# What are examples of "flat priors"?

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?

• 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 $p$?
– whuber
Nov 19, 2014 at 23:16

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.
• 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. Aug 25, 2019 at 23:03