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Questions tagged [entropy]

A mathematical quantity designed to measure the amount of randomness of a random variable.

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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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Multivariate conditional entropy

I would like to take data columns and compute the multivariate conditional entropy. For instance, suppose I have columns $A, B, C D, E$ and I want to compute the conditional entropy $H(E | A,B,C,D)$. ...
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Simple expression related to Mutual Information

One way to define the mutual information is $I(X;Y) = H(X) - H(X|Y)$ I have found it useful to look the related quantity $?(X;Y=y) = H(X) - H(X|Y=y)$ That is, we look at how much the entropy of $X$...
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Feature selection using information gain numeric features

I am trying to perform feature selection using the information gain criteria i.e. with the information.gain function in the FSelector R package and I am at a loss to what to do with my features that ...
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How is the upper bound for Normalized Mutual Information determined?

Mutual information between two clusterings $A$ and $B$ can be calculated as: $$MI(A,B)=H(A)+H(B)-H(A,B)$$ In the 10th page of this paper it is stated that $MI(A,B)$ can vary in the range $[0,\min\{H(...
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Cross - entropy for two variables with different prob. distributions

Let us say that we have given two random variables with different prob. distributions: A = [0.1, 0, 0.5, ...] B = [0.3, 0.1, 0.03, ...] What should I do when I want to compute the reformulated cross-...
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Is there any intuition behind the writing Jensen-Shannon divergence based on the entropy?

I know that Jensen-Shannon divergence between two distribution $P$ and $Q$ can be written as follow: $JS(P||Q) = H(\frac{P+Q}{2}) - \frac{H(P)}{2} - \frac{H(Q)}{2}$ But is there any intuition behind ...
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Estimate Information Entropy from Moments

I hope I'm using the right terminology below. I have access to the moments statistics of a large sample. That is, I have $\sum(x)$, $\sum(x^2)$, ..., $\sum(x^k)$. I also have access to max and min, ...
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Small calculation in Max entropy

I am wondering where the -1 comes from in the derivatives with respect to the probability. For example, these notes. For the basic example with the uniform distribution. We want to minimze $[-\int ...
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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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Correct number of grayscale levels when computing entropy of an image

I want to compute and to compare the entropy of images. Right now I am using images with 256 grayscale levels. But should the number of level not depend on the image size? For a really small image, ...
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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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Is it possible to use Information Gain metric for CART?

I've been looking for an example of CART using Information Gain but haven't found one. This made me wonder if it was even possible, so I tried to train one manually (by hand) using the dataset below ...
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Question when using sklearn's DecisionTreeClassifier

sklearn's DecisionTreeClassifier is not behaving as I expected. From the following: ...
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How do we show the exponential distribution has maximal entropy on R+?

I’ve been looking around the internet, and having trouble finding a demonstration that the exponential distribution is maximal entropy on R+. I’d appreciate any points in the right direction.
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is there a measure for the roughness of a contour plot

There has to be a measure for the difference between "instantaneous" change of "energy" along a line in a space compared to averaged changed of energy along a line. I could take a smooth surface in ...
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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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what is the meaning or intuition of entropy (from the point of view of reinforcement learning)

Can someone give an intuition of the concept 'entropy'? I am reading maximum entropy inverse reinforcement learning and I wanted to ask what the meaning intuition of 'entropy' is. I understand ...
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Does perplexity indicate how often my model gets it right?

This answer currently has the most votes and suggests The perplexity of whatever you're evaluating, on the data you're evaluating it on, sort of tells you "this thing is right about as often as an ...
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How to measure how “good” or accurate a probability distribution is? Entropy, variance or what?

How can one measure the accuracy of the probability distribution of, say, a physical magnitude? I know one good candidate is the entropy, which measures the amount of information one has about the ...
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How can I maximise binary cross entropy loss?

I have a multi-task learning model with two binary classification tasks. One part of the model creates a shared feature representation that is fed into two subnets in parallel. The loss function for ...
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Why is the cross-entropy always more than the entropy?

I understand intuitively why cross-entropy is always bigger. However, could someone show that mathematically?
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central limit theorem in term of entropy

The usual central theorem uses iid samples. Are there a generalizing theorem to non-iid samples, using the conditional entropy of each sample given previous ones ?
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Which result for normalized mutual information is correct?

I wanted to find the normalized mutual information to validate a clustering algorithm, but I've encountered two different values depending on the library I use. In Python: ...
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Does downsampling decrease the entropy of the data?

Suppose we have an $n-dim$ time-series $X={x_1, x_2, \cdots, x_n}$ and we resample it to $m-dim$, $\hat{X}={\hat{x}_1, \hat{x}_2, \cdots, \hat{x}_m}$, where $m < n$. Can we say this downsampling ...
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On the meaning of mutual information and on how to test the convergence of an estimation

As the result of a Molecular Dynamics simulation, I have the time series of two variables, $X$ and $Y,$ and I am interested in computing the mutual information of these two random variables. I've ...
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Estimating a surprise of a word in context

What will be the best way to estimate the entropy/surprise of a word in a specific context? Let's say to compare the surprise of: context: "I watched the movie in my" word: Computer I ...
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147 views

NLP various probabilities estimators in nltk

I saw there are many types of probabilities in nltk: MLE, ELE, Laplace, Heldout, KnereserNey, Lidstone, Random, WittenBel.. What is the exact difference between them and when should I use each? My ...
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Entropy of a mixture of Gaussians

I need to estimate as fast and accurately as possible the differential entropy of a mixture of $K$ multivariate Gaussians: $$ \mathcal{H}[q] = -\sum_{k=1}^K w_k \int q_k(\textbf{x}) \log \left[\sum_{...
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Is continuous mutual information the correct analogue of the discrete version?

