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.

  • 3
    $\begingroup$ 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. $\endgroup$ – 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 $\alpha=\beta=1/2$.

  • $\begingroup$ 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. $\endgroup$ – Michael Chernick Dec 26 '16 at 2:45
  • $\begingroup$ @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. $\endgroup$ – Tim Dec 26 '16 at 8:17

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.


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