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edit approved Mar 2, 2017 at 8:21

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According to your motivation,

Numerical computation of $\rho$ is notoriously inaccurate and problematic, and so KL divergence or even $L^1$ distance are problematic as well. I obtain $f$ anyway, and I really only need to know how "far" is $\rho$ from being uniform.

And I assumed that your already obtained some information about the distribution of $f_i$, say moments. The moments of $f_i$'s are most easily obtained if you know the original $d\mu$ via simulation techniques like MCMCMCMC. Then I would suggest Wasserstein distanceWasserstein distance as a candidate since all you want is how dissimilar it looks from uniform, and Wasserstein distance is proven to perform well in this respect.

Your calculation does not have to involve $\rho_i$ but you should at least be able to sample from $d\mu$ and hence $f_i(X)$.

And the following paper verified that the convergence behavior is satisfying.

Fournier, Nicolas, and Arnaud Guillin. "On the rate of convergence in Wasserstein distance of the empirical measure." Probability Theory and Related Fields 162.3-4 (2015): 707-738.

According to your motivation,

Numerical computation of $\rho$ is notoriously inaccurate and problematic, and so KL divergence or even $L^1$ distance are problematic as well. I obtain $f$ anyway, and I really only need to know how "far" is $\rho$ from being uniform.

And I assumed that your already obtained some information about the distribution of $f_i$, say moments. The moments of $f_i$'s are most easily obtained if you know the original $d\mu$ via simulation techniques like MCMC. Then I would suggest Wasserstein distance as a candidate since all you want is how dissimilar it looks from uniform, and Wasserstein distance is proven to perform well in this respect.

Your calculation does not have to involve $\rho_i$ but you should at least be able to sample from $d\mu$ and hence $f_i(X)$.

And the following paper verified that the convergence behavior is satisfying.

Fournier, Nicolas, and Arnaud Guillin. "On the rate of convergence in Wasserstein distance of the empirical measure." Probability Theory and Related Fields 162.3-4 (2015): 707-738.

According to your motivation,

Numerical computation of $\rho$ is notoriously inaccurate and problematic, and so KL divergence or even $L^1$ distance are problematic as well. I obtain $f$ anyway, and I really only need to know how "far" is $\rho$ from being uniform.

And I assumed that your already obtained some information about the distribution of $f_i$, say moments. The moments of $f_i$'s are most easily obtained if you know the original $d\mu$ via simulation techniques like MCMC. Then I would suggest Wasserstein distance as a candidate since all you want is how dissimilar it looks from uniform, and Wasserstein distance is proven to perform well in this respect.

Your calculation does not have to involve $\rho_i$ but you should at least be able to sample from $d\mu$ and hence $f_i(X)$.

And the following paper verified that the convergence behavior is satisfying.

Fournier, Nicolas, and Arnaud Guillin. "On the rate of convergence in Wasserstein distance of the empirical measure." Probability Theory and Related Fields 162.3-4 (2015): 707-738.

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created Mar 1, 2017 at 20:13

Henry.L

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According to your motivation,

Numerical computation of $\rho$ is notoriously inaccurate and problematic, and so KL divergence or even $L^1$ distance are problematic as well. I obtain $f$ anyway, and I really only need to know how "far" is $\rho$ from being uniform.

And I assumed that your already obtained some information about the distribution of $f_i$, say moments. The moments of $f_i$'s are most easily obtained if you know the original $d\mu$ via simulation techniques like MCMC. Then I would suggest Wasserstein distance as a candidate since all you want is how dissimilar it looks from uniform, and Wasserstein distance is proven to perform well in this respect.

Your calculation does not have to involve $\rho_i$ but you should at least be able to sample from $d\mu$ and hence $f_i(X)$.

And the following paper verified that the convergence behavior is satisfying.

Fournier, Nicolas, and Arnaud Guillin. "On the rate of convergence in Wasserstein distance of the empirical measure." Probability Theory and Related Fields 162.3-4 (2015): 707-738.