Bayesian Optimal Pricing

Pricing is a common problem faced by businesses, and one that can be addressed effectively by Bayesian statistical methods. We'll step through a simple example and build the background necessary to extend get involved with this approach.

Transcript

1. Bayesian Optimal Pricing Chad Scherrer Senior Data Scientist, Metis June 5, 2018
2. Optimal Pricing • Given historical sales data… • Model demand across prices • Use this to set future prices
3. Disclaimer • Real life: • Lots of products • Competitors • Income fluctuations • Consumer psychology • This is a toy example on fake data • I am not an economist!!
4. Model Setup • An economist may write • Statistically, it’s more like this: • Or, in the usual GLM form: Q = aPc <latexit sha1_base64="OZogS+coDW04A+FlKHrgGwk2c9g=">AAAB7nicbVDLSgNBEOyNrxhfUY96GAyCp7DrRS9C0IvHBMwDkjXMTmaTIbOzy0yvEJZ8hBcPinj1E/wOb978FCePgyYWNBRV3XR3BYkUBl33y8mtrK6tb+Q3C1vbO7t7xf2DholTzXidxTLWrYAaLoXidRQoeSvRnEaB5M1geDPxmw9cGxGrOxwl3I9oX4lQMIpWatauKKnes26x5JbdKcgy8eakVDn+qH0DQLVb/Oz0YpZGXCGT1Ji25yboZ1SjYJKPC53U8ISyIe3ztqWKRtz42fTcMTm1So+EsbalkEzV3xMZjYwZRYHtjCgOzKI3Ef/z2imGl34mVJIiV2y2KEwlwZhMfic9oTlDObKEMi3srYQNqKYMbUIFG4K3+PIyaZyXPbfs1Wwa1zBDHo7gBM7AgwuowC1UoQ4MhvAIz/DiJM6T8+q8zVpzznzmEP7Aef8BWz6RIQ==</latexit> <latexit 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5. Inference with PyMC3 • Build the model: • Sample: • Default is No-U-Turn Sampler (“NUTS”)
6. Check results We took 1000 samples - what’s going wrong?
7. Source of the Problem
8. Second Attempt: Centering log E[Q|P] = ↵ + (log P log P0) <latexit sha1_base64="ku00EG7pfDNqpNFGlzYzTyx1V6g=">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</latexit> <latexit sha1_base64="FYm/MXlIKfGsrxTn8jYs+BgeoX0=">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</latexit> <latexit sha1_base64="FYm/MXlIKfGsrxTn8jYs+BgeoX0=">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</latexit> <latexit sha1_base64="XQYJgXa0pO2MqG+ZyuTI13CjiAc=">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</latexit>
9. Much Better!
10. Improved Posterior Correlation
11. Trace plots
12. Forest Plots
13. Expected Sales
14. Bayesian Model Diagnostics: Posterior Predictive Checks • Sample “replicated” observed data for each posterior sample • How do these compare to original data? • Example: Bayesian p-values • Should be uniform for out-of-sample data • For more detail, evaluate each cdf (next slide)
15. CDF Posterior Predictive Check
16. Initial Solution
17. Sensitivity Analysis • How confident are we in this result? • PPC doesn’t apply here • Maybe we can just bootstrap? • Unfortunately, it’s not so simple! Bootstrap Results
18. Bootstrapping the Mean of a Cauchy
19. Building Replications • Now we need a way to “map” over replications…
20. Using Replications
21. Replicated Price Optimization
22. Replicated Posterior Parameters
23. Analysis • Start with • Set and solve: • Diverges as ! ⇡ = (P K)µ log µ = ↵ + (log P log P0) <latexit sha1_base64="d1e7wc6f45UKUNtBZNc6HkSSQTg=">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</latexit> <latexit 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24. An Alternate Approach • We now have a way to optimize when parameters are given • Let’s do this for each posterior sample, and aggregate • Heavy tail causes instability of mean • Median is scale-invariant • So…
25. Much Better!
26. Final Thoughts • “Max-of-means” makes intuitive sense… • But intuition can be wrong! • Heavy tails can have dramatic effect on estimation of means • Always check results with PPC or replications!
27. Thank you! Check out my 2-part blog series: https://cscherrer.github.io/post/max-profit/