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I understand that marginal-likelihood can be derived as answered here. Quoting the same proof from MATHEMATICS FOR MACHINE LEARNING book (9.3.5) Page 312, enter image description here

The same book mentions that we can derive this using multiplication of two Gaussians (which is not used in the above derivation) (6.5.2)(Page 201) enter image description here

But, as you can see we need to have same random variable $x$ to apply the above rule, MML book mentioned in Chapter 9 that we can change $y$ in form of $\theta$ and then use above rule. (Page 312 again) enter image description here

I am not able to convert $y$ in form of $\theta$ as suggested here and derive the marginal likelihood. Can someone help on this?

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