Data augmentation,frequentist estimation,and the Bayesian analysis of multinomial logit models |
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Authors: | Steven L Scott |
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Institution: | (1) Department of Psychiatry, University of California, San Francisco, 330 Laurel Way, Mill Valley, California 94941-4046, USA;(2) Department of Neuropsychiatry, University of South Carolina, Columbia, South Carolina, USA;(3) Tri-Valley Community Foundation, Pleasanton, California, USA;(4) School of Social Work, University of California, Berkeley, California, USA |
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Abstract: | This article describes a convenient method of selecting Metropolis– Hastings proposal distributions for multinomial logit
models. There are two key ideas involved. The first is that multinomial logit models have a latent variable representation
similar to that exploited by Albert and Chib (J Am Stat Assoc 88:669–679, 1993) for probit regression. Augmenting the latent
variables replaces the multinomial logit likelihood function with the complete data likelihood for a linear model with extreme
value errors. While no conjugate prior is available for this model, a least squares estimate of the parameters is easily obtained.
The asymptotic sampling distribution of the least squares estimate is Gaussian with known variance. The second key idea in
this paper is to generate a Metropolis–Hastings proposal distribution by conditioning on the estimator instead of the full
data set. The resulting sampler has many of the benefits of so-called tailored or approximation Metropolis–Hastings samplers.
However, because the proposal distributions are available in closed form they can be implemented without numerical methods
for exploring the posterior distribution. The algorithm converges geometrically ergodically, its computational burden is minor,
and it requires minimal user input. Improvements to the sampler’s mixing rate are investigated. The algorithm is also applied
to partial credit models describing ordinal item response data from the 1998 National Assessment of Educational Progress.
Its application to hierarchical models and Poisson regression are briefly discussed. |
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Keywords: | |
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