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I have a joint distribution which can be written down as follows: Where D is the observed data. w, n and r are the model parameters. The likelihood term P(D|w, n, r) is modelled using a Gaussian. The prior on 'w' is a zero mean normal distribution as well given by p(w|r) where 'r' is the scale factor for the noise variance and is to be also determined by the data. The prior on P(n) where n is the noise parameter and p(r) are given by Gamma distributions as conjugate priors. Now, imagine I know 'w'. So I know the mean and I want to update the prior parameters of 'n' and 'r'. Reading Bishop' book and online texts, there should be explicit updates to these parameters. So, Bishop's book on page 100, gives update equations to the Gamma parameters. However, I need to update parameters of both 'n' and 'r'. So there are two Gamma distributions and also the prior on 'w' is a conditional prior p(w|r). So, I was wondering if update equations for both 'n' and 'r' can still be derived and I was wondering if someone can give me a starting point or just some guideline on how to go about it. For example, if I want to update the $lambda$ parameter should I try and integrate out the $phi$ parameter from the joint distribution equation and then see what form the resulting expression takes (hopefully another Gamma!) |
