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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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Conditional KL Divergence in Clustering Paper

I am trying to implement the following paper on Self-taught Clustering https://www.cse.ust.hk/~qyang/Docs/2008/dwyakicml.pdf. I have the following three co-clustering functions: where p̃(Z|x̃) is Z ...
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Relative Entropy decomposition

Can the relative entropy (Kullback Leibler divergence) between multivariate distributions be decomposed into relative entropies of the different variables plus some measure of dependence between the ...
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How to measure the “variability” across a set of many (>>2) probability distributions?

Given a set of many of discrete probability distributions, is there a way I can efficiently calculate a metric that quantifies how different the entire set of these probability distributions are to ...
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KL-Diverence of Q(z|X) and P(z) in Variational Autoencoder (VAE)

I aim to understand how $D_{KL}[Q(z | X) || P(z)]$ can be converted to $\frac{1}{2} \sum_{k} (\Sigma(X) + \mu^{2}(X) - 1 - \log \Sigma(X))$, where $k$ is the dimension of the Gaussian distribution. ...
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Kullback-Leibler Divergence - Distance between two univariate gaussian distribution when their means are equal [duplicate]

We find the KL divergence between two univariate Gaussian distributions: p(x) and q(x). When the mean is same for both the distributions , Which KL divergence is larger, D(P||Q) or D(Q||P).
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Kullback-Leibler divergence with huge values? [closed]

I was playing with the KL divergence. My simple example is to calculate the divergence between two 2-dimensional normal distribution using PyTorch. The code for doing it is just below. I get an ...
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Confusion about maximising vs minimising (relative) entropy terminology and methods

I'm studying rules of inference for updating from a prior probability distribution to a posterior. One method for doing this is by maximising entropy, subject to constraints. I'm reading papers like ...
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Can we apply KL divergence to the probability distributions on different domains?

When I was reading the original paper of t-SNE, I had an question whether or not we can apply KL divergence to the discrete probability distributions on different domains. In the paper, they measure ...
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Difference of notation between cross entropy and joint entropy

Although it is clear to me, how the two concepts differs, it has been difficult for me to find a notation that would make it clear, to which type of entropy we refer. From wikipedia, we can see that ...
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Kullback-Leibler Divergence as the class separation measure

I am trying to use KL divergence as the separation measure between the classes. I have the positive, negative samples for 2 distributions and want to adjust the algorithm parameters to get the best ...
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Solution of $D_{KL}(q_\phi(z)||p_\theta(z))$ of Gaussian case

Following is from the original paper of concept of VAE(variational autoencoder) by Kingma,Welling 2014 B. Solution of $D_{KL}(q_\phi(z)||p_\theta(z))$ of Gaussian case The variational lower bound (...
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KL Divergence with different domains

I want to calculate KL Divergence between a normal and an exponential r.v. i.e. $$D(P||Q) = ?\\ \;\; P=N(\mu,\sigma), \;\; Q=exp(\lambda)$$ My problem is that in this case the domains of the ...
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Validate a model with Posterior Predictive Checks, Kernel Density estimation and Kullback-Leibler

I'm wondering if this scheme is a good way to validate a model. Generate new data $y_{new}$ from Posterior Predictive distribution PPC given observed data $y_{obs}$ Use Kernel Density Estimation in ...
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Derivation of ELBO upon the Existence of Conditional Latent Variable Model

I am reading the recently published paper from DeepMind, "Neural Scene Representation and Rendering" and especially its "Supplementary Materials". Following is the page 1 and it's pretty hard for ...
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Finding the value of KL divergence to determine whether one distribution is distrinct from another?

Given the KL divergence value between 2 distributions, how should someone use this to determine whether the value is significant for the distributions $P$ and $Q$ to be different? One method I can ...
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Distance between angle distributions

I want to quantify the complexity of the street network of different cities. For each city I have the angle distribution of its streets. My hypothesis that the more complex the street network, the ...
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Why minimize the KL divergence from data to the model instead of KL divergence of model to data?

