A Prior Distribution of Bayesian Nonparametrics Incorporating Multiple Distances
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1 A Prior Distribution of Bayesian Nonparametrics Incorporating Multiple Distances Brian M. Hartman 1 David B. Dahl 1 Debabrata Talukdar 2 Bani K. Mallick 1 1 Department of Statistics Texas A&M University 2 School of Management SUNY - Buffalo 03 August 2010
2 Introduction Global marketing managers are deeply concerned with predicting how well their products will sell. As the world has become more global, it is important to use relevant information from other countries when predicting diffusion. We examine: how to measure the distance between two countries how to best cluster the countries into groups to improve the predictions. with a focus on improving the sales predictions.
3 Sampling Model The Bass Diffusion Model (Bass, 1969) is commonly used in the new product diffusion literature. We begin with the hierarchical setup first presented in Talukdar (2002): [ y in (t) = [α in M i (t) Y in (t 1)] αin pin qin α i p i q i p in + q in Y in (t 1) α in M i (t) α i + π αn X T = pi αin θ α + π pn + X T qi pin θ p + + π qn X T qin θ q ( X T β ) αi ( = X T β ) π αi ( pi X T β ) + π pi π qi qi ] exp [ɛ in (t)] π αin π pin π qin
4 Model Possibilities Option 1: Standard β i = β β N(µ 0, Σ 0 ) Option 2: Overparameterized β i β j β i N(µ 0, Σ 0 ) Option 3: Clustering β i G G G DP(α 0, G 0 ) G 0 = N(θ 0, Σ 0 ) p(β i = β j ) = p(β i = β j ) = α 0
5 Distance Justification Our method utilizes the following distance information about the countries to inform the prior clustering probabilities. Centroid Distance Bilateral Tourism Flow Bilateral Trade Flow Cultural Similarity (Hofstede indices)
6 Incorporating a Single Distance The affinity distribution (Dahl et al., 2008) rewards partitions which are formed by items which are close in terms of the distances d ij. They define the affinity of items i and j as: a ij = (1 + ɛ)d d ij (1 + ɛ)d where d is the maximum distance among all pairs in the dataset. The kernel of the p.m.f. for π in the affinity distribution is: p(π) S π exp(t (g(s) g(s 0 ))) [αγ( S )] 1+t (g(s) g(s 0))
7 Incorporating Multiple Distances The distance between country i and country j is then defined by the norm of a subset of the four distances: d(i, j) = K γ k d k (i, j) 2 k=1 where γ k is a binary variable defining if the k th covariate is included in the model.
8 Prior Specification For the joint prior of (β i, γ i ), we use: p(β i, γ i ) = Ng(β i, γ i 0, I k, ν, θ) = N(β i 0, γ i I k )Ga(γ i ν, θ) To obtain a draw from the prior distribution of β i, we integrate out the γ i value and obtain: ( 0, ν ) p(β i ) = St β i θ I k, 2ν
9 Posterior Distribution The posterior distribution for the β i values is also available with the γ i integrated out. θ n = (I k + X T X ) 1 X T y λ n = θ (y X θ n) T y p(β i X, y) = St ( β i θn, (I k + X T X ) ( ν + n ) 2 ) λ 1 n, 2ν + n
10 Prediction Comparison Centroid Cultural Tourism Trade Improvement P(Better than DP) % (1.000) % (0.988) % (0.993) % (0.995) % (0.975) % (0.978) % (0.920) % (0.895) % (0.947) % (0.960) % (0.969) % (0.917) % (0.892) % (0.880) % (0.741)
11 Posterior Clustering (Cultural and Tourism) Cluster 1 Cluster 2 Cluster 3 Cluster 4 Argentina Australia Chile Denmark Brazil Austria Portugal Finland China Belgium South Korea Norway Greece Canada Thailand Sweden Hong Kong France India Germany Malaysia Ireland Mexico Italy Philippines Singapore Netherlands Spain Switzerland United Kingdom United States
12 Conclusion Global marketing managers are keenly interested in being able to predict the sales of their new products. Our work improves the understanding of how a product is adopted allowing the managers to optimally allocate their resources by exploring how to leverage cross-country information. In addition to augmenting a current marketing model, we propose a new method for incorporating multiple distances into a nonparametric prior distribution.
13 A Prior Distribution of Bayesian Nonparametrics Incorporating Multiple Distances Brian M. Hartman 1 David B. Dahl 1 Debabrata Talukdar 2 Bani K. Mallick 1 1 Department of Statistics Texas A&M University 2 School of Management SUNY - Buffalo 03 August 2010
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