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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Does minimizing KL imply boundedness of density ratios

Suppose $q^*(\theta) = \underset{q \in \mathcal{Q}}{argmin} \,\, KL[q(\theta)||p(\theta)] = \min_{q\in\mathcal{Q}} \int_{\theta\in\Theta}q(\theta)\ln\Big( \frac{q(\theta)}{p(\theta)} \Big)d\theta$ ...
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Meaning of entropy for a multivariate data set

In an earlier question, I asked, “Any meaning to the concept of ‘Self Mutual Information?” and got a great answer - thanks. Now, this begs another question/set of questions: What does the concept of ...
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KL divergence between two multivariate Gaussians with close means and variances

KL divergence between two Gaussian distributions denoted by $\mathcal{N}(\mathbf \mu_1, \mathbf \Sigma_1)$ and $\mathcal{N}(\mathbf \mu_2, \mathbf \Sigma_2)$ is available in a closed form as: $$\...
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Kullback-Leibler Divergence vs Normalized Cross Correlation

I have couple of time series data that I want to cluster. As I was looking for ways to calculate similarity for time series data, I came across couple of different similarity methods. What I am ...
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Mathematical meaning of minimizing JS divergence about GAN [closed]

Optimization of the loss function of GAN is equivalent to minimizing Jensen Shannon divergence, and minimization of cross-entropy loss, which is often used in classification problems such as image ...
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maximizing KL divergence as the objective function

As far as I know, the most common approach to train neural networks is to minimize the KL divergence between the data distribution and the output of the model distribution which results in minimizing ...
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Derivations of Forward and Reverse KL Divergence equations

In the Forward KL, the entropy has disappeared and in the Reverse KL, the entropy has a plus sign, why are they so?
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Use boostrap to compute confidence interval of KL statistic

I have two sample distributions p and q. samples in p are maybe identically distributed to samples in q or not. this is the task. samples within one group are iid. p and q have the same cardinality. ...
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Jensen–Shannon divergence as a distance measure between nonprobabilistic objects

We are working on an optimization problem. The objective function involves distance between data points. We tried a wide variety of distance measures and found the entropy-based measures, especially ...
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Kullback-Leibler (KL) divergence cutoff value

I am performing the KL divergence method to compare distributions of variables between two groups. I have a list of variables within different categories (award types, organization types, topics, etc) ...
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Maximum likelihood and Minimizing Kullback–Leibler divergence to the ecdf?! (i.e.: finite sample statement?)

There is a known relationship stating that finding MLE is asymptotically the same is minimizing Kullback–Leibler divergence (see wiki here), or just the cross entropy. I'm wondering if there is a ...
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“Any meaning to the concept of ‘Self Mutual Information?”

** “Any meaning to the concept of ‘Self Mutual Information?” ** A blog post entitled, “Entropy in machine learning” dated May 6, 2019 (https://amethix.com/entropy-in-machine-learning/) gave a very ...
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KL divergence between samples from a unknown distribution and a Normal distribution with zero mean and unit variance

If you draw samples of unknown distribution, how can you measure the KL-divergence between the unknown distribution and a gaussian distribution with zero mean and unit variance N(0,1)? Can we use ...
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Is Kullback-Leibler distance the same as the Kullback Information Criterion?

I have seen some old papers, where they seem to be referring to Kullback-Leibler divergence as the Kullback Information Criterion (e.g. Gourieroux et al (1987) "Kullback Causality Measures"). I just ...
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Statistical estimators distance for close empirical distributions

Is it valid to argue that two empirical distributions $ p_1, p_2 $ having small Wasserstein distance $W_r(\cdot)$ for an order $ r $ will yield close MLE estimators for a statistical model ...
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145 views

Kullback Leibler Divergence between two Normal Whishart Distributions

I'm having trouble to compute the KL Divergence between two normal-Wishart distributions. KL divergence from $Q$ to $P$ is defined as: $$D_{\mathrm{KL}}(P \Vert Q) = \int p(x) \log \frac{p(x)}{q(x)}...
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Clarifying notation for the Kullback-Leibler divergence in terms of expectations

We know that for discrete variables \begin{equation} D(p(x),q(x))=\mathbb{E}_{p}\left(\log\frac{p(x)}{q(x)}\right) \end{equation} where $p(x)$ and $q(x)$ are probability mass functions. Can this be ...
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Distance measure for two probability distribution of unequal sample size

Context: I have 100 stores and these stores are divided into 10 business markets. I want to select 3 markets where each market is a good representation of the 100 stores i.e. the population. There ...
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Kullback–Leibler and the Brier score?

