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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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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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  • $\begingroup$ 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? $\endgroup$ – Tim Mar 30 '14 at 22:18
  • $\begingroup$ @Tim: do you think my example meets the conditions in your comment? $\endgroup$ – Henry Mar 30 '14 at 22:20
  • $\begingroup$ 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? $\endgroup$ – Tim Mar 30 '14 at 22:21
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    $\begingroup$ $\frac{x\pm\sqrt{x^2-4}}{2}$ has a $\pm$ and so gives two maxima $\endgroup$ – Henry Mar 30 '14 at 22:22
  • $\begingroup$ I see. thanks. Could you also address the last paragraph in my post, if possible? $\endgroup$ – Tim Mar 30 '14 at 22:22

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