When minimizing the KL divergence in machine learning, why the expression of KL is from data to model, instead of from model to data ?
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KL divergence between which distributions could be infinity

I know that KL divergence measures difference between two probability distributions. My doubt is for which of the distributions it could become Infinity, putting it in another way, ...
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What is the difference Cross-entropy and KL divergence?

Both of Cross-entropy and KL divergence are tools to measure the distance between two probability distribution. What is the difference? $$ H(P,Q) = -\sum_x P(x)\log Q(x) $$ $$ KL(P | Q) = \sum_{x} P(...
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Rate of converge of KL-divergence to posterior

Suppose you have samples from some distribution P. You have a prior distribution Q, which represents your estimate of P, and assume for now that it's parameterized the same way as P. Upon observing ...
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Difference between Empirical distribution and Bernoulli distribution

I've been studying binary cross entropy error for binary classification weight optimization. From my knowledge, Cross entropy itself quantifies divergence between two probability distributions with ...
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Why Kullback-Leibler in Stochastic Neighbor Embedding

Stochastic Neighbor Embedding (and t-SNE) relies on Kullback-Leibler divergence between the point distributions in the original and the low-dimensional space. Why? Why not any other dissimilarity ...
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Choosing an appropriate statistical distance which punishes entropy

Problem Description: I have with me experimental statistics of a system and I wish to fit a theoretical model so that the computed statistics on the model fit the experimental ones. I am using an RBM ...
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Similarity of two sets of points

PROBLEM I have one set of 10 points, $X = \{(x_1,y_1),\,\dots,\,(x_{10},y_{10})\}$ and two sets of 3 points each, $A = \{ (a_1,b_1),\, (a_2,b_2),\, (a_3,b_3) \}$ and $C = \{(c_1,d_1),\,(c_2,d_2),\,(...
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What's the maximum value of Kullback-Leibler (KL) divergence

I am going to use KL divergence in my python code and I got this tutorial. On that tutorial, to implement KL divergence is quite simple. ...
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Comparing distribution A to B and C

I have three discrete probability distributions, A, B and C. They are all measuring P(X) under different circumstances. I suspect that A is more similar to B than it is to C. I know that I can compare ...
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Kullback–Leibler divergence when one measure is a sum of diracs

In the book "Deep Learning" of Goodfellow, Bengio and Courville, section 5.5 of maximum likelihood estimation they explain a relation between the maximization of likelihood and minimization of the K-L ...
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What is the intuition regarding the different viewpoints of the Kullback-Leibler Divergence and the Kolmogorov-Smirnov Statistic? [duplicate]

In trying to understand the Kullback-Leibler Divergence I conceive it as a metric that if minimized would make the Approximation PDF Q as close as possible to the True PDF P either in the ...
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Calculating Jensen-Shannon and Kullback-Leibler for density splines

I need to find a way to calculate the JS distance (and, by necessity, the KL divergence) between continuous, non-normal (generating distribution unknown) distributions. I have many classes, and ...
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Does this bounded, continuous probability distribution have a name?

Does this bounded, continuous probability distribution over $x$ have a name? $P(x|y) \propto \big(\frac{y}{x}\big)^x\big(\frac{1-y}{1-x}\big)^{(1-x)}$ for $x, y \in (0,1)$. This comes about by ...
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If the AIC is an estimate of Kullback-Leibler Divergence, then why can AIC be negative when KL divergence is always positive?

I have read many times that the AIC serves as an estimate of the KL divergence, and I know that AIC can be a negative value (and have seen that myself). Yet, the KL divergence must always be positive. ...
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KL divergence invariant to affine transformation?

