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I'm reading section 3.5 of the PRML book, entitled Evidence approximation, and is having difficulty understanding this part: enter image description here. I don't understand how to derive (3.75) from (3.74). The author says it is because alpha and beta are sharply peaked but I don't see how it's relevant here, or is it because they are sharply peaked that the probability P(alpha, beta|t) somehoww turn into a Dirac delta function ? Thank you very much

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Indeed the assumption is that $p(\alpha,\beta|t)\approx \delta(\alpha-\hat{\alpha})\delta(\beta-\hat{\beta})$.

The point is that otherwise the maximization with respect to $\alpha,\beta$ is intractable. The other extreme is when $p(\alpha,\beta)$ is approximately uniform in $\alpha,\beta$. In this case you can write $p(\alpha,\beta|t)=\frac{p(t|\alpha,\beta)p(\alpha,\beta)}{p(t)}$ from which you can maximize $p(t|\alpha,\beta)$ instead (for example in a linear basis model).

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