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Apologies for the naive question, but I have a problem I would like to solve.

Suppose I have a two dimensional likelihood of the form

$L \propto \exp\{-\frac{1}{2}\} \begin{bmatrix}x & y\end{bmatrix}' C^{-1}(\theta) \begin{bmatrix}x & y\end{bmatrix}$,

where the covariance matrix depends on some parameter $\theta$.

I would like to maximise this likelihood with respect to $\theta$. But I have some practical cases where I want to ignore the $x-x$ and $y-y$ parts (e.g. their too noisy).

Does it make sense to just take the mixed $x-y$ part in the likelihood

$L' \propto \exp\{-\frac{1}{2}\} x A(\theta) y$ and maximise?

What are the general dangers of this?

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  • $\begingroup$ Restated without matrices, this asks: “can I find $\theta$ maximizing $E[a_\theta X^2+ b_\theta XY+c_\theta Y^2]$ by maximizing $E[b_\theta XY]$?” The general answer is obviously no, but there might be something more useful to say with more detail on the application. $\endgroup$
    – user225256
    Commented Sep 19, 2023 at 7:58
  • $\begingroup$ Thanks for your answer @MattF. ! I am not looking for the same maximum $\theta$. I am looking to know if there is a good $\theta$ with the second way. Suppose for example if X, Y are correlated noisy variables, but the individual independent noises for each of these is high (but not that high that it does not matter in the covariance). Studying the X-Y term can be useful in this case. But my question is: does it make sense? Can I have a meaningful answer? $\endgroup$
    – Fellow99
    Commented Sep 19, 2023 at 8:26
  • $\begingroup$ I’m not sure what you mean by noisy. Can we model this with normal $X$ and $Y$ with correlation $\theta$? Or with $X=e^U$, $Y=e^V$ where $U$ and $V$ are normal with correlation $\theta$? $\endgroup$
    – user225256
    Commented Sep 19, 2023 at 12:39

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