Questions tagged [entropy]
A mathematical quantity designed to measure the amount of randomness of a random variable.
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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 ...
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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.
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 $\min_i$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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 $H(E | A,B,C,D)$. ...
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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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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(...
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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-...
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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 ...
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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, ...
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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 ...
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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 ...
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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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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 ...
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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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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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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.
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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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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 ...
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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 ...
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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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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 ...
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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 ...
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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?
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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 ?
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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:
...
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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 ...
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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 ...
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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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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 ...
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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_{...
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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 ...
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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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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 ...
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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 ...
