Chapter 15 Stochastic Choice The sequence of coverage

Chapter 15: Stochastic Choice The sequence of coverage is: q q q q Key Terminology The Brand Switching Matrix Zero-Order Bernoulli Model Population Heterogeneity Markov Chains Learning Models (Not Covered) Purchase Incidence Negative Binomial Model This chapter follows the development in Lilien, Gary L. and Philip Kotler (1983) Marketing Decision Making. New York: Harper and Row. Mathematical Marketing Slide 15. 1 Stochastic Choice

Key Chapter Terminology q Consumer Panel Data q Stationarity q Purchase Incidence Data q Brand Switching Data Mathematical Marketing Slide 15. 2 Stochastic Choice

A Typical Record in Consumer Panel Data q q q Household ID Date and Time Household Member Buying Household Member Using Product Purchased q q q Package Size Price Promotion Information Local Media Feature Where Bought Collected via diaries, scanners in the home, or all local stores have scanners Equivalent data for online behavior are called clickstream data Mathematical Marketing Slide 15. 3 Stochastic Choice

Definition of Stationarity For a parameter we have t = t´ for all t, t´ = 1, 2, …, T. Mathematical Marketing Slide 15. 4 Stochastic Choice

Purchase Incidence Data r 0 f 0 1 f 1 2 f 2 ··· T f. T Total Mathematical Marketing Number of Households n Slide 15. 5 Stochastic Choice

Brand Switching Data Purchase Occasion Two A 10 5 10 25 B 8 12 5 25 C Mathematical Marketing C A Purchase Occasion One B 10 10 30 50 Slide 15. 6 Stochastic Choice

Three Kinds of Probabilities What are the differences between the following types of probabilities? q Joint Probability q Marginal Probability q Conditional Probability A A 1 2 B Mathematical Marketing B 3 4 Slide 15. 7 Stochastic Choice

Three Kinds of Probabilities What are the differences between the following types of probabilities? q Joint Probability – Pr(A 1 and B 2) = 2/10 q Marginal Probability – Pr(A 1) = (1+2)/10 q Conditional Probability – Pr(A 2 | A 1) = 1/3 A A 1 2 B Mathematical Marketing B 3 4 Slide 15. 8 Stochastic Choice

Notation for the Three Kinds of Probabilities q Joint Probability q Marginal Probability q Conditional Probability Mathematical Marketing Slide 15. 9 Stochastic Choice

Bayes Theorem Pr(A, B) = Pr(B | A) · Pr(A) = Pr(A | B) · Pr(B) Mathematical Marketing Slide 15. 10 Stochastic Choice

Bayesian Terminology Conditional Probability or Likelihood Prior Probability Posterior Probability Normalizing Constant Mathematical Marketing Slide 15. 11 Stochastic Choice

Combinations (Order Does Not Matter) The number of combinations of T things taken r at a time is given by this expression What is T!? Mathematical Marketing Alternative notation - Slide 15. 12 Stochastic Choice

The Zero-Order Property Pr(A, B, A, A, B, ···) = p · (1 - p) · p (1 - p) · ··· So overall, r purchases of A out of T occasions would be Mathematical Marketing Slide 15. 13 Stochastic Choice

The Zero-Order Property How many ways are there of “r out T” happening? Mathematical Marketing What is the probability of any one of them happening? Slide 15. 14 Stochastic Choice

Zero-Order Homogeneous Bernoulli Model Joint Probabilities Occasion Two A Occasion One Mathematical Marketing B A p 2 p (1 - p) B (1 - p) p (1 - p)2 Slide 15. 15 Stochastic Choice

Zero-Order Homogeneous Bernoulli Model Probabilities Conditional on Occasion 1 Occasion Two A Occasion One Mathematical Marketing B A p (1 - p) B p (1 – p) Slide 15. 16 Stochastic Choice

Population Heterogeneity q p itself is a random variable that differs from household to household q We assume p is distributed according to the Beta distribution, which acts as a mixing distribution q We call this the prior distribution of p Pr(p) = c 1 p -1(1 - p) -1 Mathematical Marketing Slide 15. 17 Stochastic Choice

Likelihood or Conditional Probability of r Purchases out of T Occasions Pr(r, T | p) = c 2 pr (1 - p)T- r with c 2 = Mathematical Marketing Slide 15. 18 Stochastic Choice

