Why do T prior and likelihood make a bimodal posterior?

Question

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.

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