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16 views

Information theory without normalization

I'd like to know if there is a way anyone knows of for doing information theory with unnormalized densities. Specifically, I hav two log likelihoods $\phi(x), \psi(x)$ and so I can write: $p(x) = ...
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0answers
14 views

Kullback Leibler divergence “efficient” upper bound

For a distribution of N values, how can I efficiently upper-bound the largest divergence between all non-negative distributions over the same random field? For example, for all distributions of a ...
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0answers
18 views

Want to findout satistical distance unsing another procedure except kullback liebler divergence

i want to find the distance between two pdf(pdf is calculated using kernel density estimator from two random data set of different size ) .Is there any alternative and efficent way to calculate ...
2
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2answers
67 views

Why KL-Divergence uses “ln” in its formula?

I notice in KL-Divergence formula a $ln$ function is used: $${D_{KL}}(P||Q) = \sum\limits_i {P(i)} \ln \frac{{P(i)}}{{Q(i)}},$$ where $i$ is a point and $P(i)$ the true discrete probability ...
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1answer
58 views

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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0answers
16 views

lower bound on square root of KL divergence

$\mu_1$ and $\mu_2$ are two probability measures on $\Omega$. Assume they have probability mass functions $f_1$ and $f_2$. I wonder if the following is a correct usage of Jensen' inequality: $$ ...
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0answers
26 views

Difference between two data distributions

I have the following problem: I have two datasets: A and B. From each of the datasets, I extracted the same features. I also know the data labels. So, I would like to perform 10-fold CV to see how a ...
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0answers
30 views

Occupancy octree metrics (Kullback-Leibler)

As I'm currently working on scan matching for outdoor environments I was wondering about the best metric to compare two occupancy octrees (one resulted from the scan matching and one ground truth ...
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3answers
120 views

Analysis of Kullback-Leibler divergence

Let us consider the following two probability distributions ...
1
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0answers
51 views

Neuroscience Equations

I am trying to understand a neuroscience article by Karl Friston. In it he gives three equations that are, as I understand him, equivalent or inter-convertertable and refer to both physical and ...
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1answer
90 views
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110 views

Under what conditions will Kullback-Leibler divergence/mutual information be infinity?

For two perfectly correlated Gaussian variables, the mutual information between them, and thus the KL divergence between the product of the marginal distributions and the joint distribution, is ...
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0answers
40 views

How to estimate similarity between several probability distributions?

I have several set of probability distributions. I want to reliably estimate consistency across distributions inside each set. Literature contains methods to compare two distributions: ...
0
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0answers
40 views

sufficient statistic and KL-divergence: Confusion with an equation

I am reading a paper, which talks about minimising KL-divergence of any arbitrary distribution over a family of exponential distribution. So, given a distribution $p$, we want to compute its ...
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0answers
43 views

computing KL divergence: M projections for arbitrary distributions

Background I have a generative model for a process that can be described as follows: $$ y = t(x, w) + e $$ where $x$ and $y$ observations of a set of random variables which are related by a ...
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0answers
41 views

Kullback-Leibler divergence Pareto Distribution

What is the Kullback-Leibler divergence for a Pareto Distribution? Given $p(x)$ = $ \alpha$ $\frac{x^{\alpha}_{min,1}}{x^{a+1}}$.
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3answers
2k 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) + ...
3
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1answer
130 views

KL divergence between an uninformative (?) Gaussian and a Gaussian

I have to calculate the KL divergence between a distribution $q$ and a prior distribution $p$, both of which are univariate Gaussians, i.e. $KL(q|p), q \sim \mathcal{N}(\mu, \sigma^2), p \sim ...
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0answers
62 views

Problem with Kullback–Leibler divergence criteria

I am using Kullback–Leibler divergence criteria for comparing my estimation and true density functions, but I have zero value on my estimation function when I have a testing set of size 10000, mostly ...
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2answers
314 views

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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0answers
133 views

Kullback-Liebler Divergence function in R package “FNN” returns NAN for certain data? Why?

I am using the "FNN" package to calculate the Kullback-Liebler Divergence between two numeric vectors: require(FNN) load("~/xy.RData") KL.divergence(x,y) (Link ...
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0answers
56 views

Quantify the information lost given by the Kullback-Leibler divergence measure

Consider there are $N$ individuals and these measure a quantity $X\in \mathbb{R}^{N\times M}$ where $M$ is the number of measurements and let $P(X)$ denote a probability distribution over $X$. The ...
1
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1answer
120 views

How to test whether a series data follow Ornstein-Uhlenbeck process (OU process)?

I have some measures which seems to have some mean-reverting properties and I'm wondering whether they can be modeled as Ornstein-Uhlenbeck process (OU process). And actually I quite expect it because ...
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1answer
33 views

How to use Kullback–Leibler divergence to help me choose the best distribution function?