Both seem to be quite obviously about prediction and sort of map one probability distribution onto another one. Whereas with the DKL (https://en.wikipedia.org/wiki/Kullback%E2%80%...
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KL divergence between two Asymmetric Laplace distributions?

Consider two asymmetric Laplace distribution $L_1(\mu_1,\sigma_1,\tau_1)$ and $L_2(\mu_2,\sigma_2,\tau_2)$ where \begin{equation} L(x;\mu,\sigma,\tau) =\frac{\tau(1-\tau)}{\sigma} \begin{cases} ...
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Which KL Divergence is larger D(P|Q) or D(Q|P)?

From the perspective of information theory, I understand how D(P|Q) is non-negative and why the KL divergence is asymmetric, i.e. $D(P|Q) \neq D(Q|P)$, given two gaussian univariate gaussian ...
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Maximum likelihood as minimizing the dissimilarity between the empirical distriution and the model distribution

I am reading Ian Goodfellow "Deep Learning" book. At page 128 it says One way to interpret maximum likelihood estimation is to view it as minimizing the dissimilarity between the empirical ...
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Computing KL Divergence for distributions over sets

I have a distribution over a set of (hundreds of) discrete terms, and I'd like to describe the difference between I see a couple of options, and none seems really attractive: Take the KL divergence ...
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597 views

Gradients of KL divergence and ELBO for variational inference

When doing variational inference, due to intractability we typically maximize the evidence lower bound (ELBO) instead of minimizing Kullback-Leibler divergence (KLD) between our approximate and exact ...
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Comparing two coin tossing experiments using Kullback-Leibler divergence

In a coin-tossing, suppose two people have conducted two separate experiments to find out the probability of head $p_H$ of the same coin. They both are Bayesians and have started from the prior $$...
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How is Akaike Information Criterion related to Information theory?

How is the Akaike Information Criterion (AIC) related to Information theory ? I mean from the equation (below), it is not at all intuitive how information theory comes into picture. Also is AIC ...
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How to solve for the minimum KL Divergence when the distribution is discrete?

Say we have a simple case of $p(x,y)$ is a 3x3 matrix: $$\begin{bmatrix} 1/6 & 0 & 0 \\ 1/6 & 3/6 & 0 \\ 0 & 0 & 1/6 \end{bmatrix}$$ And $q(x,y)=...
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Use Effect Size VS p-Value when determining the best fitting distribution?

I have a large sample size of about 50,000 I want to determine what theoretical distribution fits the distribution of my samples the best. What I did is to fit all distributions I know to the data ...
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Proof help for multivariate mutual information as a sum of entropies

I'm following this paper on ICA and I got to equation (1) describing the multivariate mutual information contrast function as a sum of entropies. $J(Y) = \int p(y_1,...y_D)log(\frac{p(y_1,...y_D)}{p(...
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Variational inference with dependent variables

Classical variational inference uses mean field theory because of its computational benefits, i.e. assumes that the latent variables are independent. For gaussian distribution, it wants to find a ...
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Different optimization behaviours on delta vs non-delta targets

I built a simple classification model that is required to predict a probability distribution for a set of two available classes. The target distributions are not necessarily delta distributions. i.e ...
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The expected log-Likelihood in Kullback Leibler Divergence

Given a true normal distribution $g(x)$ with mean $\mu_G$ and variance $\sigma_G$, and a model $f(x)$, the KL divergence involves the expected log-likelihood $\mathbb{E}_G[log f(x|\theta]$. The ...
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Why does variational bayes use $KL(Q || P)$ and not $KL(P || Q)$

In variational Bayes, we approximate the intractable posterior $P(Z | X)$ with a tractable $Q(Z)$ and minimize $KL(Q || P)$. Why do we not minimize $KL(P || Q)$ instead?
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Weigh Kullback Leibler Divergence with P entropy?

