# MAP, MLE and parametrised data

It is often said that maximum likelihood is used to obtain estimates of distrubtion's parameters. However, what is unclear is whether it will produce consistent estimate parameters other than those of a distribution. For instance, let's assume and the instead of supplying a data matrix $$X$$ into a likelihood function $$L(\theta|X)$$ (that $$\theta$$ is a parameters' vector) we supply it some function of $$X$$ and $$\theta$$ $$f(X, \theta)$$. Will estimates of $$\theta$$ still be consistent?

On the other hand, let's assume that we have two vectors of parameters $$\theta_1, \theta_2$$, and $$\theta_2 \sim g(.)$$. Then, by employing the formula for the probability of product of events, we can write $$L(\theta_1, \theta_2|X) = L(\theta_1|\theta_2,X)g(\theta_2)$$. Are estimates of $$\theta_1, \theta_2$$ obtained by maximising $$L(\theta_1|\theta_2,X)g(\theta_2)$$ consistent?

• The question is unclear. Are you using "some function" $f(X,\theta)$ to replace the likelihood $L(\theta|X)$ (in which case you might be thinking of GEE or M-estimators), or is $f(X,\theta)$ "supplied instead of the data matrix" $X$ into the likelihood, i.e. $L(\theta|f(X,\theta))$, in which case you're just dealing with a change of variables. Commented Jul 8 at 22:57

It's not consistent for any function, e.g. $$f(x)=0$$. But, it's consistent if the function is one-to-one and doesn't depend on the parameters, as outlined in here. Your example also violates the second condition.

I can't quite follow your second question. If you explain more about why you multiplied $$L$$ ang $$g$$, and what $$f$$ is, I might have some ideas.

• I've updated the question. I hope it is clearer now. It's just MAP with some data transformation parametrised by $\theta_2$. Commented Jan 24, 2020 at 17:47