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Referring to Pattern Recognition and Machine Learning by Bishop(Page 367, Section 8.1):

Such models have particularly nice properties if we choose the relationship between each parent-child pair in a directed graph to be conjugate, and ...

As a newcomer to Bayesian statistics, I'm a little confused about the concept of conjugate relationship here. The wiki of conjugate prior states that:

In Bayesian probability theory, if the posterior distributions p(θ | x) are in the same probability distribution family as the prior probability distribution p(θ), the prior and posterior are then called conjugate distributions

My question is: are the two conjugations here same? In other words, does the quotation from Bishop book entail that all the parent-child pairs are in the same probability distribution family?

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Yes, they are the same. In a mentioned graphical models, the child-parent relationships are between prior distributions for the parameters of some distribution and the distribution itself. So what Bishop means in here, is using conjugate priors. What he says is basically that if we choose conjugate priors, then we end up with closed-form solutions for the posterior distributions, what tremendously simplifies the computations. If the concept of conjugate is still not clear for you, check Can anyone explain conjugate priors in simplest possible terms?, or other threads tagged as .

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