ids]
+    plt.legend(handles, labels, loc='upper right')