Questions tagged [entropy]

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

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Geometric distribution and entropy

According to wikipedia, among all discrete probability distributions supported on $\{1, 2, 3, ... \}$ with given expected value $\mu$, the geometric distribution X with parameter $p = \frac{1}{ \mu} $ ...
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Generate data that matches a frequency distribution while preserving the original spatial structure [closed]

I am dealing with a 3D array containing values representing the "importance" of each voxel. For my analysis, I would like to synthesize n new arrays from my original array to have a ...
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Difference between mutual and conditional information

I have studied about the basic probability theory. Now I am studing about entropy from information theory point of view from Bose's Information Theory, Coding and Cryptography. I have not understood ...
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Does autocorrelation in time series make meaningless the conditional entropy?

The conditional entropy of ${\displaystyle Y}$ given ${\displaystyle X}$ is defined as $${\displaystyle \mathrm {H} (Y|X)\ =-\sum _{x\in {\mathcal {X}},y\in {\mathcal {Y}}}p(x,y)\log {\frac {p(x,y)}{p(...
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A question about the information content in the entropy formula

One intuition in the entropy definition is that there is an inverse relationship between the information content of an event and its probability. This makes sense since learning an event which has a ...
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How can I show a mathematical proof of entropy in clasification tree?

I am trying to understand the splitting criteria in the classification tree. How can I show that for $p_1,p_2,..,p_n$ these functions attaining their maximum and minimum? $g(p_1,p_2,...,p_n) = Σp_i(1-...
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Does the entropy of a random variable change under a linear transformation?

Let $X$ be a random variable. If $Y=aX+b$, where $a,b \in \mathbb{R}$, is the entropy of $Y$ the same as the entropy of $X$?
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Information loss

Let's imagine we have some binary random variable $Y$ and some other continuous variable $X\in\mathbb{R}$ and we have some sample of size $n$. Suppose we want to determine the relationship between $Y$ ...
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Joint entropy of a bivariate Gamma probability density function

In Nadarajah and Kotz, 2009 (https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-39/issue-1/Four-Bivariate-Distributions-with-Gamma-Type-Marginals/10.1216/RMJ-2009-39-1-231....
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How to quantify information in a time series?

I have a time series of daily values for one year and I want to find out which slice of the time series contains maximum information considering a slice is of 150 values. I calculated Shannon's ...
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The correct way to form a PDF from probabilities of means (to calculate Entropy later)

The Question I have a probabilistic belief. I have 4 normal distributions, and I believe in each one a different amount. As follows: $$ P(M=ceramic)=0.9987 \\ P(M=aluminium)=0.0013 \\P(M=plastic)=0 \\ ...
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How the entropies of the random variables in conditional entropy affect its value?

We know that the conditional entropy $H(Y|X)\to 0$ as $X$ determines the value of $Y$. Now, I have the intuition of that $H(Y|X)<H(Y|Z)$ if $H(X)<H(Z)$. With this, and from the first statement, ...
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Why is entropy sometimes written as a function with a random variable as its argument?

Why do people sometimes write the entropy as a function on a random variable? For example, in some class notes I am seeing: $$ \sum_X Q(X) \log Q(X) = - H_Q(X) $$ I realize this is not to be taken as ...
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Copula entropy: calculation is borked?

I came across a pretty cool paper whose idea makes a lot of sense to me. Ma, Jian, and Zengqi Sun. "Mutual information is copula entropy." Tsinghua Science & Technology 16.1 (2011): 51-...
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How can I code the sequence of [1,2,3,4,5,6,6,5] in 2.5 or 3 bits?