I'm wondering if it makes sense to weight Kullback-Leibler Divergences to highlight divergences on highly distinguishing features. I am however not very well versed in Information Theory, and would ...
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KL Divergence of two standard normal arrays

I generated two 9000,1 np arrays with a = np.random.standard_normal(9000) b = np.random.standard_normal(9000) Then I check the KL Divergence with ...
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Expectation of Log Likelihood Function with given parameters Proof

I have been looking for AICc value derivation employing Kullback-Leibler distance but as a result of my search I got stuck with expectation of loglikelihood. In the link loglikelihood is given as $InL(...
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How to show an alternate data processing inequality concerning KL divergence between conditionals?

Suppose $(Y,X) \sim F \in \mathcal{P(\mathbb{R^d})}$. Consider an arbitrary transformation $f$ that acts on $X$. My intuition is that the following should be a result in information theory: $$ \...
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Meaning of expectation with respect to a function?

This might be a trivial question, but I've come across a paper where some expectation is said to be taken with respect to some pdf. See example: How am I to interpret this, and is there some ...
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Variational inference with deterministic dependencies between variables

Suppose I have a probabilistic graphical model shown in the picture, in which all variables are binary, $c_1$ and $c_2$ are observed, and I want to use mean-field variational inference to estimate ...
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KL divergence minimization

While reading Unsupervised Data Augmentation for Consistency Training, I came across an equation that describes the minimization of KL divergence. $$\min_\theta \mathscr{J}_{UDA}(\theta) = \mathbb{E}...
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Expectation of the log-likelihood under the posterior

Suppose $L(X \mid \theta)$ is a likelihood function, i.e., a probability distribution over $X \in \mathcal{X}$ indexed by a parameter $\theta \in \Theta$. Suppose further we have a prior $\pi(\theta)$,...
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225 views

Measure the distance between two probability transition matrices

I have a probability transition matrix $P$ that contains some values very close to zero. I want to sparsify this matrix by taking the k largest values for each row and setting the others to zero. For ...
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How to minimise and expectation with respect to a parameter?

Suppose that $X$ is a random variable with distribution $G$. Let $H(X;\theta)$ be a parametric function with $\theta \in \Theta \subset {\mathbb R}^p$. I want to maximize the function $$\varphi(\...
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How to use KL divergence to compare two distributions?

I am trying to model the probability distribution of a multi-dimensional dataset where all the values are discrete. Suppose the training data (represented by T) is of the shape (m, n) where n is the ...
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KL divergence of a uniform prior and a custom posterior

So I was reading the Google's paper on VQ-VAE and have stumbled upon the derivation of KL divergence of the uniform prior and the given distribution: $$q(z=k \mid x)=\left\{\begin{array}{ll}{1} & ...
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How to prove KL(q(z)|p(z)) = E_q(z) [ log f(z) ] where f is the CDF of p?

As title. It was used in https://arxiv.org/abs/1905.10549 without proving.
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Markov type bound based on KL divergence?

Given two discrete distribution $p, q$ on some universe $U$, if I know they have a bounded KL divergence, say some number $c$, can I say anything about how much each point in the universe differs in ...
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Why is Kullback-Leilbler divergence a better metric for measuring distance between two probability distributions than squared error? [duplicate]

I know that KL-divergence is a metric that is more suitable when we want to measure the distance between numbers which a probability form. However, I am still confused what is the benefit of using KL-...
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A divergence that can be extended to logistic functions?

If I have data $\{(x_i, y_i)\}_{i=1}^n$ where the dependent variable is binary $(y_i = 0,1)$ I can model it using a logistic function: $$f(x; \alpha, \beta) = \frac{1}{1 + e^{-(\alpha + \beta x)}}$$ ...
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Relation Between Wasserstein Distance and Relative Entropy

Consider the Wasserstein metric of order one $W_1$ (aka the Earth Movers Distance). I would like to know whether it is possible to link $W_1$ and relative entropy and what this would mean intuitively. ...

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