3
$\begingroup$

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.

$\endgroup$
12
$\begingroup$

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).

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.