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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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Expectation of multivariate gaussian w.r.t. other multivariate gaussian

I want to calculate the Kullbach Leibler Divergence of two multivariate Gaussians as in KL divergence between two multivariate Gaussians. At one point one has to solve the following expression (...
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KL divergence between sample and true (multivariate normal) distribution

I was wondering, whether there is a possible interpretation of the KL-Divergence between sample and true distribution in terms of probabilities. E.g. given $P=\mathcal{N}\left(\mu,\Sigma\right)$ and $...
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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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What scenario corresponds to choosing the “true distribution” $p$ in $\textsf{KL}(p\parallel q)$?

I understand that when you think about changing $q$ in the Kullback-Leibler divergence $\textsf{KL}(p\parallel q)$, this corresponds to trying to find the distribution that minimizes information loss ...
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rewriting ELBO to highlight the role of priors

I am reading this paper which rewrites ELBO. I am stuck in verifying the mathematics used for doing the rewriting. Essentially, the paper writes the KL term involved in ELBO as follows (equations 13 ...
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compute the KL divergence between two datasets

I have two datasets $D1$ and $D2$ in two different feature spaces $\mathcal{X}_{1} \in \Re^{m}$ and $\mathcal{X}_{2} \in \Re^{n}$. Further assume that the datasets have different number of data points....
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KL divergence of models with both continuous and discrete variables

When $x$ is discrete, KL divergence is $D_{KL}(P||Q)=\sum\limits_{x}P(x)\log \frac{P(x)}{Q(x)}$, when $x$ is continuous, $D_{KL}(P||Q)=\int\limits_{x}p(x)\log \frac{p(x)}{q(x)}dx$. However, when the ...
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Relation between the AIC and the Kullback-Leibler Divergence

I am searching a formal derivation of the Akaike Information Criterion from the Kullback-Leibler Divergence. Can you show me one, or point me toward a book/article in which this is done? Here I set ...
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A different proof for KL divergence non-negativity

KL divergence's non-negativity can be proved in many ways. One could use the inequality $\log x \leq x - 1$ as a main step in the proof, another one could leverage the property of concave of the ...
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clustering with KL divergence

Recently I read some papers which related to clustering. The paper is https://xifengguo.github.io/papers/ICONIP17-DCEC.pdf In this paper, they calculate Loss function as KL Divergence, KL(P||Q). ...
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Are there two motivations for Bayesian information criteria?

Are there two motivations for all these Bayesian information criteria? I am only aware of the motivation of "expected out-of-sample prediction score." Let the in-sample data be $y$ and the parameter ...
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50 views

Bayesian consistency in compact uncountable parameter space

Let $p(y_i \mid \theta)$ be the likelihood we are using of a single data point, $p(\theta)$ be the prior, and $f(y_i)$ the true distribution of the data. Also, let $\theta_0$ be the parameter that ...
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Expectation of KL between Categorical and samples from a Dirichlet

Is there a nice closed form expression for $\mathbb{E}_{\theta' \sim Dir(\alpha)} KL (Cat(x; \theta)|| Cat(x;\theta')$, where $Dir(\alpha)$ is the Dirichlet distribution with concentration parameters $...
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Kullback–Leibler Divergence of two 2-dimensional probability distributions

I need to calculate a KL divergence (and others) in the following question. But I have a hard time to understand the meaning of this syntax. Isn't P a function of (x,y)? I wasn't able to find any ...
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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 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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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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125 views

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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3answers
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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) = \...