Questions tagged [kullback-leibler]

An asymmetric measure of distance (or dissimilarity) between probability distributions. It might be interpreted as the expected value of the log likelihood ratio under the alternative hypothesis.

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KL divergence between two univariate Gaussians

I need to determine the KL-divergence between two Gaussians. I am comparing my results to these, but I can't reproduce their result. My result is obviously wrong, because the KL is not 0 for KL(p, p). ...
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Intuition on the Kullback-Leibler (KL) Divergence

I have learned about the intuition behind the KL Divergence as how much a model distribution function differs from the theoretical/true distribution of the data. The source I am reading goes on to say ...
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KL divergence between two multivariate Gaussians

I'm having trouble deriving the KL divergence formula assuming two multivariate normal distributions. I've done the univariate case fairly easily. However, it's been quite a while since I took math ...
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Why do we use Kullback-Leibler divergence rather than cross entropy in the t-SNE objective function?

In my mind, KL divergence from sample distribution to true distribution is simply the difference between cross entropy and entropy. Why do we use cross entropy to be the cost function in many machine ...
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Kullback–Leibler vs Kolmogorov-Smirnov distance

I can see that there are a lot of formal differences between Kullback–Leibler vs Kolmogorov-Smirnov distance measures. However, both are used to measure the distance between distributions. Is there a ...
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Differences between Bhattacharyya distance and KL divergence

I'm looking for an intuitive explanation for the following questions: In statistics and information theory, what's the difference between Bhattacharyya distance and KL divergence, as measures of the ...
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An adaptation of the Kullback-Leibler distance?

Look at this picture: If we draw a sample from the red density then some values are expected to be less than 0.25 whereas it is impossible to generate such a sample from the blue distribution. As a ...
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Measures of similarity or distance between two covariance matrices

Are there any measures of similarity or distance between two symmetric covariance matrices (both having the same dimensions)? I am thinking here of analogues to KL divergence of two probability ...
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Kullback-Leibler divergence WITHOUT information theory

After much trawling of Cross Validated, I still don't feel like I'm any closer to understanding KL divergence outside of the realm of information theory. It's rather odd as somebody with a Math ...
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What is the relationship between the GINI score and the log-likelihood ratio

I am studying classification and regression trees, and one of the measures for the split location is the GINI score. Now I am used to determining best split location when the log of the likelihood ...
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Can someone prove the following connection between Fisher information metric and the relative entropy (or KL divergence) in a purely mathematical rigorous way? $$D( p(\cdot , a+da) \parallel p(\cdot,... 2answers 4k views What is the advantages of Wasserstein metric compared to Kullback-Leibler divergence? What is the practical difference between Wasserstein metric and Kullback-Leibler divergence? Wasserstein metric is also referred to as Earth mover's distance. From Wikipedia: Wasserstein (or ... 2answers 9k views Kullback-Leibler divergence - interpretation [duplicate] I have a question about the Kullback-Leibler divergence. Can someone explain why the "distance" between the blue density and the "red" density is smaller than the distance between the "green" curve ... 3answers 11k views Calculate the Kullback-Leibler Divergence in practice? I am using KL Divergence as a measure of dissimilarity between 2 p.m.f. P and Q.$$D_{KL}(P||Q) = \sum_{i=1}^N \ln \left( \frac{P_i}{Q_i} \right) P_i=-\sum P(X_i)ln\left(Q(X_i)\right) + \...
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I am comparing two distributions with KL divergence which returns me a non-standardized number that, according to what I read about this measure, is the amount of information that is required to ...
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Interpretation of Radon-Nikodym derivative between probability measures?

I have seen at some points the use of the Radon-Nikodym derivative of one probability measure with respect to another, most notably in the Kullback-Leibler divergence, where it is the derivative of ...
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How to calculate Kullback-Leibler divergence/distance?

I have three data sets X, Y and Z. Each data set defines the frequency of an event occurring. For example: Data Set X: E1:4, E2:0, E3:10, E4:5, E5:0, E6:0 and so on.. Data Set Y: E1:2, E2:3, E3:7, ...
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Hypothesis testing and total variation distance vs. Kullback-Leibler divergence

In my research I have run into the following general problem: I have two distributions $P$ and $Q$ over the same domain, and a large (but finite) number of samples from those distributions. Samples ...
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Distance between two Gaussian mixtures to evaluate cluster solutions

I'm running a quick simulation to compare different clustering methods, and currently hit a snag trying to evaluate the cluster solutions. I know of various validation metrics (many found in cluster....
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Is it possible to apply KL divergence between discrete and continuous distribution?

I am not a mathematician. I have searched the internet about KL Divergence. What I learned is the the KL divergence measures the information lost when we approximate distribution of a model with ...
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Why does the Bayesian posterior concentrate around the minimiser of KL divergence?

Consider the Bayesian posterior $\theta\mid X$. Asymptotically, its maximum occurs at the MLE estimate $\hat \theta$, which just maximizes the likelihood $\operatorname{argmin}_\theta\, f_\theta(X)$. ...
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Population stability index - division by zero

Population stability index quantifies the change of a distribution of a variable by comparing data samples in two time periods. It is very commonly used to measure shifts in scores. It is calculated ...
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Jensen Shannon Divergence vs Kullback-Leibler Divergence?

I know that KL Divergence is not symmetric and it cannot be strictly considered as a metric. If so, why is it used when JS Divergence satisfies the required properties for a metric? Are there ...
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Estimate the Kullback Leibler (KL) divergence with monte carlo

I want to estimate the KL divergence between two continuous distributions f and g. However, I can't write down the density for either f or g. I can sample from both f and g via some method (for ...
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Why is there a E in the name EM algorithm?

I understand where the E step happens in the algorithm (as explicated in the math section below). In my mind, the key ingenuity of the algorithm is the use of the Jensen's inequality to create a lower ...
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Multivariate time series model evaluation with conditional moments

Consider multivariate time series models that estimate potentially time-varying conditional means, variances, and correlations (one type of model might be a VAR(p)+Garch(1,1)+DCC Gaussian Copula model)...
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Kullback–Leibler divergence between two Wishart distributions

The result is shown in: [1] W.D. Penny, KL-Divergences of Normal, Gamma, Dirichlet, and Wishart densities, Available at: www.fil.ion.ucl.ac.uk/~wpenny/publications/densities.ps But could anyone help ...
I've been working on building some test statistics based on the KL-Divergence, $$D_{KL}(p \| q) = \sum_i p(i) \log\left(\frac{p(i)}{q(i)}\right),$$ And I ended up with a ...