How do I prove: the posterier is proportional to the product of the prior distribution and the likelihood function?

Asked Nov 22 '18 at 13:53

Active Nov 22 '18 at 14:05

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In the book of pattern recognition and machine learning equation (1.66) says:

Using Bayes’ theorem, the posterior distribution $\bf{w}$ for is proportional 
to the product of the prior distribution and the likelihood function.

$$ p(\bf{w}|\bf{x},\bf{t}, \alpha, \beta) \propto p(\bf{t}|\bf{w},\bf{x}, \beta) p(\alpha|\bf{w}) $$

I do know that Bayes's theorem is $P(A,B)=P(A|B)P(B)=P(B|A)P(A)$ but I don't quite know how to derive this so many variable case. Besides, what is the proportion constant, why the remaining is a constant?

edited Nov 22 '18 at 14:05

asked Nov 22 '18 at 13:53

an offer can't refuse

Is there any dependence/independence between variables? – Carlos Campos Nov 22 '18 at 14:20
@CarlosCampos It seems that $\bf{x}$ and $\bf{t}$ are datat, $\alpha$ is the parameter of the prior probability and $\beta$ is the variance of the Gaussian distribution – an offer can't refuse Nov 22 '18 at 14:34

How do I prove: the posterier is proportional to the product of the prior distribution and the likelihood function?

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