I'm a little confused why everyone writes a flat prior as $f(\theta) \propto c$. 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.
Solved – Why are flat priors said to be proportional to a constant
bayesiannotationprior
Best Answer
I beg to disagree with the answer given by pche8701: the main reason a flat prior is introduced (in an improper setting) as $f(\theta)\propto c$ or $f(\theta)\propto 1$ which is equivalent but more rigorous is that (i) any constant $c$ leads to the same posterior distribution and (ii) there is no principled way to choose a value for the constant $c$ since a constant density integrates to infinity. It simply cannot be normalised. Hence the qualificative of improper, since it is not a probability density. This explains for instance why improper priors cannot be used in model choice, because the constant $c$ then gets in the way.
While this may appear as a limiting case of a Uniform distribution, exploiting the analogy may lead to paradoxes and contradictions. A flat prior is not a Uniform distribution, but a $\sigma$-finite measure. (Again, I take the convention that one would not use the "flat" denomination in a compact setting since the "uniform" denomination would then become appropriate.)