Patrick Ding, Guido Imbens, Zhaonan Qu, Yinyu Ye
arXiv 12 Jul 2024 · Statistics — Methodology · 1 citations (OpenAlex)
arXiv:2407.09371 · PDF · DOI · OpenAlex · Extracted main text
Probit models are useful for modeling correlated discrete responses in many disciplines, including consumer choice data in economics and marketing. However, the Gaussian latent variable feature of probit models coupled with identification constraints pose significant computational challenges for its estimation and inference, especially when the dimension of the discrete response variable is large. In this paper, we propose a computationally efficient Expectation-Maximization (EM) algorithm for estimating large probit models. Our work is distinct from existing methods in two important aspects. First, instead of simulation or sampling methods, we apply and customize expectation propagation (EP), a deterministic method originally proposed for approximate Bayesian inference, to estimate moments of the truncated multivariate normal (TMVN) in the E (expectation) step. Second, we take advantage of a symmetric identification condition to transform the constrained optimization problem in the M (maximization) step into a one-dimensional problem, which is solved efficiently using Newton's method instead of off-the-shelf solvers. Our method enables the analysis of correlated choice data in the presence of more than 100 alternatives, which is a reasonable size in modern applications, such as online shopping and booking platforms, but has been difficult in practice with probit models. We apply our probit estimation method to study ordering effects in hotel search results on Expedia's online booking platform.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Burgette, L. F. and Nordheim, E. V (2012) The trace restriction: An alternative identification strategy for the bayesian multinomial probit model | 0.811 | 4 | 2 | 100% |
| bien_sparse_2011 | unmatched citation key bien_sparse_2011 | 0.737 | 3 | 2 | 100% |
| 3 | Cunningham, J. P., Hennig, P., and Lacoste-Julien, S (2013) Gaussian Probabilities and Expectation Propagation | 0.693 | 6 | 1 | 100% |
| 4 | Natarajan, R., McCulloch, C. E., and Kiefer, N. M (2000) A monte carlo em method for estimating multinomial probit models | 0.644 | 2 | 2 | 100% |
| 5 | Burgette, L. F., Puelz, D., and Hahn, P. R (2021) A symmetric prior for multinomial probit models | 0.585 | 3 | 1 | 100% |
| minkaExpectationPropagationApproximate2001a | unmatched citation key minkaExpectationPropagationApproximate2001a | 0.585 | 3 | 1 | 100% |
| 7 | Minka, T (2005) Divergence measures and message passing | 0.511 | 2 | 1 | 100% |
| bickel_covariance_2008 | unmatched citation key bickel_covariance_2008 | 0.405 | 1 | 1 | 100% |
| 9 | Blei, D. M., Kucukelbir, A., and McAuliffe, J. D (2017) Variational Inference: A Review for Statisticians | 0.405 | 1 | 1 | 100% |
| friedman2008sparse | unmatched citation key friedman2008sparse | 0.405 | 1 | 1 | 100% |
Showing the top 10 of 17 scored citations. 4 of these could not be matched to a bibliography entry, so only the citation key is shown.
arXiv econ.EM papers that cite this one, ranked by how heavily they lean on it.
| Citing paper | Intensity | Mentions | Sections | |
|---|---|---|---|---|
| 1 | Learning Correlated Reward Models: Statistical Barriers and Opportunities | 0.000 | 1 | 1 |