I saw in some lecture the fact that as the number of data points N goes to infinity, the prediction of the Bayesian method goes to the prediction of the MLE. Can someone explain what exactly this sentence means, and why is it true?



The MLE is known to be consistent under specific conditions, that means that the estimate converges (either in probability or almost surely) to the true value of the parameter $\theta_0$.

Bayesian parameter estimation updates the posterior of $f(\theta)$ and makes it narrower and narrower around $\theta_0$. In the end, you obtain a Dirac on $\theta_0$. The only assumption is that prior $f(\theta)$ was not zero for $\theta_0$.

Thus, you can see that both methods converge under specific conditions to the real value $\theta_0$. Therefore, they converge to the same results.

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