Ellipsoidal Methods for Adaptive Choice-based Conjoint Analysis (CBCA)

Transcription

1 Ellipsoidal Methods for Adaptive Choice-based Conjoint Analysis (CBCA) Juan Pablo Vielma Massachusetts Institute of Technology Joint work with Denis Saure Operations Management Seminar, Rotman School of Management, Toronto, Canada. December, 2016.

2 Motivation: (Custom) Product Recommendations Feature SX530 RX100 Zoom 50x 3.6x Weight ounces 7.5 ounces Feature TG-4 G9 Waterproof Yes No Weight 7.36 lb 7.5 lb We recommend: Feature TG-4 Galaxy 2 Waterproof Yes No Viewfinder Electronic Optical Ellipsoidal Methods for Adaptive CBCA 1 / 27

3 Towards CBCA-Based Recommendations Individual preference estimates with few questions Adaptive Questions: Fast question selection Pick next question to reduce uncertainty Quantify estimate variance Favorable properties for future: Intuitive geometric model (e.g. Robust Opt.) Parametric model Feature SX530 RX100 Zoom 50x 3.6x Weight ounces 7.5 ounces Feature TG-4 Galaxy 2 Waterproof Yes No Viewfinder Electronic Optical Feature TG-4 G9 Waterproof Yes No Weight 7.36 lb 7.5 lb We recommend: Ellipsoidal Methods for Adaptive CBCA 2 / 26

4 Choice-based Conjoint Analysis Feature Chewbacca BB-8 Wookiee Yes No Droid No Yes Blaster Yes No 0 1 1A = x 2 0 I would buy toy Product Profile x 1 x 2 Ellipsoidal Methods for Adaptive CBCA 3 / 26

5 MNL Preference Model Utilities for 2 products, d features U 1 = x = X d i=1 i x 1 i + 1 U 2 = x = X d i=1 i x 2 i + 2 part-worths product profile noise (gumbel) Utility maximizing customer: x 1 x 2, U 1 U 2 Noise can result in response error: P x 1 x 2 = e x1 e x1 + e x2 Ellipsoidal Methods for Adaptive CBCA 4 / 26

6 Next Question To Reduce Variance : Bayesian Prior Distribution of Feature SX530 RX100 Zoom 50x 3.6x Weight ounces 7.5 ounces Bayesian Update Posterior Distribution Feature TG-4 Galaxy 2 Waterproof Yes No MCMC Bayesian Update Posterior Distribution Viewfinder Electronic Optical Update uses MNL response error Question Selection (even knowing answer): Enumeration Ellipsoidal Methods for Adaptive CBCA 5 / 26

7 Next Question To Reduce Variance : Polyhedral Toubia, Hauser and Simester, 04 Polyhedron Containing Feature SX530 RX100 Zoom 50x 3.6x Weight ounces 7.5 ounces Geometric Update Posterior Polyhedron Feature TG-4 Galaxy 2 Waterproof Yes No Viewfinder Electronic Optical Geometric Update Posterior Polyhedron Update ignores response error Question Selection: (Multi-Obj.) Discrete Optimization Ellipsoidal Methods for Adaptive CBCA 6 / 26

8 Outline Objective: Combine Bayesian and polyhedral methods into intuitive geometric approach 1. Review of geometry of polyhedral method 2. Incorporating response error = ellipsoids 3. MIP based near-optimal question selection to reduce variance measure (D-efficiency) 4. Optimal one-step look-ahead moment-matching approximate Bayesian approach (OOLMMABA?) Ellipsoidal Method Ellipsoidal Methods for Adaptive CBCA 7 / 26

9 Polyhedral Method

10 Preference Model and Geometric Interpretation Utilities for 2 products, d features, logit model U 1 = U 2 = x = X d i=1 i x 1 i + 1 x = X d i=1 i x 2 i + 2 part-worths product profile noise (gumbel) Utility maximizing customer Geometric interpretation of preference for product 1 without error 2 x 1 x 2 0 x 1 x 2, U 1 U 2 Ellipsoidal Methods for Adaptive CBCA 9 / 26 1

11 Polyhedral Method: Ask Question and Update Geometric prior for 2 1st geometric x 15 3 / x 46 x 2 2 3rd 2ndgeometric posterior for =? x 1 x 2 0? x 5 x 6 0 x 3 x 4 0 Ellipsoidal Methods for Adaptive CBCA 10 / 26 1

12 Polyhedral: Estimation and Question Selection Good estimator Question? for? Center Postchoice Choice of ellipsoid balance symmetry Ellipsoidal Methods for Adaptive CBCA 11 / 26 1

13 Incorporating Response Error

14 Distributions and Credibility Ellipsoids Prior distribution of 90% confidence/credibility ellipsoid N(µ, ) ( µ) 0 1 ( µ) apple r Ellipsoidal Methods for Adaptive CBCA 13 / 26

15 Answers with Error: Logit Probabilities Likelihood Function Question/Answer P x 1 x 2 = e x1 e x1 + e x2 x 1 x 2 Ellipsoidal Methods for Adaptive CBCA 14 / 26

