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Given data $D$ you assume a predictive model parametrized by some parameters. You may then seek to do MLE or MAP estimate to determine those parameters. If you do MLE, then only likelihood function is involved but if you use MAP then both likelihood and prior function are involved.

David Mackay says in Ch 45 on Gaussian Processes that as far as predictive performance is concerned for MAP prediction only choice of likelihood and prior function matters and not choice of the parameters. Why latter does not matter?

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  • $\begingroup$ For predicting future values, only the mean function $m(x)$ and exact values $Y_1,Y_2,...$ are used. It does not matter how exactly the function $y(x;w)$ is parameterized. $\endgroup$
    – Md Ashiqur Rahman
    Commented Aug 1, 2021 at 11:02

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