Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Use of Cauchy Prior Distributions for Bayesian Logistic
  Regression   In logistic regression, separation occurs when a linear combination of the
predictors can perfectly classify part or all of the observations in the
sample, and as a result, finite maximum likelihood estimates of the regression
coefficients do not exist. Gelman et al. (2008) recommended independent Cauchy
distributions as default priors for the regression coefficients in logistic
regression, even in the case of separation, and reported posterior modes in
their analyses. As the mean does not exist for the Cauchy prior, a natural
question is whether the posterior means of the regression coefficients exist
under separation. We prove theorems that provide necessary and sufficient
conditions for the existence of posterior means under independent Cauchy priors
for the logit link and a general family of link functions, including the probit
link. We also study the existence of posterior means under multivariate Cauchy
priors. For full Bayesian inference, we develop a Gibbs sampler based on
Polya-Gamma data augmentation to sample from the posterior distribution under
independent Student-t priors including Cauchy priors, and provide a companion R
package in the supplement. We demonstrate empirically that even when the
posterior means of the regression coefficients exist under separation, the
magnitude of the posterior samples for Cauchy priors may be unusually large,
and the corresponding Gibbs sampler shows extremely slow mixing. While
alternative algorithms such as the No-U-Turn Sampler in Stan can greatly
improve mixing, in order to resolve the issue of extremely heavy tailed
posteriors for Cauchy priors under separation, one would need to consider
lighter tailed priors such as normal priors or Student-t priors with degrees of
freedom larger than one.

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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: acc-phys
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: physics
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The categories of this paper are: solv-int
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The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci