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I would like to show that the expression $ \frac{E\,\left[\, p^{t+1}\left(1-p\right)^{\left(n-t\right)}\right]}{E\left[\, p^{t}\left(1-p\right)^{\left(n-t\right)}\right]} $ , where $p$ is random on $[0, 1] $ and $n$ and $t \leq n$ are positive integers, is increasing in $t$ (for any $n$).

The background of this question is as follows. I'm currently starting to learn Bayesian statistics and I'm trying to understand some simple models. The above expression arises when there is a Bernoulli experiment with unknown probability $p$. Suppose that after $n$ trials we have drawn $t$ successes. It seems very intuitive that the larger $t$, the higher the posterior expectation of $p$ should be. However, the posterior expectation can be expressed as the expression above and I do not find a straightforward way to show that this expression is increasing in $t$.

I assume that this is a simple textbook problem, but all I can find are treatments in which a specific prior distribution of $p$ is assumed. I however would like to see a solution which allows $p$ to have any distribution on $[0, 1]$, as long as the above expectations are defined.

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