I'm interested in the mutual information of two continuous random variables $X$ and $Y$. Shannon defined differential entropy as $h(X) = -\int p_X(x)\log p_X(x) dx$, where $p_X$ is the probability ...
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How to compare two point estimates between two samples

I have two samples, composed by independent observations and a set of features. Each sample can be represented as a binary matrix ($A_{ij} \in \{0, 1\}$) of observations x features. I want to find ...
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How to select value of k in K-NN when using Transfer Entropy?

I was reading about Transfer Entropy and I came across the estimators used to calculate TE, one of them being the Kraskov Estimator: $ T_{X \rightarrow Y} = \frac{p(Y_{n+1}, Y(k)_{n}, X(l)_{n})*log(...
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How does Noise affect the results of Transfer Entropy?

I was reading about Transfer Entropy and came across this package: https://cran.r-project.org/web/packages/TransferEntropy/TransferEntropy.pdf The code in the package: ...
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What is “Entropic Capacity”?

I found this term on the Keras blog website, quoted below Your main focus for fighting overfitting should be the entropic capacity of your model --how much information your model is allowed to ...
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Priors in Shannon and Rényi entropies

[Note: Cross-posted at Math StackExchange] I am new to information theory and currently working with Shannon and Rényi entropies. Given the pdf $p_{\theta}(x)$ of a random variable $x$, that is ...
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Naive Bayes: Understanding the Entropy equation

I am trying to understand the entropy equation: -p1*log2(p1) - p2*log2(p2) - pn*log2(pn) Specifically why do we multiply each log by the probability? In the ...
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Meaning and interpretation of Transfer Entropy

I am a first-year undergrad student and I have been reading about Transfer Entropy for my research. Although I understand the math behind I am not really sure what the value means. For example, I run $...
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Entropy or co-occurence matrix to compute the randomness of gray scale images?

I hope this is the right place to ask this question. I have an algorithm that outputs gray scale images (not normalized). These images oftentimes contain a lot of random noise and sometimes ...
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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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Mutual information between continuous $Z$ and $g(Z)$ for differentiable $g$?

I have a continuous random variable $Z$ and a differentiable function $X = g(Z)$. Is the mutual information between $X$ and $Z$ necessarily $\infty$ or 0? Are there any examples of differentiable ...
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Computing joint entropy from marginal distributions

I have distributions of N random variables (supposed conditionally independent) consequently, the joint distribution is the multiplication of all the distributions. I want to compute the joint ...
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Can the differential entropy be negative infinity?

Define the (differential) entropy for density $f$ as $$ H(f) :=-\int_{0}^{1} f(x) \log_{2}(f(x)) dx \, .$$ I am trying to find a Lebesgue measurable $f$ defined on $[0,1]$ such that $f\geq 0, \...
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Shannon entropy with regards to independent random variables

I had a question regarding a question on Shannon entropy I came across. It has to do with representing entropy in the form of their probability distributions, but let me elaborate. Here's the specific ...
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Cross-entropy for comparing images

Suppose we have two greyscale images which are flattened to 1d arrays: $y=(y_1, y_2, \ldots, y_n)$ and $\hat{y} = (\hat{y}_1, \hat{y}_2, \ldots, \hat{y}_n)$ with pixel values in $[0,1]$. How exactly ...
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Conditions Mutual Information and Confounding Effect

Given that conditional mutual information (CMI) I(A,B |C) is the information shared between A, and B given C, does this consider the confounding effect -if any - that C introduces? In other words, ...
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Interpretation of entropy for continuous distribution?

"Entropy" roughly captures the degree of "information" in a probability distribution. For discrete distributions there is a far more exact interpretation: The entropy of a discrete random variable ...
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Distribution of $-\log f_X(X)$

Say, $X \in \mathbb{R}^n$ (with $n > 1$) has a density $f_X(x)$. What can we say about the distribution of $$ Y = -\log f_X(X)? $$
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How to compute joint entropy of high-dimensional data?

Normally, I compute the (empirical) joint entropy of some data, using the following code: ...
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Trying to implement the Jensen-Shannon Divergence for Multivariate Gaussians

Given two multivariate Gaussian distributions $P \equiv \mathcal{N}(\mu_p, \Sigma_p)$ and $Q \equiv \mathcal{N}(\mu_q, \Sigma_q)$, I am trying to calculate the Jensen-Shannon divergence between them. ...
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Calculating conditional mutual information for textual features

I am reading this paper regarding the usage of conditional mutual information in textual data. I am unable to understand or see how cmi between certain features would enrich a classification algorithm ...