16 Bayesian Update and Geometric Updates Prior distribution Answer likelihood Posterior distribution Prior ellipsoid Question/Answer Posterior ellipsoid Ellipsoidal Methods for Adaptive CBCA 15 / 26

17 Computational Comparison of Updates Gaussian prior and 90% credibility ellipsoid, 100 inst. 12 features, 2 profiles and 5 questions Sample 1 true β and simulate MNL responses with it Polyhedral Ellipsoidal Feasible β Distance (scaled) Gaussian Volume ( ˆi 0 k k 2 0 k 0 k 2 2 Ellipsoidal Methods for Adaptive CBCA 16 / 26

18 Question Selection: Optimizing D-Efficiency

19 D-Efficiency and Posterior Covariance Matrix x 1 x 2 approx. N µ 1, 1 D-Efficiency: p =2 proportional to expected volume of posterior ellipsoid f x 1,x 2 := E,x 1 / x 2 det( i ) 1/p N(µ, ) x 1 x 2 approx. N µ 2, 2 Evaluating = multi-dim integration Ellipsoidal Methods for Adaptive CBCA 18 / 26

20 Back to Question Selection: Property Trade-off ( µ) 0 1 ( µ) apple r Choice balance: Minimize distance to center µ x 1 x 2 µ Postchoice symmetry: Maximize variance of question x 1 x 2 0 x 1 x 2 Ellipsoidal Methods for Adaptive CBCA 19 / 26

21 D-efficiency Simplification for CBCA D-efficiency = Non-convex function distance: d := µ x 1 x 2 f(d, v) variance: v := x 1 x 2 0 x 1 x 2 of Can evaluate f(d, v) with 1-dim integral Ellipsoidal Methods for Adaptive CBCA 20 / 27

22 Optimization Model min f(d, v) s.t. µ x 1 x 2 = d x 1 x 2 0 x 1 x 2 = v A 1 x 1 + A 2 x 2 apple b linearize x k i xl j x 1 6= x 2 x 1,x 2 2 {0, 1} n Ellipsoidal Methods for Adaptive CBCA 21 / 27

23 Piecewise Linear Approximation D-efficiency = Non-convex function distance: d := µ x 1 x 2 variance: v := x 1 x 2 0 x 1 x 2 of f(d, v) Can evaluate f(d, v) with 1-dim integral Piecewise Linear Interpolation MIP formulation Ellipsoidal Methods for Adaptive CBCA 22 / 27

24 Putting Everything Together: Ellipsoidal Method

25 MIP-based Adaptive Questionnaires Feature SX530 RX100 Zoom 50x 3.6x Weight ounces 7.5 ounces Feature TG-4 G9 Waterproof Yes No Weight 7.36 lb 7.5 lb We recommend: Feature TG-4 Galaxy 2 Waterproof Yes No Viewfinder Electronic Optical E Y,X 1,X 2 cov Y,X 1,X 2 Optimal one-step look-ahead moment-matching approximate Bayesian approach. Ellipsoidal Methods for Adaptive CBCA 24 / 27

26 Optimal One-Step Look-Ahead Prior distribution Feature SX530 RX100 Zoom 50x 3.6x Weight ounces 7.5 ounces min f x 1,x x1,x 2 2 Feature TG-4 Galaxy 2 Waterproof Yes No Viewfinder Electronic Optical N µ i, i Feature TG-4 Galaxy G9 2 Waterproof Yes No Viewfinder Weight Electronic 7.36 lb Optical 7.5 lb Solve with MIP formulation Ellipsoidal Methods for Adaptive CBCA 25 / 27

27 Moment-Matching Approximate Bayesian Update Answer likelihood Prior distribution Posterior distribution N µ i, i Feature TG-4 Galaxy 2 Waterproof Yes No Viewfinder Electronic Optical approx. N µ i+1, i+1 µ i+1 = E y, x 1,x 2 i+1 =cov y, x 1,x 2 1-dim integral Ellipsoidal Methods for Adaptive CBCA 26 / 27

28 Computational Experiments 16 questions, 2 options, 12 features Simulate MNL responses with known Question Selection MIP-based using CPLEX and open source COIN-OR solver Knapsack-based geometric Heuristic by Toubia et al. Time limits of 1 s Metrics: Estimator variance = det cov Y,X 1,X 2 1/2 Estimator distance = E Y,X 1,X 2 2 Computed for true posterior with MCMC Ellipsoidal Methods for Adaptive CBCA 27 / 27

29 Results for 12 Features, 1 s time limit Estimator Variance Estimator Distance Heuristic (Avg. = 0.04 s, Max = 0.61s) COIN-OR (Avg. = 0.93 s, Max = 1s) CPLEX (Avg. = 0.21 s, Max = 0.48s) Ellipsoidal Methods for Adaptive CBCA 28 / 27

30 Summary Messages: Always choose Chewbacca! Polyhedral Geometric Bayesian Question selection and update with optimization and limited sampling (1-dim integrals) Point estimation and credibility region Improvements in point estimation, reduction of uncertainty and precision of credibility region Also works for more profiles, attribute levels, etc. Future: Combination and comparison with fully Bayesian Combine with polyhedral updates Computational improvements Field experiments Ellipsoidal Methods for Adaptive CBCA 29 / 26