Properties of an unusual estimator

Question

I came across a paper in which the authors are interested in parameter estimation but they use an unusual estimator that I never encounter before. Given a parameter $\theta$ to estimate and a likelihood function $\mathcal{L}(\theta)$ given a certain sample, the authors calculate the following estimate $\hat{\theta}$: $$ \hat{\theta} = \frac{\int \theta\, \mathcal{L}(\theta)\, d \theta}{\int \mathcal{L}(\theta)\, d \theta} $$ In other words, that estimator seems to calculate some kind of mean value for the parameter as if the likelihood function was the probability density function for that parameter (which we know it is not...)

My question is just: have anyone ever encounter that estimator before? What are its properties in terms of consistency, efficiency, etc...

Thanks in advance.

Answer 1

Under a weak improper prior $\pi (\theta) \propto 1$ the posterior is $\pi (\theta | x ) \propto \mathcal{L} (\theta)$ and then the normalising constant is just $\int \mathcal{L}(\theta) d\theta$. Hence, the posterior expectation is \begin{align*} E(\theta|x) & = \int \theta \pi(\theta | x) d\theta \\ & = \int \frac{\theta \mathcal{L}(\theta)}{\int \mathcal{L}(\theta) d\theta} d\theta \\ & = \frac{\int \theta \mathcal{L}(\theta) d\theta}{\int \mathcal{L}(\theta) d\theta} \end{align*}

So $\hat{\theta}$ is can be interpreted as the posterior expectation of $\theta$ under the improper prior $\pi(\theta) \propto 1$. Wikipedia tells me this is the ''generalised Bayes estimator''. Unfortunately, I don't know much about the generalised bayes estimator but know that we know it's a thing then someone else on stackexchange can help us find out more about it!