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I have two GMMs that I used to fit two different sets of data in the same space, and I would like to calculate the KL-divergence between them.

Currently I am using the GMMs defined in sklearn (http://scikit-learn.org/stable/modules/generated/sklearn.mixture.GMM.html) and the SciPy implementation of KL-divergence (http://docs.scipy.org/doc/scipy-dev/reference/generated/scipy.stats.entropy.html)

How would I go about doing this? Do I want to just create tons of random points, get their probabilities on each of the two models (call them P and Q) and then use those probabilities as my input? Or is there some more canonical way to do this within the SciPy/SKLearn environment?

Andrew Latham
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  • The closed form doesn't exist. Take a look at this paper to get approximation of it. http://scholar.google.co.kr/scholar?cluster=17600982039879101400&hl=ko&as_sdt=0,5&authuser=1 – emesday Sep 28 '14 at 08:10

1 Answers1

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There's no closed form for the KL divergence between GMMs. You can easily do Monte Carlo, though. Recall that KL(p||q) = \int p(x) log(p(x) / q(x)) dx = E_p[ log(p(x) / q(x)). So:

def gmm_kl(gmm_p, gmm_q, n_samples=10**5):
    X = gmm_p.sample(n_samples)
    log_p_X, _ = gmm_p.score_samples(X)
    log_q_X, _ = gmm_q.score_samples(X)
    return log_p_X.mean() - log_q_X.mean()

(mean(log(p(x) / q(x))) = mean(log(p(x)) - log(q(x))) = mean(log(p(x))) - mean(log(q(x))) is somewhat cheaper computationally.)

You don't want to use scipy.stats.entropy; that's for discrete distributions.

If you want the symmetrized and smoothed Jensen-Shannon divergence KL(p||(p+q)/2) + KL(q||(p+q)/2) instead, it's pretty similar:

def gmm_js(gmm_p, gmm_q, n_samples=10**5):
    X = gmm_p.sample(n_samples)
    log_p_X, _ = gmm_p.score_samples(X)
    log_q_X, _ = gmm_q.score_samples(X)
    log_mix_X = np.logaddexp(log_p_X, log_q_X)

    Y = gmm_q.sample(n_samples)
    log_p_Y, _ = gmm_p.score_samples(Y)
    log_q_Y, _ = gmm_q.score_samples(Y)
    log_mix_Y = np.logaddexp(log_p_Y, log_q_Y)

    return (log_p_X.mean() - (log_mix_X.mean() - np.log(2))
            + log_q_Y.mean() - (log_mix_Y.mean() - np.log(2))) / 2

(log_mix_X/log_mix_Y are actually the log of twice the mixture densities; pulling that out of the mean operation saves some flops.)

Danica
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  • Hi Dougal, I am trying to use gym_js function that you defined to figure out the robustness of my model but not sure what the n_samples is doing here and how to interpret the return value? If I use all dataset that I have to make a models gmm_p and gmm_q with same number of clusters and feed into this function can I confirm the robustness of my model based on the js metric here? Thanks. – Mojgan Mazouchi Jun 18 '19 at 05:08
  • I came here to see if I can calculate the js divergence between two distributions and it looks like I could (the code adapted from this answer lies as answer here: https://stats.stackexchange.com/questions/345915/trying-to-implement-the-jensen-shannon-divergence-for-multivariate-gaussians) – Holi Jul 27 '19 at 13:47