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Interdependences of Products in Market Baskets: Comparing the Conditional Restricted Boltzmann Machine to the Multivariate Logit Model

Harald Hruschka ORCID logo EMAIL logo

Abstract

We analyze market baskets of individual households in two consumer durables categories (music, computer related products) by the multivariate logit (MVL) model, its finite mixture extension (FM-MVL) and the conditional restricted Boltzmann machine (CRBM). The CRBM attains a vastly better out-of-sample performance than MVL and FM-MVL models. Based on simulation-based likelihood ratio tests we prefer the CRBM to the FM-MVL model. To interpret hidden variables of conditional Boltzmann machines we look at their average probability differences between purchase and non-purchases of any sub-category across all baskets. To measure interdependences we compute cross effects between sub-categories for the best performing FM-MVL model and CRBM. In both product categories the CRBM indicates more or higher positive cross effects than the FM-MVL model. Finally, we suggest appropriate future research based on larger and more detailed data sets.


Corresponding author: Harald Hruschka, University of Regensburg Faculty of Business Economics and Management Information Systems, Regensburg, 93053, Germany, E-mail:

Appendices

A Conditional Probabilities of the Investigated Models

For the MVL model the conditional probability of a purchase of product j can be written as:

(4) P ( y j = 1 ) = 1 1 + exp ( ( α j + x T β . j + y T V . j ) )

For the FM-MVL model with S segments we obtain the following expression for the conditional probability of a purchase of product j:

(5) P ( y j = 1 ) = s = 1 S π s 1 1 + exp ( ( α j s + x T β s . j + y T V s . j ) )

Parameters of this model are segment-specific. π s denotes the posterior probability of belonging to segment s.

For the CRBM we obtain the following expressions for the conditional probabilities of purchases given hidden variables and for hidden variables given purchases (Li et al. 2015):

(6) P ( y j = 1 | h ) = 1 1 + exp ( ( α j + β . j x + k = 1 K W j k h k ) )

(7) P ( h k = 1 | y ) = 1 1 + exp ( ( γ k + j = 1 J W j k y j ) )

y j denotes the binary purchase indicator for product j, h k the binary kth hidden variable.

B Estimation of the MVL Model

Maximum likelihood estimation of the MVL model requires computation of the so-called normalization constant in every iteration that is obtained by summing over 2 J possible market baskets. Only when expression (1) is divided by the normalization constant a proper probability results. For 30 products we would have to deal with more than 1.0 × 109 possible market baskets. Because of the impracticality of this approach we resort to maximum pseudo-likelihood (MPL) estimation. In a simulation study Bel et al. (2018) compare MPL to maximum likelihood estimation for a maximum number of 12 alternatives. These authors conclude that MPL estimation leads to negligible efficiency losses only.

The pseudo-probability P ˜ j for product j is defined as probability of y j conditional on the observed basket y j , i.e., basket y without product j:

(8) P ˜ j P ( y | y j ) P ( y | y j ) + P ( y ˜ | y j )

Basket y ˜ corresponds to the observed basket y except for product j, whose purchase indicator is flipped, i.e.,  y ˜ j = 1 y j .

MPL estimation is feasible, because the normalization constant drops out in expression (8). Moreover, it is straightforward as the pseudo-likelihood function has only one local maximum. For the MVL model the pseudo-probability P j for product j in basket y is given by (Besag 1972, 1974):

(9) P ˜ j = exp ( y j ( α j + x T β . j + y T V . j ) ) 1 + exp ( α j + x T β . j + y T V . j )

The log pseudo-likelihood LPL of basket y is obtained by summing the logs of pseudo-probabilities across all products

(10) L P L = j = 1 J log ( P ˜ j )

C Estimation of the FM-MVL Model

We assign households to mixture components (i.e., segments) by the Gibbs sampling approach of Shi et al. (2005) replacing the intractable log likelihood value of a basket by its log pseudo likelihood value like in Dippold and Hruschka (2013a) as part of the estimation process. In each iteration, one MVL model specific to the households currently assigned to a segment is estimated by MPL. We start from 100 initial random allocations of households to segments, as the FM-MVL model may be subject to local optima. We choose the solution leading to the best log likelihood value for the estimation sample determined by the Gibbs sampling procedure explained in section 2.3.

D Estimation of the RBM and CRBM

We estimate the RBM and the CRBM by the contrastive divergence (CD) algorithm of Hinton (2002) which approximates the log likelihood. For the CRBM we extend the CD algorithm by adding gradients for the coefficients in β and δ k . Because of the existence of local optima we start the CD algorithm 100 times with random initial coefficient values. Just like for the FM-MVL model we choose the solution of a RBM or CRBM attaining the best log likelihood value for the estimation data using the Gibbs sampling procedure explained in section 2.3.

E Simulation-Based Computation of the Likelihood Ratio Test

The likelihood ratio statistic LRT with LL 1 and LL 0 as log likelihood values for the alternative and the null model can be written as:

(11) L R T = 2 ( L L 1 L L 0 )

The simulation-based approach for the LRT (Lewis et al. 2011) consists of three steps:

  1. Generate S bootstrap samples from the null model.

  2. For each bootstrap sample fit both the null model and the alternative model, determine the likelihood values of these models by the Gibbs sampling procedure explained in section 2.3 and compute the LRT statistic.

  3. The null model is rejected if the proportion of the test statistics for the bootstrap samples which are greater than the test statistic for the estimation data exceeds a prespecified significance level.

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Received: 2020-10-08
Accepted: 2020-11-07
Published Online: 2020-11-23

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