I read in this tutorial on page 20 that $KL$ divergence is invariant to affine transformation, but I think it is incorrect. Say we have two 1D normal distributions $P_{1}(x) = \mathcal N(\mu_{1}, \...
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KL Divergence between parallel lines

I am trying to understand the example described in the WGAN paper about learning parallel lines with various divergences. More specifically the setup is as follows: Let $Z \sim [0, 1]$ the uniform ...
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Simulating KL Divergence between Cauchy RV and the MLE estimate of the RV - Multimodality seems wrong

I'm working on a (what I think is a fairly simple, straightforward) explanation of how it's really hard to approximate distributions with fat tails accurately in the tails. I started looking at an ...
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What constitutes a large KL divergence?

I have 2 gamma distributions $X_1 \sim Ga(13,1) \\ X_2 \sim Ga(3,1)$ Where we are defining our gamma distribution probability density function for a random variable $X \sim Ga(a,b)$ to be $f(X) = \...
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Generating distributions specifically distanced from the ground truth - (Betting strategy simulation)

Thanks in advance for your patience and helpful answers. I am new to cross validated so hopefully I can word this correctly. I am developing a betting strategy and I'd like to use a rigorous testing ...
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Connection between log predictive density and Kullback-Leibler information measure

I have read in Bayesian Data Analysis by Andrew Gelman that log predictive density can be used to compare Bayesian models due to its connection to the Kullback-Leibler information. The log ...
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Calculate the Kullback-Leibler Divergence for these 2 Gamma distributions

I have 2 models $P \sim Ga(115,1329.914) \\ Q \sim Ga(140,650.6775)$ and I'm looking to calculate the K-L divergence of these 2. $D_{KL}(P||Q) = \int_\infty ^\infty p(x) log \frac{p(x)}{q(x)}\,dx$...
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How to picture EM algorithm and KL-divergence geometrically?

In reading up on the Expectation-Mmaximization algorithm on Wikipedia, I read this short and intriguing line, under the subheading "Geometric Intuition": In information geometry, the E step and the ...
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Why KL divergence is non-negative?

Why is KL divergence non-negative? From the perspective of information theory, I have such an intuitive understanding: Say there are two ensembles $A$ and $B$ which are composed of the same set of ...
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191 views

Upper bound on KL divergence

Is there a maximum (unique?) to the KL divergence between discrete distributions p & q, with the restriction that q is a proper probability distribution? I know KL is unbounded from above when q ...
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2answers
516 views

Why don't we use a symmetric cross-entropy loss?

Machine learning classifiers often use the cross-entropy $\mathbb{H}[p,q]$, where $p$ is the true distribution (often a delta) and $q$ is the predicted distribution over classes (or can at least be ...
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Distance or divergence for ordinal distribution

Measures like KL divergence can be symmetrized (into JS divergence). Bhattacharyya distance serves a similar function. Either is well-suited to both continuous distributions and discrete (e.g. ...
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need help in understanding a research paper… specifically related to KL divergence

In this paper https://dl.acm.org/citation.cfm?id=2002654 specifically section 2.4. I understood that Review rating is modelled as random variable with guassian distribution with mean as wt * or. I ...
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Comparing ELBO of a VAE for different samples

I am lacking of an interpretation of the evidence lower bound (ELBO), when comparing two different samples $x_1, x_2 \sim X$. Writing the marginal log-likelihood as the sum of lower variational bound ...
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How to justify a KL divergence when a distribution contains continuous and discrete components

Like in contamination models, some distributions have discrete component. e.g. $p(x) := (1 - \epsilon) q(x) + \epsilon \delta_{x_0}(x)$ In these distributions, is there a way to justify a definition ...
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Algorithm for highly correlated columns in fixed dictionary for Kullback-Leibler divergence problem?

According to this paper, l1-regularized linear least-squares problem with fixed dictionary is efficient. But if the columns of dictionaries are in highly correlated, LARS-Lasso algorithm provides ...
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Estimating the Kullback-Leibler divergence between exact distribution and its M-projection

My question is quite general. But, I prefer to illustrate it with the actual simple problem I am facing. Suppose that I have a naive Bayesian network (a star-like model). I want to marginalize out ...