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In Bayesian probability, does the prior distribution $\pi(\theta)$ only depend on $\theta$? For example, suppose the prior distribution of the unknown parameter $\theta$ is binomial. Then does $$ \pi(\theta) = \binom{n}{\theta} p^{\theta} (1-p)^{n-\theta}$$

Whereas if $f(\theta|x_1)$ is binomial then $$f(\theta|x_1) = \binom{n}{x_1} p^{x_1}(1-p)^{n-x_1}$$

Is this correct?

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    Your second equality doesn't make sense. If you'd written either $f(x_1\mid\theta) = \dbinom{n}{x_1} \theta^{x_1}(1-\theta)^{n-x_1}$ or $f(x_1\mid p) = \dbinom{n}{x_1} p^{x_1}(1-p)^{n-x_1}$ it would make some sense, but starting with $f(\theta\mid x_1)$ leaves me wondering what you mean, and then putting an expression with no "$\theta$" in it on the other side of the "$=$" is weird, and it's hard to even guess what you mean. The word "whereas" usually means you're calling attention to a contrast, but I don't know what you have in mind there either.2012-05-04
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    @MichaelHardy: I am talking about the posterior distribution $f(\theta| x_1)$.2012-05-04
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    Oh ok it should be $\theta$. But the first is correct?2012-05-04

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Your first statement might be correct, though I doubt it. It is possible that the parameter $\theta$ is an integer in the range $[0,n]$ with that prior distribution, though normally you would give an explicit value for $p$ or describe it as a hyperparameter. But I am guessing you actually expect $\theta$ to be a real number in the range $[0,1]$.

If your second statement is supposed to be an expression for the likelihood or posterior probability (density) of the parameter $\theta$ given the observation $x_1$, then it is almost certainly wrong, as it does not seem to vary with $\theta$.

My guess is that in fact $\theta$ is intended to be a parameter of the binomial distribution where $$\Pr(X_1=x_1|\theta,n) = \binom{n}{x_1} \theta^{x_1} (1-\theta)^{n-x_1}$$ and with $\theta$ having a (perhaps improper) prior distribution on $[0,1]$.

A common example chosen for the prior in this case, because it is part of a conjugate family, would be a beta distribution proportional to $$\theta^{\alpha-1}(1-\theta)^{\beta-1}$$ for some given $\alpha $ and $ \beta$, in which case the posterior density, given an observation of $x_1$ out of $n$, is proportional to $$\theta^{x_1+ \alpha-1}(1-\theta)^{n-x_1+\beta-1}$$ i.e. from the same family.