Move explanation to Medium blog post

app.py

@@ -1,15 +1,12 @@
 """Streamlit entrypoint"""
 
-import base64
 import time
 
 import numpy as np
 import streamlit as st
-import sympy
 
 from helpers.thompson_sampling import ThompsonSampler
 
-eta, a, p, D, profit, var_cost, fixed_cost = sympy.symbols("eta a p D Profit varcost fixedcost")
 np.random.seed(42)
 
 st.set_page_config(
@@ -27,265 +24,12 @@ st.set_page_config(
 st.title("Dynamic Pricing")
 st.subheader("Setting optimal prices with Bayesian stats 📈")
 
-# (0) Intro
-st.header("Why care about dynamic pricing? 💭")
-
-st.markdown("""Dynamic pricing aims to actively adapt product prices based on insights about 
-customer behaviour.  \n
-This pricing strategy has proven exceptionally effective in a wide range of industries: from 
-e-commerce (e.g., eBay) to airlines (e.g., Delta Airlines) and from retail (e.g., Walmart) to 
-utilities (e.g., Tampa Electric) as it allows to **adjust prices to changes in demand patterns**.""")
-st.markdown("""It is hardly surprising that recent times of extraordinary uncertainty and volatility
-caused a surge in adoption of dynamic pricing strategies with an [estimated 21% of e-commerce 
-businesses reportedly already using dynamic pricing](https://www.statista.com/statistics/1174557/dynamic-pricing-ecommerce-companies-worldwide/) 
-and an additional 15% planning to adopt the strategy in the upcoming year.""")
-st.markdown("""To find a big success story, we should look no further than Amazon who (on average)
-change their products' prices once every 10 minutes. They attribute roughly [25% of their e-commerce
-profits](https://dzone.com/articles/big-data-analytics-delivering-business-value-at-am)
-to their pricing strategy.""")
-st.markdown("""In what follows we'll discuss the working principles & challenges in the field of
-dynamic pricing, followed by an interactive demo and some final considerations.""")
-
-# (1) Basics
-st.header("Back to 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 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. 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 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 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("""Note that fixed costs (e.g., rent, insurance, etc.), per definition, don't vary when
-demand or price changes. Therefore, fixed costs have no influence on the behavior of dynamic pricing
-algorithms.""")
-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""")
-st.markdown("So can we kick back & relax now?  \nWell, there are a few issues with what we just did.")
-
-# (2) Dynamic demand curves
-st.header("The demands they are a-changin' 🎸")
-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_:
-    contents = file_.read()
-    data_url = base64.b64encode(contents).decode("utf-8")
-
-st.markdown(
-    f'<img src="data:image/gif;base64,{data_url}" alt="dynamic demand">',
-    unsafe_allow_html=True,
-)
-st.markdown("""Now, you may think we can solve this issue by periodically re-estimating the demand 
-curve.  \nAnd you would be very right! But also very wrong as this leads us nicely to the 
-next issue.""")
-
-# (3) Constrained data
-st.header("Where are we getting this data anyways? 🖥️")
-st.markdown("""So far, we have assumed that we get (and keep getting) data on demand levels at
-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 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 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) 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.""")
-st.image("assets/images/realistic_demand_latent_curve.png")
-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? 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**: 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
-actually do what we set out to do: set our price at an optimal level.""")
-
-# (4) Thompson sampling explanation
-st.header("Enter: Thompson Sampling 📊")
-st.markdown("""As we mentioned, this is a well-known problem in statistics. So luckily for us, 
-there is a pretty neat solution in the form of **Thompson sampling**!""")
-st.markdown("""Basically instead of estimating one demand function based on the data available to 
-us, we will estimate a probability distribution of demand functions or simply put, for every 
-possible demand function that fits our functional form (i.e. constant elasticity) 
-we will estimate the probability that it is the correct one, given our data.""")
-st.markdown("""Or mathematically speaking, we will place a prior distribution on the parameters
-that define our demand function and update these priors to posterior distributions via Bayes rule,
