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?
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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
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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.)
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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
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2$\begingroup$ en.wikipedia.org/wiki/Prior_probability#Improper_priors $\endgroup$– Glen_bApr 14, 2020 at 23:55