If I calculate the entropy for the following sequence: [1,2,3,4,5,6,6,5] I get the entropy of 2.5 but I am wondering how can I actually do the encoding with 2.5 or 3 bits. Does it mean I need 3 bits ...
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Detecting behaviour change in a recommender system

Suppose I want to track a recommender system's live performance. The task isn't exactly to detect outliers, but to detect if the system started behaving differently, looking at the output only - an ...
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Why is the cross-entropy loss function used in logistic regression? [duplicate]

Just wanted to double check: I understand that we can't use the squared error cost function (as we do in linear regression) for the cost function of a logistic regression model as the sigmoid function ...
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Generalization of Burg's Maximum Entropy Theorem

Burg's Theorem characterizes the form of an entropy-maximizing time series, subject to constraints on the autocorrelation. More precisely, the theorem states that the autoregressive Gaussian process $...
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Differential entropy/LDDP with F-distributed data

I am working with a dataset where many variables are an F-distribution, particularly distributed like the black and gray curves in this graph (taken from Wikipedia). I am interested to calculated ...
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How can I prove that the conditional entropy of $X$ given $Y$ is $0$ if and only if $Y = f(X)$?

I want to show that: $$ H(X|Y) = 0 \iff Y=f(X) $$ Where $H(X|Y)$ is the average conditional entropy of the discrete random variable $X$ over all values of the discrete random variable $Y$, and $f$ is ...
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Correct notation for the probability of an event in entropy

I am looking at the formula of entropy on Wikipedia, where $P(X)$ is a probability mass function. \begin{equation} H(X) = -\sum_{i=1}^{n}P(x_i)log_bP(x_i) \end{equation} I got curious why they use ...
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Why is the Kullback-Leibler divergence defined with a negative sign?

I am aware of Gibbs' inequality, but I still want to know why the Kullback-Leibler divergence is defined with a negative sign. Here is my reasoning so far: Let $X$ be a Bernoulli random variable with $...
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Is the generalized entropy of certain events alway null?

In their paper: Novel Decompositions of Proper Scoring Rules for Classification, Kull and Flach wrote in section 2.2 Divergence, Entropy and Properness that when $y$ is the true class $d(p,y)=\phi(p,...
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Name for a particular family of distributions on unit simplex

Let $\Delta^n$ be the unit simplex, and let $\alpha\ge 0$ be a parameter. Is there a standard name for the following probability density function supported on $\Delta^n$? $$f_{\alpha}(p)=C(\alpha)e^{-\...
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How to conceptualise statistical entropy

Preamble There's this great YouTube video explaining the entropy of a probability distribution using the idea of "what is the minimum number of questions I need to ask?". So as an example, ...
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Measuring entropy/concentration in multivariate datasets

I am interested in measuring how concentrated or widely dispersed a certain binary property is among a population, which is defined by a number of categorical variables. For instance, let's say I have ...
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Change in Shannon Entropy in Markov Chain Process

What are the known results on the change in Shannon entropy $\Delta H_{k} = H(\vec{p}_{k}) - H(\vec{p}_{k-1})$ of the $k$-th step in a process governed by a finite state discrete time Markov chain ...
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Does logit transformation of information entropy values make sense?

I have a vector of information entropy values that range between 0 and 1 which I want to explain with some explanatory variables. I realized that the distribution of the entropy values in my dataset ...
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Relationship between log-likelihood function and entropy (instead of cross-entropy)

Negative log-likelihood function $$ -\ln f(X\mid\Theta)=-\int_{-\infty}^{\infty}\ln f(x\mid\Theta) \, \mathrm{d}x$$ Differential entropy $$ h(X) = -\int_{-\infty}^{\infty} f(x) \ln f(x) \, \mathrm{d}...
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Constructing a joint distribution from pairwise marginals

Consider a set of random variables $\{X_i\}$ with joint pdf $f(x_1 ... x_n)$. Given the marginal pdfs $f_i(x_i)$, we can construct a joint distribution $$g(x_1 ... x_n) = \prod_i f_i(x_i)$$ which has ...
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Information Entropy Problem

I cannot figure out this simple entropy problem and it is driving me crazy! From McElreath's Statistical Rethinking: Imagine instead 5 buckets and a pile of 10 individually numbered pebbles. You stand ...
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“Information” Correlation

(Let $X$ and $Y$ be random variables, sufficiently nice for my question to make sense.) $$ \text{Correlation} $$ $$ \rho(X, Y) = \dfrac{\text{cov}(X, Y)}{\sqrt{\text{var}(X)}\sqrt{\text{var}(Y)}} $$ ...
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Information gain of the root node

Recently I saw this question and answer as attached in following image Anyone can add details how this solution achieved?
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Can a likelihood's relative entropy be related to its predictive accuracy?

