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

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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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16 views

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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9 views

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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1answer
25 views

Question when using sklearn's DecisionTreeClassifier

sklearn's DecisionTreeClassifier is not behaving as I expected. From the following: ...
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24 views

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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13 views

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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1answer
13 views

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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1answer
42 views

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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6 views

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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1answer
45 views

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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39 views

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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1answer
22 views

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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17 views

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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1answer
32 views

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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1answer
27 views

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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31 views

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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19 views

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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1answer
75 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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35 views

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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33 views

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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38 views

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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29 views

Test for Uniform Distribution

The data I have has 24 data points for each observation. Each data point can take value of either zero or one. I want to test whether the 24 values come from a (discrete) uniform distribution. What ...
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41 views

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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21 views

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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49 views

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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32 views

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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16 views

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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1answer
92 views

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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2answers
73 views

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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1answer
547 views

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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29 views

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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1answer
52 views

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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125 views

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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1answer
28 views

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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1answer
385 views

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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18 views

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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2answers
161 views

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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1answer
150 views

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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1answer
357 views

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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191 views

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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13 views

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 ...
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42 views

How to prove subadditivity of joint entropy?

I've been kind of stuck on this exercise for the last hours. Heres everything I know and everything I'm allowed to use to prove it: Let $X= \{w_1, \ldots, w_n \}$ and $Y = \{v_1, \ldots, v_n \}$ be ...
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1answer
90 views

Entropy of the beta-binomial compound distribution

I have a generative process as follows: $$ x \mid \alpha \sim \textsf{Beta}\left (\alpha,\beta \right) \\ y \mid x \sim \textsf{Bernoulli}(x). $$ How does one go about calculating the Entropy of ...
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2answers
251 views

Entropy of a function of independent random variables

Suppose I have an operator (function) $f(\cdot)$ which takes three arguments $x,y,z$ all of which are independent random variables, and all of which I have access to the probability mass function (...
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23 views

Entropy of a factorised joint distribution

Suppose I have three discrete random variables $X, Y$ and $Z$. Their joint distribution factorises as so: $$ P(X,Y,Z) = P(X)P(Y)P(Z) $$ i.e. they are fully independent variables. Now suppose I want ...
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9 views

Entropy of a binary matrix distributed according to an Indian Buffet Process?

Given that a probability mass function gives the probability that a discrete random variable is exactly equal to some value, and that we do not possess a PMF for the Indian Buffet Process - is it ...
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1answer
117 views

Entropy of a matrix with Bernoulli distributed (binary entries) row-vectors

The entropy $H[x]$ of a Bernoulli distributed binary random variable $x$ is given by : $$ H[x]=−θlnθ−(1−θ)ln(1−θ) $$ where $$ p(x=1∣θ)= \theta \\ p(x=0∣θ)=1−θ $$ Now, suppose I have a vector as so: ...
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59 views

How to Calculate the multi scale approximate entropy for a financial time series

I want to calculate the multiscale approximate entropy for the financial time series. The code which I got to calculate approximate entropy is as follows: approx_entropy(ts, edim = 2, r = 0.2*sd(ts), ...
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84 views

Measure of neatness of a file tree [closed]

I'm working on a way to calculate the "neatness" of any file system tree. Features I'm considering include: depth of directory from root neatness of child directories number of files/child ...
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1answer
86 views

Was logistic regression based on Boltzmann distribution from statistical mechanics?

Ever since seeing the logistic distribution for the first time many years ago, I always thought of it as an application of the Boltzmann distribution. Whoever developed it may had seen the Boltzmann ...