# Why do T prior and likelihood make a bimodal posterior?

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.

• I'm a bit confused, where in the link does the poster show that the likelihood and prior are t, 2df distributed? Sep 11 at 17:21
• 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. Sep 12 at 2:53
• 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. Sep 12 at 12:18

• for $$t$$ data and prior, sufficiently far apart, the posterior is bimodal