# Combining two Beta disributions

I want to combine two beta distributions to find the the posterior distribution as the precision-weighted combination of the prior and the likelihood distributions. As the prior and likelihood I have two beta distributions. How do i combine them? Want to know about all the theoretical techniques available. Is there any article which may help me in the analysis. Even guiding me with a weblink to any article would also help!

Regards, Bik

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It is unclear exactly what you are asking for.

If your prior distribution is $B(\alpha,\beta)$ then the prior density is proportional to $x^{\alpha-1} (1-x)^{\beta-1}$.

Your likelihood function is perhaps proportional to $x^{\gamma} (1-x)^{\delta}$.

So you multiply these together and your posterior density is proportional to $x^{\alpha+\gamma-1} (1-x)^{\beta+\delta-1}$ which means your posterior distribution is $B(\alpha+\gamma,\beta+\delta)$.

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Hello @Henry! Firstly, thanks a ton for helping me. Let me give more explanation on my query: I am trying to combine two beta distributions B1 (a1,b1) and B2 (a2,b2) using spreadsheet to get another alpha (a3) and beta (b3) for a combined beta distribution. Using this technique I intend to combine expert judgement and data to recalibrate my model. One of the option which I have is to take a weighted average of the paramaters (a1,b1,a2 and b2) to arrive at a3 and b3. Just wanted to inquire if there are other techniques available. Looking forwark for your response... –  Bik Feb 15 '13 at 13:57
What is supposed to be distributed with a Beta distribution? The parameter of another distribution? –  Henry Feb 15 '13 at 19:28
Yes... you are right... –  Bik Feb 16 '13 at 8:02