Why are flat priors said to be proportional to a constant? 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.
 A: As dsaxton said in his comment, it is a convenient way of saying you are working with a flat shaped distribution without having to specify the parameters of the flat prior (e.g. the upper and lower limits of the uniform distribution). 
Writing it explicitly as $f(\theta) = c$ would be technically incorrect since the assumption is then that $f(\theta)$ goes on forever in both directions: $(-\infty,\infty)$. You are saying it is literally just the straight line extending to infinity. You cannot have this in probability (well you can, but that is known as an improper prior - a prior distribution which does not integrate to unity, but this is also a bit contentious).
So in general writing it as $f(\theta) \propto c$ removes the ambiguity of having it need to integrate to unity at this point in time - during model specification (since the function is not fully specified yet, it is not equality), and also saves you having to fully parameterise the density in some form, $\mathcal{U}(a,b)$. It leaves things more flexible, and general, without violating any principles of probability. 
A: There are various kinds of 'flat' or 'noninformative' prior distributions. A couple of specific examples may help to illustrate both the intuitive idea and technical aspects.
Beta-Binomial. Suppose you have binomial data, such as $x$ successes out of $n$ independent trials, where unknown parameter $\theta$ is the probability of success on any one trial. The binomial likelihood can be written as 
$f(x|\theta) = {n \choose x}\theta^x(1-\theta)^{n-x}$ and considered as a function of $\theta$ for data $x.$ Often the 'norming constant' ${n \choose x},$
which does not contain $\theta$ is omitted, and one writes
$f(x|\theta) \propto \theta^x(1-\theta)^{n-x},$ where the symbol $\propto$ (read "proportional to") indicates the absence of the norming constant.
In this situation one may choose the minimally informative prior distribution $\mathsf{Unif}(0,1) \equiv \mathsf{Beta}(1,1),$ which has density function.
$f(\theta) \propto 1 = \theta^0(1-\theta)^0.$  When the norming constant is omitted,
this may be called the 'kernel' of the prior distribution.
Multiplying the kernels of the prior and likelihood, one has the posterior kernel 
$$p(\theta|x) \propto p(\theta) \times p(x|\theta)\\
\propto \theta^{1-1}(1-\theta)^{1-1}\times \theta^x(1-\theta)^{n-x},\\
\propto \theta^{1+x-1}(1-\theta)^{1+n-x -1},$$ 
where we recognize the last expression as the kernel of the distribution
$\mathsf{Beta}(1+x, 1+n-x).$
For technical reasons, some people prefer to use the Jeffreys noninformative prior $\mathsf{Beta}(.5, .5)$ with kernel $f(\theta) \propto
\theta^{.5-1}(1-\theta)^{.5-1},$ instead of the uniform prior. Then
the posterior posterior distribution turns out to be
$\mathsf{Beta}(.5+x. .5+m-x).$ In many cases, the practical difference
between the two priors is very small and inconsequential. But with
the Jeffreys prior, it would be incorrect to write $f(\theta) \propto 1.$
So not every 'noninformative' prior should be written with $\propto 1.$
Gamma-Poisson. Suppose your data are Poisson counts, such as $x_1, x_2, \dots x_n$ with $t = \sum_{i=1}^n x_i,$ where the unknown parameter $\lambda$ is the Poisson rate. The Poisson likelihood can be written as 
$f(x|\lambda) \propto \prod_{i=1}^n \lambda^{x_i}e^{-\lambda} \propto \lambda^te^{-n\lambda}.$
An informative prior might be $\mathsf{Gamma}(\text{shape}=4,\text{rate}=1/3),$ which has kernel $f(\lambda) \propto \lambda^{4-1}e^{-(1/3)\lambda}.$ Then
the kernel of the posterior distribution is given by
$$f(\lambda|x) = f(\lambda)\times f(x|\lambda)
\propto \lambda^{4-1}e^{-(1/3)\lambda} \times \lambda^t e^{-n\lambda}\\
\propto \lambda^{[4+t]-1}e^{-[(1/3)+n]\lambda}.$$
where we recognize the last expression as the kernel of
$\mathsf{Gamma}(4+t, (1/3)+n).$
If we want a noninformative prior distribution--that has very little effect on the posterior distribution--we can reduce the shape parameter to very near 0 and the rate parameter
to very near 0. There is no distribution $``\mathsf{Gamma}(0,0),"$ but we can
imagine such an 'improper' gamma distribution.  [In a computer application that requires 'legal' parameters, we might use something like 0.0001 for each parameter.] The consequence would be
that the likelihood function (essentially) becomes the posterior distribution.

A: 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.)
