Logistic regression: Difference between revisions

Revision as of 18:33, 29 May 2024

Revision as of 02:10, 8 June 2024

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[[File:Logistic-sigmoid-vs-scaled-probit.svg|right|300px|thumb|Comparison of [[logistic function]] with a scaled inverse [[probit function]] (i.e. the [[cumulative distribution function|CDF]] of the [[normal distribution]]), comparing <math>\sigma(x)</math> vs. <math display="inline">\Phi(\sqrt{\frac{\pi}{8}}x)</math>, which makes the slopes the same at the origin. This shows the [[heavy-tailed distribution|heavier tails]] of the logistic distribution.]]

In a [[Bayesian statistics]] context, [[prior distribution]]s are normally placed on the regression coefficients, for example in the form of [[Gaussian distribution]]s. There is no [[conjugate prior]] of the [[likelihood function]] in logistic regression. When Bayesian inference was performed analytically, this made the [[posterior distribution]] difficult to calculate except in very low dimensions. Now, though, automatic software such as [[OpenBUGS]], [[Just another Gibbs sampler|JAGS]], [[PyMC3PyMC]], [[Stan (software)|Stan]] or [[Turing.jl]] allows these posteriors to be computed using simulation, so lack of conjugacy is not a concern. However, when the sample size or the number of parameters is large, full Bayesian simulation can be slow, and people often use approximate methods such as [[variational Bayesian methods]] and [[expectation propagation]].

==="Rule of ten"===