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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?

Thanks!

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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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