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There many literature that is talking about the posterior binomial distribution. I understand the concept but this is what is confused me:

In general the posterior probability is given by

P(theta | x) = P(x | theta) * P( theta ) / P(x) ... (1)

where P(theta) is the prior distribution of theta. If you have a look on the pattern recognition and machine learning book at page 71 when he is discussing the posterior distribution of binomial R.V . You will see that he is using the annotation like this Beta(mu | a,b) to denote P(theta).

Now, the posterior probability for the Binomial Distribution, is the product of binomial and it conjugate probability; the Beta distribution. I understand the intuitive sense but what is confused me the annotation. According to equation (1) the prior probability is not a conditional probability, but in that book, it's written as Beta(mu | a,b). So, according to this book the equation 1 became

P (mu | m,l,a,b) = P(m | N,mu) * P (mu | a,b) / X.. (2)

Where N = m + l which is the number of samples in sample space and a and b are fictitious samples. X is a proportion that is not related to mu, so we can ignore them when we want to optimize this equation according to mu.

How can I relate equation (2) to the general formula in equation (1)?

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    The author is using fairly common notation: In $\mathrm{Beta}(\theta | a, b)$ the symbol $|$ indicates that fixed _parameters_ of the beta distribution follow; no conditioning is involved (unless in a more intricate model, $a$ and $b$ become random 'hyperparameters'). [It might have been better to write $\mathrm{Beta}(\theta ; a, b),$ but this use of $|$ is not unusual.] In other places, within a few pages, the author uses $|$ to indicate actual conditioning. – 2017-02-05
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    @BruceET What do you mean by fixed parameters? you mean something like constants? How can I avoid this confusion in the future? I mean how can I distinguish between | indicates conditional and | indicates fixed parameters? – 2017-02-05
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    Yes, fixed parameters for the prior distribution are _constants_, not random variables. To avoid confusion pay attention to which symbols are constants and which are random variables. Only two possibilities, shouldn't be an insurmountable obstacle. – 2017-02-05

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