Suppose I have some prior $\pi(\theta)$, from which I draw $N$ samples, each having parameter $\theta_i$. These $\theta_i$'s are known to me. Suppose that one of these samples (unknown to me which) ...
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Bounds on the difference between entropies of two distributions?

Have there been any results regarding bounds on $\left| H(\hat{\pi})-H(\pi) \right|$ where $\hat{\pi}$ and $\pi$ are the empirical and true probability distribution?
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Clarification of variational autoencoders

I've been spending almost a month trying to understand VAE. I was reading a bunch of tutorials, and first it made sense, and seemed straightforward. Then I was experimenting with it, it produced weird ...
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Measure of segregation that takes population-level proportions into account (modified entropy/Theil index)

I am interested in a measure of segregation which allows for multiple groups and is maximized when group levels reflect the population as a whole (rather than equality of group size). The best I have ...
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CNN: quantitatively evaluate the activation of a filter

When an image is fed into a CNN, it would pass through different layer of different filters. The visualization of these filters look like: Here we can safely claim that the 1st filter is more ...
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Entropy of multivariate Pearson Type II

I was checking the entropy of multivariate Pearson Type II for maximizing Renyi entropy. It would be great if someone came up with this question or advise some reliable sources.
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Minimising KL divergence between two distributions

say we want to approximate a distribution $p(x)$ with $q(x|\theta)$. We do not know the distribution $p(x)$ but we can draw samples from $p(x)$. The KL divergence between the two distributions is $$ \...
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Why doesn't the entropy decrease with an increasing number of observations?

I'm trying to think more about entropy. I have the following toy example: Consider a coin flip. Case 1: I think p_h = 0.5 The entropy of this is 0.5 ln(0.5) x 2 = ln(0.5) Case 2: I don't know what p_h ...
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Statistically, is it effective to design exams to get harder as you answer correctly?

I think we have all taken exams where as you answer more questions correctly, the exam gets harder. Intuitively, this is obvious why...If an IQ test had questions for 3rd graders, everyone would get ...
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Information gain for unequally sized time series

I have some time series data, $\mathbf{x}_{1:T} = \{ x_1, \dots, x_T \}$ where the observation at time $t$, $X_t$, is a continuous random variable. Let $Y_t$ denote a discrete random variable at time $...
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Is there a metric to identify whether samples are uniformly distributed when number of samples is small?

Suppose I have a small number of samples drawn from an unknown distribution $\{X_1,X_2,...,X_n\}$, where $0\le X_i \le L$, and $3\le n \le10$. I want to identify a metric to understand how far these ...
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entropy regularization in generative model

I am wondering if it is possible to use entropy as a regularization in a generative model. For example, in the conjugate model where $x_i \in X$ is observed data and generated from a Normal ...
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Log probabilities versus squared probabilities (entropy vs Gini)

The advantage of log probabilities over direct probabilities, as discussed here and here, is that they make numerical values close to $0$ more easy to work with. (my question, instead of the links, ...
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When would you use purity as a measure of external validity over entropy? [closed]

This question particularly pertains to text clustering. I've not really found anything on why one would use purity over entropy or vice versa. Could someone explain this to me?
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How does Information Gain work?

I know the algorithm upon which the information gain is based. But how are we sure that after a split, its entropy will always decrease i.e. on dividing the data into smaller chunks, the entropy will ...
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Maximum Entropy Discrete Distribution

In Pattern Recognition and Machine Learning the author uses Lagrange multipliers to find the discrete distribution with maximum entropy. Entropy is defined by; $$H=-\sum_i p(x_i)\ln(p(x_i))$$ and the ...
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What is the effect of number of points on calculation of mutual information between two variables?

I was trying to calculate mutual information between two variables as follows: mutual_info_score(x,y) This essentially creates a 2d histogram and evaluates the ...

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