Multimodal prior

Question

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 - Reinstate Monica, Dec 25 '16 at 18:05

score 6 · Accepted Answer · answered Dec 25 '16 at 22:08

6

Any distribution can be used as prior. The simplest example of commonly used bimodal prior is Jeffreys beta prior with parameters $\alpha=\beta=1/2$.

answered Dec 25 '16 at 22:08

Tim

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From a practical standpoint, before collecting data what type of problem lends itself to using a bimodal prior? I am sure there are good answers to this. It may be what the OP had in mind. – Michael R. Chernick Dec 26 '16 at 02:45
@MichaelChernick I don't have a general answer to this as this is not a general practice. In this particular case it's a uninformative prior of such form. – Tim Dec 26 '16 at 08:17

score 1 · Answer 2 · answered Dec 28 '16 at 08:07

Tim's example is a natural answer to the question (and to address Michael Chernick's comment, the Beta(1/2,1/2) prior is both noninformative and allowing for estimating probabilities close to zero or one). I have however an additional answer to the question which is that any prior is bimodal when using the right reparameterisation!

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.

Multimodal prior

2 Answers2