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From Stan reference,

The default is to randomly generate initial values between -2 and 2 on the unconstrained support

It seems to me that it makes more sense to randomly generate initial values from the prior. A priori, I'm saying that I believe that these values make the most sense. Starting an MCMC chain from somewhere that has a possibly minuscule a priori probability doesn't intuitively make sense to me.

What is the reason for Stan's default?

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The biggest problem with drawing from the prior is if a user is using a rather flat prior. For example, if a user is using a logistic regression model and they don't want the prior to have much of an effect on the posterior, they may choose to make the prior a normal distribution with a standard deviation of 100. Taking a draw from this prior will, with very high probability, lead to numerical errors (i.e., NaN) in computing the likelihood.

As such, the Stan authors are betting that values in the range $[-2, 2]$ are less likely to lead to computational errors than a draw from the prior.

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    $\begingroup$ I like that you say that this is a "bet" $\endgroup$ – TrynnaDoStat Jul 23 '19 at 0:10
  • $\begingroup$ @TrynnaDoStat: isn't that the proper definition of probability in Bayesian statistics: acceptable betting odds? $\endgroup$ – Cliff AB Jul 23 '19 at 0:11

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