-thus obtaining a posterior distribution for our demand function""")
-st.markdown("""Thompson sampling then entails just sampling a demand function out of this 
-distribution, calculating the optimal price given this demand function, observing demand for this
-new price point and using this information to refine our demand function estimates.""")
-st.image("assets/images/flywheel_1.png")
-st.markdown("""So:  \n
-👉 When we are **less certain** of our estimates, we will sample more diverse demand functions, 
-which means that we will also explore more diverse price points. Thus, we will **explore**.  \n
-👉 When we are **more certain** of our estimates, we will sample a demand function close to 
-the real one & set a price close to the optimal price more often. Thus, we will **exploit**.""")
-
-st.markdown("""With that said, we'll take another look at our constrained data and see whether
-Thompson sampling gets us any closer to the optimal price of €4.24""")
-st.image("assets/images/realistic_demand_latent_curve.png")
-st.markdown("""Let's start working our mathemagic:  \n
-We'll start off by placing semi-informed priors on the parameters that make up our 
-demand function.""")
-
-st.latex(f"{sympy.latex(a)} \sim N(μ=0,σ=2)")
-st.latex(f"{sympy.latex(eta)} \sim N(μ=0.5,σ=0.5)")
-st.latex("sd \sim Exp(\lambda=1)")
-st.latex(f"{sympy.latex(D)}|P=p \sim N(μ={sympy.latex(a*p**(-eta))},σ=sd)")
-
-st.markdown("""These priors are semi-informed because we have the prior knowledge that 
-price elasticity is most likely between 0 and 1. As for the other parameters, we have little
-knowledge about them so we can place a pretty uninformative prior.""")
-st.markdown("If that made sense to you, great. If it didn't, don't worry about it")
-
-st.markdown("""Now that are priors are taken care of, we can update these beliefs by incorporating
-the data at the €7.5, €10 and €11 price levels we have available to us.""")
-st.markdown("The resulting demand & profit curve distributions look a little something like this:")
-st.image(["assets/images/posterior_demand.png", "assets/images/posterior_profit.png"])
-
-st.markdown("""It's time to sample one demand curve out of this posterior distribution.  \n
-The lucky curve is:""")
-st.image("assets/images/posterior_demand_sample.png")
-st.markdown("This results in the following expected profit curve")
-st.image("assets/images/posterior_profit_sample.png")
-st.markdown("""And eventually we arrive at a new price: €5.25! Which is indeed considerably closer
-to the actual optimal price of €4.24""")
-st.markdown("Now that we have our first updated price point, why stop there?")
-st.markdown("""With \"pure\" Thompson sampling, we would sample a new demand curve (and thus price
-point) out of the posterior distribution every time. But since we are mainly interested in seeing
-the convergence behavior of Thompson sampling, let's simulate 10 demand points at this fixed €5.25
-price point.""")
-st.image("assets/images/updated_prices_demand.png")
-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.")
-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 €4.04 which is eerily close to
-the actual optimum of €4.24""")
-
-# (5) Extra topics
-st.header("Some things to think about 🤔")
-
-st.markdown("""Because we have purposefully kept the example above quite simple, you may still be
-wondering what happens when added complexities show up.  \n
-Let's discuss some of those concerns FAQ-style:""")
-
-st.subheader("👉 Isn't this constant-elasticity model a bit too simple to work in practice?")
-st.markdown("Brief answer: usually yes it is.")
-st.markdown("""Luckily, more flexible methods exist.  \n
-We would recommend to use Gaussian Processes. We won't go into how these work here but the main idea
-is that it doesn't impose a restrictive functional form onto the demand function but rather lets
-the data speak for itself.""")
-
-with open("assets/images/gaussian_process.gif", "rb") as file_:
-    contents = file_.read()
-    data_url = base64.b64encode(contents).decode("utf-8")
-
-st.markdown(
-    f'<img src="data:image/gif;base64,{data_url}" alt="gaussian process">',
-    unsafe_allow_html=True,
-)
-st.markdown("""If you do want to learn more, we recommend these links: 
-[1](https://distill.pub/2019/visual-exploration-gaussian-processes/),
-[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 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")
-st.markdown("You can tune the other two steps as much as you like too.")
-st.image("assets/images/flywheel_3.png")
-
