Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [entropy]

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

0
votes
0answers
21 views

Am I using conditional entropy formula correctly?

I want to use the formula for a measure of complexity of a system: $$C(X) = H(X) - \sum_{x \in X}{H(x\mid X-x)}$$ where $x$ is a subpart of the system $X,$ and $H(X)$ is shannon entropy. Then I have ...
1
vote
1answer
18 views

Is there a probability distribution function (PDF) that maximizes entropy for a given mode value?

I want to have a PDF that maximizes entropy for a given mode value. I searched in Maximum entropy probability distribution but here we have maximization done over a certain moment constraint.
0
votes
0answers
16 views

Information Gain property

Studying about information gain I found in the web (from the presentation of a lecture) that $IG(C|X) = IG(X|C)$ is it true? How I prove it?
0
votes
0answers
14 views

Conditional entropy of an outcome

Given three discrete random variables $X$, $Y$ and $Z$, the conditional entropy $$ H(X|Y,Z) = \sum_{X}\sum_{Y}\sum_{Z} p(x,y,z) \ \text{log}\frac{p(y,z)}{p(x,y,z)} $$ If I want to calculate the ...
0
votes
0answers
51 views

Can information entropy be computed on arbitrary set of non-negative numbers smaller than 1?

I have a set of non-negative numbers smaller than 1 which do not sum to 1. These numbers will be used in some weighted sum. I'm wondering in this case can we still measure the degree of concentration ...
0
votes
0answers
11 views

Is Entropy conserved under invertible (but probabilistic) mappings?

I know that under a one-to-one mapping, the entropy is conserved. This was answered very well here: Is entropy conserved under invertible mappings? I am having a doubt about this being applicable to ...
2
votes
1answer
82 views

Interpretation of spectral entropy of a timeseries

The tsfeatures package for R has an entropy() function. The vignette for the package describes it as: The spectral entropy is ...
2
votes
1answer
29 views

Measure of randomness that maps well to “hard to guess”?

It was recently pointed out to me that entropy does not necessarily map well to "hard to guess". E.g.: consider a distribution where there's a very high (say, 90%) chance you select a single, ...
0
votes
0answers
29 views

Entropy/Measure of Knowledge for probability intervals

I have a system which outputs probability interval tuples over the decision set $\{d_1, d_2, d_3\}$, such that a tuple $([\min_1, \max_1], [\min_2, \max_2], [\min_3, \max_3])$ indicates that $\min_i$ ...
0
votes
0answers
18 views

joint entropy and mutual information

If we have joint entropy of 10 bits between distribution A and B, while mutual information is 2 bits. Can we say there is 0.4% of useful communication between two distributions? Would this be proper ...
0
votes
0answers
18 views

What does Joint entropy really tells us?

I have two sequences of data A and B. A = [sunny, sunny, cloudy, rainy, sunny, sunny] with entropy 1.25 bits b = [hot, hot, cold, cold, cold, hot] with entropy 1 bits I have calculated joint ...
2
votes
0answers
34 views

Why is unbiased estimation from a sample only possible for certain properties?

I was thinking about finding some monotonic measure of entropy of a sample from a continuous distribution (2D, btw), but couldn't think of any such without making assumptions. Why is it that one can ...
0
votes
0answers
45 views

Conditional Entropy in the context of Gaussian Processes

I've got a question regarding the conditional entropy of a discrete random variable. According to this paper the conditional entropy of a Gaussian random variable conditioned on a set of variables can ...
0
votes
0answers
17 views

Multivariate Conditional Entropy as a test of correlation between random variables

I use the word columns to mean the data from which a random variable can be estimated. It is a sample of a random variable. I am working with $N$ columns of weakly correlated data. Furthermore, I ...
2
votes
2answers
33 views

Estimating Differential entropy from unbiased samples of a probability distribution

If I can get N unbiased samples $x$ from $p(x)$ how can I approximate the Differential entropy: $$H(X) = -\displaystyle\int_{x} p(x)\log p(x) dx$$ I'm not very knowledgeable in statistics so I'm not ...
0
votes
0answers
24 views

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 ...
1
vote
1answer
61 views

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 $H(E | A,B,C,D)$. ...
0
votes
0answers
16 views

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 $X$...
0
votes
0answers
65 views

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 ...
1
vote
1answer
50 views

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(...
0
votes
0answers
30 views

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-...
0
votes
0answers
52 views

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 ...
2
votes
1answer
91 views

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, ...
0
votes
1answer
21 views

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 ...
2
votes
1answer
36 views

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 ...
0
votes
0answers
6 views

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, ...
0
votes
0answers
22 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 ...
1
vote
0answers
24 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 ...
0
votes
1answer
25 views

Question when using sklearn's DecisionTreeClassifier

sklearn's DecisionTreeClassifier is not behaving as I expected. From the following: ...
3
votes
0answers
90 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.
0
votes
0answers
17 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 ...
2
votes
1answer
87 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 ...
0
votes
1answer
65 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 ...
0
votes
0answers
12 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 ...
0
votes
1answer
79 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 ...
1
vote
0answers
90 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 ...
3
votes
1answer
81 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?
0
votes
0answers
20 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 ?
3
votes
1answer
162 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: ...
0
votes
1answer
40 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 ...
0
votes
0answers
73 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 ...
0
votes
0answers
20 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 ...
0
votes
1answer
323 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 ...
1
vote
0answers
68 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_{...
0
votes
0answers
78 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 ...
2
votes
0answers
42 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 ...
1
vote
0answers
45 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(...
1
vote
0answers
29 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: ...
1
vote
0answers
111 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 ...
1
vote
0answers
38 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 ...