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I am taking ML course and I am confused about some derivations of math

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Could you explain the two steps I marked on the slides? For the first step, I thought $P(beta|X,y) = \frac{P(X,y|beta)P(beta)}{P(X,y)}$ but I don't know the further steps to reach the next step. Maybe I am confused of the conditional notation and semicolon notation.

tesio
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For step 1: you're on the right track, but the derivation is easier if we leave everything conditional on $X$ $$p(\beta| X,y) = \frac{p(y| X, \beta) p(\beta | X)}{p(y |X)}$$ Now, the two key steps. First, $p(\beta | X) = p(\beta)$ if $X$ and $\beta$ are independent (why is this the case?). Second, $p(y |X)$ is constant in $\beta$. How does multiplying a function by a constant affect the location of its maximum?

For step 2: simply substitute the expressions given for $p(y|X, \beta)$ and $p(\beta)$. There will be additional terms that don't depend on $\beta$. How does adding a constant to a function affect the location of its maximum?

Luke
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