Let $X,Y$ be input and output (observed) continuous variables in $\mathbb{R}$. Let $\{y_1,...,y_n\}$ be the set of $n$ observations. Is there a name for the estimator $\hat x = \int_{x \in X} x \,p(y_1,y_2,...,y_n|x) \,dx $ (mean of likelihood)? I tried googling but I could not find its name.


You aren't strictly taking the "mean" of the likelihood, because the Likelihood function isn't a probability distribution over x. It isn't even a probability distribution anyway, but assuming you have a likelihood function that you can normalize into a PDF then it would be the probability of $Y$ not of $X$. This is a likelihood weighted average of $X$.

I think you have come across an ad hoc Bayes estimate here. If we note that

$P(X|Y) = \frac{P(Y|X)P(X)}{P(Y)} \propto L(Y|X) P(X)$

Then by simply normalising the likelihood into a PDF you are creating the Posterior distribution using uniform priors for $X$. This may or may not be a sensible thing to do.

By then integrating out this distribution you are taking the expected value of $X$ under the posterior distribution of $X$. This is therefore the posterior mean estimator, also the Minimum Mean Square Estimator (MMSE).


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