I am now using Exponential distribution to model the the time intervals of a sequence of random events.Since I can choose several different lamdas for this model , I want to find out which lamda of ...
1
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1answer
56 views

KL divergence minimisation equation

I am looking at some literature on KL divergence minimisation and am having trouble understanding the derivation of the second order moment. So, if we have a distribution from the exponential family, ...
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0answers
24 views

Modeling a list with a tunable degree of disorder/shuffling

Imagine we have a list of ordered numbers $L = (1, 2,\dots, N)$. I want to add an arbitrary amount of "disorder" to that list. For instance: Adding a little bit of disorder would permute a few ...
1
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1answer
56 views

Renyi divergence identity

I'm reading the paper, T. van Erven and P. Harremoës, Rényi Divergence and Kullback-Leibler Divergence, arXiv 1206.2459 on the Renyi divergence, and I'm trying to make sense of "Example 1". I think ...
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0answers
47 views

Formal statistical test for comparing likelihood distributions obtained via MCMC

I am trying to formally compare the distribution of the likelihood values generated using two different models with marginal posterior values of the parameters obtained using MCMC in order to assess ...
2
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0answers
179 views

Disadvantages of the Kullback-Leibler divergence

I'm working on a calibration problem which involves the usage of the Kullback-Leibler divergence as an error between some empirical distribution $p$ and a theoretical distribution $q$. In the model, ...
2
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0answers
74 views

Does relative Kullback-Leibler divergence exist?

Suppose I have two multivariate normal distributions. I have computed the KL divergence ($d_{LK}(N_1, N_2)$). Is there a way to measure a relative divergence between these two distributions? For ...
3
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1answer
413 views

KL-divergence between two categorical/multinomial distributions gives negative values?

If $$P = [0,0.9,0,0.1]$$ $$Q = [0,1,0,0]$$ Then $$KL(P||Q) = 0 + \ln(0.9/1)\cdot0.9 + 0 + 0 = -0.094$$ This shouldn't be possible from the Gibbs inequality. What am I misunderstanding?
3
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1answer
155 views

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 ...
3
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1answer
101 views

G-test statistic and KL divergence

According to Wikipedia, the G-test statistic is "proportional to the Kullback–Leibler divergence of the empirical distribution from the theoretical distribution." To get the relationship between $ G $ ...
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2answers
427 views

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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1answer
341 views

Symmetrised Kullback - Leibler divergence

I have trouble understanding KL divergence, where P is probability mass function of true distribution of data and Q is the approximation of P. The definition of KL divergence is: ...
3
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1answer
601 views

Kullback-Leibler divergence of two normal distributions

I was recently trying to find a way to compute the KL-divergence between 2 populations that are normally distributed using the mean and variance of each population. But I found several different ...
3
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1answer
109 views

Select best distance for feature selection

Suppose I have matrix $X \in R^{n \times m}$, where $n$ is the number of individuals and $m$ is the number of features and $X[i,j] \in \{0,1\}$; $1$ indicates that the individual $i$ has the feature ...
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1answer
2k views

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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2answers
2k views

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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2answers
614 views

Symmetric Kullback-Leibler divergence OR Mutual Information as a metric of distance between two distributions?

I need some metric of divergence of two distributions. (They are complex and don't fit with exponential family, normal, log-normal, power-law. Maybe some mixture of that, but I'm not feeling right ...
4
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3answers
1k views

Is the square root of the symmetric Kullback-Leibler divergence a metric?

It is well known that the square root of the Jensen-Shannon divergence is a true metric, but how about the square root of symmetric KL: D(P||Q)+D(Q||P)? I have reasons to believe that it also is a ...
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0answers
27 views

Is there any meta-approach for variable selection based of measures of similarity between each two variables?

Is there any meta-approach ( or mayby I should say universal approach which works with different measures ) for variable selection which is based on similarity matrix which every entry ...
2
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0answers
190 views

Comparing divergence from uniform distributions with differing supports (discrete)

Imagine we have a potentially biased coin and a potentially biased six-sided die and we want to know which is more biased than the other. Firstly, is this a reasonable goal? Could it make sense to ...
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2answers
900 views

Bhattacharyya distance for histograms

One of the ways to measure the similarity of two discrete probability distributions is the Bhattacharyya distance. In computer vision, for example, it is used to evaluate the degree of similarity ...
4
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0answers
84 views

Orthogonal intersection in a Riemannian manifold

Let $S$ be the set of all probability distributions on $\mathbb{R}$ and $S_n=\{p_\theta\}$ be an $n$ dimensional submanifold of parameterized family of probability distributions on $\mathbb{R}$ where ...
3
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1answer
147 views

Orthogonal intersection of linear family and exponential family

I asked the following question in MSE for which I couldn't get any answer yet. I thought this would be a better place for that question. In statistical maniolds ...
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2answers
260 views

Connection between Fisher metric and the relative entropy

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) \| p(.,a) ) ...
2
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1answer
410 views

Calculating Hellinger Divergence from Results of Kernel Density Estimates in Matlab

Using the ksdensity function in matlab returns a density estimation in the form of 2 vectors f and xi. Where f are the density values and xi the corresponding points for the density values. How do I ...
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0answers
207 views

Kullback-Leibler vs Hellinger Distance

I am working on this problem in which I have a dataset of n-dimensional examples that come from different and unknown distributions. Given a new sample, I wish to find k examples from the dataset that ...
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130 views

Estimating parameters using Kullback-Leibler or Kolmogorov-Smirnoff via Nelder-Mead

I want to find the parameters of a model which specifies a set of classification probabilities, for say M classes. (I'll use the parameters in another model later.) Given a set of parameters ...