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Let $X_i$ be the distance between two nodes in a graph.

$X_i \sim \mathrm{NB}(r,p)$ where $p \sim \mathrm{Beta}(\alpha, \beta)$.

After the posterior hyperparameters are obtained, $\mathrm{Beta}(\alpha,\beta) – \mathrm{Beta}(\alpha’,\beta’)$ could be found which would correspond to the change in expected distance.

Is it possible to find out how unlikely a given $\mathrm{Beta}(\alpha,\beta) – \mathrm{Beta}(\alpha’,\beta’)$ (change in expected distance) is?

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    $\begingroup$ It's not really clear what your question is. Perhaps you could make some edits to improve it. Also, please enclose your math with dollar signs to render it properly. $\endgroup$
    – jerad
    Feb 27 '13 at 23:05
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There might be a closed-form way to solve this problem, but I think the simplest way is a Monte Carlo. Perform $k$ draws from each of the beta distributions, and perform the subtraction. Now you have $k$ samples from the distribution $Beta(\alpha, \beta)-Beta(\alpha^\prime, \beta^\prime)$. The probability of a value in some range is the number of draws in that range divided by $k$.

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