# Multimodal prior

In Bayesian method, a posterior can be either unimodal or multimodal. But, I cannot find any multimodal prior case yet.

I wonder if it is possible, and there is any case that is using multimodal prior.

• The prior used to assess the next study (event, set of evidence, whatever) is the posterior from the previous Bayesian analysis (unless this is the very first one). Thus, if a posterior can be multimodal, next time's prior should be multimodal. – gung Dec 25 '16 at 18:05

Any distribution can be used as prior. The simplest example of commonly used bimodal prior is Jeffreys beta prior with parameters $\alpha=\beta=1/2$.
Indeed consider the reparameterisation from $\mathbb{R}^+$ into the unit interval $(-1,1)$: $$f:\ t\longrightarrow \dfrac{t^a}{1+t^a}\qquad a>0$$ which has for Jacobian $$J(x)=\frac{\text{d}\,f^{-1}(x)}{\text{d}\,x}=\frac{1}{a}\dfrac{x^{\frac{1}{a}-1}}{(1-x)^{\frac{1}{a}+1}}$$ with explosive behaviours at $x=0$ and $x=1$. Therefore, given a prior $\pi_1$ on ${\Theta}=\mathbb{R}^+$, the reparameterisation from $\theta$ to $\zeta=f(\theta)$ leads to a prior $$\pi_2(\zeta)=\pi_1\left\{\zeta^\frac{1}{a}\big/(1-\zeta)^\frac{1}{a}\right\}\times \frac{1}{a}\dfrac{\zeta^{\frac{1}{a}-1}}{(1-\zeta)^{\frac{1}{a}+1}}$$which may also enjoy asymptotes in zero and 1 if $a$ is large enough. If not, faster concentrating transforms should fill the job.