Invoking Bayes Theorem Mathematical Marketing Slide 15. 19 Stochastic Choice

Posterior Probabilities The Posterior The Likelihood The Prior The posterior probabilities look like a beta distribution that depends on r and T: * = + r and * = + T - r. Mathematical Marketing Slide 15. 20 Stochastic Choice

Touching Data We assert without proof that So for example, for r = 1 and T = 3: Mathematical Marketing Slide 15. 21 Stochastic Choice

Testing the Model So for each of the triples AAA, AAB, ABA, ABB, BAA, BAB, BBA, BBB, we can use Minimum Pearson Chi Square and use As the objective function Mathematical Marketing Slide 15. 22 Stochastic Choice

Markov Models q Single Period Memory (vs. Bernoulli model with zero memory) q Stationarity q Characterized by a • Transition Matrix and an • Initial State Vector Mathematical Marketing Slide 15. 23 Stochastic Choice

Single Period Memory in Markov Chains Define yt as the brand chosen on occasion t. With Markov Chains we have Pr(yt = j | yt-1, yt-2, ···, y 0) = Pr(yt = j | yt-1). Mathematical Marketing Slide 15. 24 Stochastic Choice

Stationarity in Markov Chains Pr(yt = j | yt-1 ) = Pr(yt = j | yt -1) for all t, t Mathematical Marketing Slide 15. 25 Stochastic Choice

Transition Matrix Occasion t + 1 A Occasion t A B B . 7. 5 . 3. 5 The elements of the transition matrix are the Pr(k | j) such that Mathematical Marketing Slide 15. 26 Stochastic Choice

Initial State Vector m(0) is a J by 1 vector of shares at “time 0”. A typical element provides the share for brand j, Mathematical Marketing Slide 15. 27 Stochastic Choice

The Law of Total Probability and Discrete Variables An additive law that loops through all the ways that an event (like A) could happen Mathematical Marketing Slide 15. 28 Stochastic Choice

Law Applied to Market Shares of brand 1 at time (1) Pr(Buy k given a previous purchase of 2) Pr(Previous Purchase of 2 at time 0) Mathematical Marketing Slide 15. 29 Stochastic Choice

Summation Notation and Matrix Notation for Law of Total Probability Mathematical Marketing Slide 15. 30 Stochastic Choice

Two Markov Models before We Seek Variety Zero –order homogeneous Bernoulli Mathematical Marketing Superior-Inferior Brand Model Slide 15. 31 Stochastic Choice

Variety Seeking Model 1. What values go in the cells of the above transition matrix that are marked with the “-”? 2. What does the model predict for Pr(AAA)? 3. How could we estimate the model from the 8 triples, AAA, AAB, ABA, …, BBB? Mathematical Marketing Slide 15. 32 Stochastic Choice

Purchase Incidence Data The goal is to predict or explain the number of households who will purchase our brand r times Or the number of Web surfers who will visit our site r times or purchase at our site r times Or in general the number of population members who will exhibit a discrete behavior r times Mathematical Marketing Slide 15. 33 Stochastic Choice

Purchase Incidence Data r 0 f 0 1 f 1 2 f 2 ··· T f. T Total Mathematical Marketing Number of Households n Slide 15. 34 Stochastic Choice

Straw Man Model – Binomial We collect the panel data for T weeks and assume one purchase opportunity per week The r+1 st term from expanding (q + p)T where q = 1 - p How would you test this model? Mathematical Marketing Slide 15. 35 Stochastic Choice

Straw Man Model - Poisson We let T p 0 Mathematical Marketing But hold Tp = Slide 15. 36 Stochastic Choice

Poisson Prediction Equation Mathematical Marketing Slide 15. 37 Stochastic Choice

Negative Binomial Distribution Named after the terms in the expansion of (q - p)-r Can arise from q A binomial where the number of tosses is itself random q A Poisson where changes over time due to contagion q A Poisson where varies across households with the gamma distribution Mathematical Marketing Slide 15. 38 Stochastic Choice

NBD Prediction Equation Where the function (not gamma distribution) is defined as (q) acts like a factorial function for non-integers, i. e. if q is an integer then (q) = (q + 1)! Mathematical Marketing Slide 15. 39 Stochastic Choice