For example, for $p$ as the parameter to a binomial or bernoulli, or a Poisson, what would a flat prior $P$ be? What does it mean to be "flat" - does this refer to diffuse?

  • 1
    $\begingroup$ Literally any continuous distribution could be a "diffuse" prior: just find a one-to-one map from the parameter space to $[0,1]$ and apply the distribution's quantile function to that! Thus, to make progress, one has to identify a set of favored parameterizations. For instance, for the Binomial would you use the chance of success or the log odds of success for $p$? $\endgroup$
    – whuber
    Nov 19, 2014 at 23:16

1 Answer 1


The term "flat" in reference to a prior generally means $f(\theta)\propto c$ over the support of $\theta$.

So a flat prior for $p$ in a Bernoulli would usually be interpreted to mean $U(0,1)$.

A flat prior for $\mu$ in a normal is an improper prior where $f(\mu)\propto c$ over the real line.

"Flat" is not necessarily synonymous with 'uninformative', nor does it have invariance to transformations of the parameter. For example, a flat prior on $\sigma$ in a normal effectively says that we think that $\sigma$ will be large, while a flat prior on $\log(\sigma)$ does not.

With flat priors, your conditional posterior will be proportional to the likelihood (possibly constrained to some interval/region if the prior was). (In this case MAP and ML will normally correspond, though if we're taking the flat prior over some region, it might change that.)

  • 1
    $\begingroup$ I'm a little confused why everyone writes $f(\theta) \propto c$. In this instance couldn't they just write $f(\theta)=c$? A uniform distribution always a has a constant density function, and AFAICT having a flat prior distribution means having a uniform distribution. Is it because of the case where the set of possible parameter values is infinite? I don't know how a uniform distribution over the entire real line would work or be defined. $\endgroup$ Aug 25, 2019 at 23:03
  • 2
    $\begingroup$ en.wikipedia.org/wiki/Prior_probability#Improper_priors $\endgroup$
    – Glen_b
    Apr 14, 2020 at 23:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.