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Can unimodal prior and unimodal sampling distributions lead to a multimodal posterior distribution?

The Bayes rule tells us that $$ f(y|x) = f(x|y) f(y) / f(x). $$ which, I think, implies a unimodal prior and unimodal sampling distributions can only lead to a unimodal posterior distribution.

If $y$ is multidimensional, can the marginal posterior distribution of some component of $y$ be multimodal? (Related question: can a unimodal multivariate distribution has some multimodal marginal distribution? If all marginal distributions are unimodal, can the multivariate distribution be multimodal?)

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1 Answer 1

up vote 4 down vote accepted

It is easy enough to find an example where the posterior distribution is not unimodal even if both the prior distribution and the likelihood function are unimodal.

E.g. if $X$ is distributed with a Cauchy distribution with unknown median $m$ so with density $f(x|m)=\frac{1}{\pi(1+(x-m)^2)}$ and $m$ has a prior distribution which has a standard Cauchy density $p(m) = \frac{1}{\pi(1+m^2)}$ then the posterior density for $m$ given an observation $X=x$ is $$p(m|x)=\frac{x^2+4}{2\pi (1+m^2)(1+(x-m)^2)}$$ which for $|x|\gt 2$ is bimodal with maximum densities at $m=\frac{x\pm\sqrt{x^2-4}}{2}$ and a local minimum at $m=\frac{x}{2}$.

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Thanks! In $f(y|x)\propto f(x|y)f(y)$, if $f(x|y)$ and $f(y)$ are both unimodal in $y$, will $f(y|x)$ be unimodal? –  Tim Mar 30 '14 at 22:18
@Tim: do you think my example meets the conditions in your comment? –  Henry Mar 30 '14 at 22:20
it seems yes. But for clarification, Do one local max and one local min count as bimodal? Isn't bimodal defined to be two local maxima? –  Tim Mar 30 '14 at 22:21
$\frac{x\pm\sqrt{x^2-4}}{2}$ has a $\pm$ and so gives two maxima –  Henry Mar 30 '14 at 22:22
I see. thanks. Could you also address the last paragraph in my post, if possible? –  Tim Mar 30 '14 at 22:22

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