-st.subheader("👉 Changing prices has a huge impact. How can I mitigate this during experimentation?")
-st.markdown("There are a few things we can do to minimize risk:")
-st.markdown("""👉 **A/B testing**: You can do a gradual roll-out of the new pricing system where a
-small (but increasing) percentage of your transactions are based on this new system. This allows you
-to start small & track/grow the impact over time.""")
-st.markdown("""👉 **Limit products**: Similarly to A/B testing, you can also segment on the 
-product-level. For instance, you can start gradually rolling out dynamic pricing for one product 
-type and extend this over time.""")
-st.markdown("""👉 **Bound price range**: Theoretically, Thompson sampling in its purest form can 
-lead to any arbitrary price point (albeit with an increasingly low probability). In order to limit
-the risk here, you can simply place a upper/lower bound on the price range you are comfortable
-experimenting in.""")
-st.markdown("""On top of all this, Bayesian methods (by design) explicitly quantify uncertainty. 
-This allows you to have a very concrete view on the variance of our demand estimates""")
-
-st.subheader("👉 What if I have multiple products that can cannibalize each other?")
-st.markdown("Here it really depends")
-st.markdown("""👉 **If you have a handful of products**, we can simply reformulate our objective while 
-keeping our methods analogous.  \n
-Instead of tuning one price to optimize profit for the demand function of one product, we tune N 
-prices to optimize profit for the joint demand function of N products. This joint demand function 
-can then account for correlations in demand within products.""")
-st.markdown("""👉 **If you have hundreds, thousands or more products**, we're sure you can imagine that 
-the procedure described above becomes increasingly infeasible.  \n
-A practical alternative is to group substitutable products into "baskets" and define the "price of
-the basket" as the average price of all products in the basket.  \n
-If we assume that the products in baskets are subtitutable but the products in different baskets are
-not, we can optimize basket prices indepedently from one another.  \n
-Finally, if we also assume that cannibalization remains constant if the ratio of prices remains 
-constant, we can calculate individual product prices as a fixed ratio of its basket price.  \n""")
-st.markdown("""For example, if a "burger basket" consists of a hamburger (€1) and a cheeseburger 
-(€3), then the "burger price" is ((€1 + €3) / 2 =) €2. So a hamburger costs 50% of the burger price 
-and a cheeseburger costs 150% of the burger price.  \n
-If we change the burger's price to €3, a hamburger will cost (50% * €3 =) €1.5 and a cheeseburger
-will cost (150% * €3 =) €4.5 because we assume that the cannibalization effect between hamburgers & 
-cheeseburgers is the same when hamburgers cost €1 & cheeseburgers cost €3 and when hamburgers cost
-€1.5 & cheeseburgers cost €4.5""")
-st.image("assets/images/cannibalization.png")
-
-st.subheader("👉 Is dynamic pricing even relevant for slow-selling products?")
-st.markdown("""The boring answer is that it depends. It depends on how dynamic the market is, the
-quality of the prior information, ...""")
-st.markdown("""But obviously this isn't very helpful.  \nIn general, we notice that you can already 
-get quite far with limited data, especially if you have an accurate prior belief on how the demand
-likely behaves.""")
-st.markdown("""For reference, in our simple example where we showed a Thompson sampling update, we 
-were already able to gain a lot of confidence in our estimates with just 10 extra demand 
-observations.""")
-
-# (6) 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
+st.markdown("""In this demo you will see  \n
+👉 How Bayesian demand function estimates are created based on sales data  \n
+👉 How Thompson sampling will generate concrete price points from these Bayesian estimates  \n
+👉 The impact of price elasticity on Bayesian demand estimation""")
+st.markdown("""You will notice:  \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
@@ -293,6 +37,8 @@ increasingly larger and increasingly more variable (as their variance is a const
 absolute value). This causes our demand posterior to become increasingly wider and thus Thompson
 sampling will lead to more exploration.
 """)
+st.markdown("""If you are looking for more insights into how dynamic pricing is done in practice,
+check out our blog post here: https://medium.com/ml6team/dynamic-pricing-in-practice-99fe2216a93d""")
 
 thompson_sampler = ThompsonSampler()
 demo_button = st.checkbox(