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From The Elements of Statistical Learning I know that logistic regression parameters can be fit using the Newton Raphson (NR) method on the likelihood function. So then why would you ever want to use MCMC?

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    $\begingroup$ The Markov Chain Monte Carlo method is used for Bayesian inference. $\endgroup$ – Michael R. Chernick Mar 28 '17 at 16:29
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If you're doing a fully Bayesian treatment of logistic regression you could use MCMC, but if you're just doing a frequentist maximum likelihood (possibly penalized) it would be very unnecessary to use MCMC. One big difference is with the Bayesian regression you need to sample from the posterior of your parameters, while with the frequentist you just need a point estimate so that's an optimization problem rather than a sampling problem. And NR's great for that.

Note that you can use sampling for optimization (see for example the connection between Metropolis-Hastings and simulated annealing) but if you've got a closed form for derivatives available in general that will work way better.

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MCMC is usually used to fit complicated models. For example, if the response has some missing values in a logistic regression model.

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    $\begingroup$ Welcome to the site, @Chris. At present it is more of a comment than an answer by our standards. Can you expand on it? We can also turn it into a comment. $\endgroup$ – gung - Reinstate Monica Sep 28 '17 at 15:59

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