In this post, the author shows that when a likelihood and prior are both T-distributed with $2$ degrees of freedom, the posterior is bimodal. The given reason is that

The two modes persist - the extra mass in the tails means each distribution finds the other's mode more plausible and so the average isn't the best "compromise".

The distributions are

$$y \sim T(\text{df} = 2, \mu = \mu_0)$$

$$\mu_0 \sim T(\text{df}=2, \mu = 10)$$

What exactly does it mean that each distribution finds the other's mode more plausible? Is this something peculiar about the T distribution or can this be generalized elsewhere?

Code to reproduce the results available here.

  • $\begingroup$ I'm a bit confused, where in the link does the poster show that the likelihood and prior are t, 2df distributed? $\endgroup$
    – AdamO
    Sep 11 at 17:21
  • 2
    $\begingroup$ For a 2D problem, you can plot the log prior and log likelihood together to see why this happens. Ben Bolker tweeted the plots here. You can see that for that example, the sum of the log prior and log likelihood results in a bimodal log posterior. $\endgroup$ Sep 12 at 2:53
  • $\begingroup$ The simplest explanation is that the $t$ likelihood is the inverse of a polynomial. If the data contains enough outliers and a few observations, this likelihood will be multimodal. $\endgroup$
    – Xi'an
    Sep 12 at 12:18

1 Answer 1


The post shows a "Bayesian surprise" setting, where the data are surprisingly far from the center of the prior. In this setting, the location and shape of the posterior is sensitive to the rate of decrease of the tails of the prior and likelihood.

  • for Normal data and prior, the posterior is Normal and is between the data mean and the prior mean
  • for $t$ data and prior, sufficiently far apart, the posterior is bimodal
  • for Laplace data and prior, the posterior is flat across a wide interval
  • and other things for other examples

I have an interactive version here, and Tony O'Hagan analysed the location problem completely in JASA in 1990


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