Best way to compare 2D pdf and 2D histogram

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I previously asked a related question but got no reply.

So more generally: I am generating points from a 2D multivariate distribution, I then generate a 2D histogram of these points using different methods and settings. I want to find the best histogram (i.e. the one that matches the true 2D multivariate distribution the best).

What is the best way to do this?

I have tried things like MSE and mutual information but a problem I encounter is that large binsizes often give me a small total error, but these are obviously less informative than histograms made with small binsizes, so I feel like I am missing something.

Thanks for any help,

M.