Estimating datapoint from log-likelihood function evaluations

Imagine that we are given multiple evaluations of a likelihood function on a datapoint for several samples of model parameters (coming from their prior), and this datapoint is hidden from us. Under which conditions is it possible to recover or obtain a good estimate of the datapoint?

• It's unclear what you mean by invertible or even by "likelihood," because the likelihood function is either not considered to be a function of the data points or it's considered to be a function of both the data and the parameters. See stats.stackexchange.com/questions/2641. For instance, if we view the likelihood as being a real-valued function of the parameters, then it can be inverted only when there's a single parameter. Please edit your post to explain the meaning. – whuber Sep 5 '19 at 20:47
• I just want to give a definition of the Likelihood function to clarify, hoping it will be useful. Given a sample of observations X1,...,Xn generated by a certain pdf, The likelihood function is defined as the joint probability density function expressed as a function of the parameters given the observed data. So the likelihood does the opposite of the pdf which is a function of the observations X1,...,Xn given the parameters. Which is why is often denoted as $L(\theta; X1,...,Xn)$ as you see it is a function of the parameter or parameter vector (depending on the situation) $\theta$ given the X – Fr1 Sep 6 '19 at 1:56
• I was thinking about the following.. if you are given a theoretical (absurde) sample whose n observations are always the same observation x and the parameter $\theta$ is fixed and known, and you have the value of the likelihood which is (assume!) the product of the marginal pdfs, then since the PDFs are always the same under the previous assumptions, you end up with an equation like $(\prod pdf(x) )^{n} = Likelihood$ where pdf is a function of $\theta$ and the unknown observation $x$. Then you can try to solve for x depending on the shape of the distribution – Fr1 Sep 6 '19 at 2:22
• But notice that we are “denaturing” the sense of the likelihood because we are given the parameters and we are trying to retrieve the obs.. we are taking de facto the value of the joint density for the sample (given the parameter), and trying to find the observations. So it like we are inverting the joint pdf of the sample given the parameters, to retrieve the observations. How many of them can we retrieve, well we have just one equation.. so.. I would say that at most we can’t invert and find one (i.e. we can invert to solve for multiple unknown observations). – Fr1 Sep 6 '19 at 2:29
• notice that in my scenario we fix the datapoint $X_i$ and we evaluate the likelihood function $L_i(X_i|\theta)$ on this single datapoint at a bunch of different sampled values of parameters $\theta$. So, eventually we have a set $\{L_i(X_i|\theta^{(s)})\}_{s=1}^{S}$ – Dionysis M Sep 6 '19 at 8:16