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

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What is the relationship between Shannon Entropy and Approximate Entropy

I prefer a worded explanation because I am not a math expert. I know it Approximate Entropy is derived as an aproximation of Kolmogorov Entropry so the relationship between Kolmogorov Entropry and ...
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26 views

Why constrain mean and standard deviation when proving Gaussian is maximum differential entropy pdf?

I'm reading Bishop's Pattern Recognition and Machine Learning. In chapter 1.6: Information Theory (page 53) when trying to derive the maximum differential entropy pdf from the definition of continuous ...
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14 views

Difference between Weighted Average Entropy and Adjusted Mutual Information (for evaluating Clustering)

I was advised by my team leader to use this weighted average entropy to evaluating the performance of my dbscan clustering algorithm, and its mathematical formulation is: Scikit provides what many ...
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Continuous variable evaluation in decision trees

I was going through the C4.5 and ID3 algorithms used to construct a decision tree. Was wondering if there is an efficient way to compute information gain from a continuous variable (during the step ...
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2answers
25 views

Entropy of a block of characters

I have a question about the following statement about entropy: If a source provides us with a sequence chosen from 4 symbols (say A, C, G, T), then the maximum average information per symbol is 2 ...
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52 views

How to split a decision tree when information gains of all attributes are zero?

The textbook tells us that we should choose an attribute with the maximum information gain to split a decision tree. My question is what if all information gains are zero? Should we stop splitting or ...
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85 views

Gini index vs entropy

If I have a discrete probability distribution $p$ with $K$ classes Gini index = $\sum_{K}$$p_k$(1-$p_k$) Entropy = -$\sum_{K}$$p_k$log$p_k$ Per 'The Elements of Statistical Learning', ...
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10 views

Model comparision

I am comparing supervised vs unsupervised method. I have computed entropy from the confusion matrix of the 10-fold crossvalidated output. I have also applied external criteria to unsupervised methods ...
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21 views

Constructing Random Forests for binary classification by minimizing entropy

I'm looking to perform a binary classification using random forests, but I do not quite understand how to minimize the entropy of the data / what tests I should run on the nodes to do so. I'm fairly ...
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19 views

Likelihood Ratio as statistical test for Transfer Entropy with MatLab

I estimated a Transfer Entropy value TE(XY). Now I want to establish the statistical significance of the estimated value. Therefore I used the method of shuffled surrogates to estimate a shuffled ...
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1answer
24 views

Significance test for entropy?

Is there any way to test the difference of entropy given frequency table? For example, let's say we have dice 1 and dice 2, and we experimented with them and they showed up like ...
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43 views

Why is this a good approximation of cross entropy?

Given a random variable $X=x_i$ for $i=1,...,n$, with true distribution $p(x)$ and approximate distribution $q(x)$, its cross entropy is given by $$H(p,q) = -\sum_{i=1}^np(x_i)\log q(x_i).$$ However ...
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24 views

Simple measure for comparison of rainfall regularity

In meteorology we have the concept of monthly rainfall, which is just the sum of daily rainfall over that month. Now, given this extreme example: Situation 1: ...
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13 views

Entropy weighted Naive Bayes performs poorer than regular Naive Bayes?

I have a text classification problem, where there are many different classes, and the text to be classified is very short (about 1 sentence each): ...
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1answer
33 views

Compare skewness of many distributions with few observations

I have a dataset with page view data for about 500,000 users, divided into two groups. Each user can visit up to 5 pages, each as many or as few times as they want. So for each user, I have the ...
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1answer
59 views

Conceptual question on mutual information and entropy

What does mutual information (MI) convey? Looking for good reference books on information theory
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1answer
38 views

Conceptual questions on Entropy and estimation

Learning Informative Statistics: A Nonparametric Approach paper presents an approach to parameter estimation by entropy minimization. There are other related works "Minimum-entropy estimation in ...
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1answer
26 views

do information entropy probabilities have to sum to one?

My understanding of information entropy is that it requires the input probabilities to sum to 1. So, for a sequence a,a,b,b you then have -([1/2 log2 1/2] + [1/2 log2 1/2]) = 1 Are there versions of ...
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1answer
34 views

Differentiating Shannon's entropy [closed]

Can somebody please show the steps of how differentiation of Shannon's entropy yields the following result? $H = -\sum_{l=0}^{L-1} p(l)\log_2[p(l)]$ The result of differentiating is $H_m = ...
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26 views

What does Class Complexity mean in Weka?

When running Weka on my dataset, in the results printout I get the following rows: ...
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51 views

Neuroscience Equations

I am trying to understand a neuroscience article by Karl Friston. In it he gives three equations that are, as I understand him, equivalent or inter-convertertable and refer to both physical and ...
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15 views

General reference to Rényi entropy

Is there a book or a review article serving as a good general reference to Rényi entropy, its applications and related concepts?
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45 views

Calculation of relative information gain for multiple splits in decision trees

As it well known, we can calculate Relative Information gain (RIG) as follows: $RIG = \frac{H(x) - H(x|a)}{H(x)}$. In binary decision trees we calculate $H(x|a)$ for univariate split for variable ...
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127 views

Feature selection : how to select the Information Gain threshold?

I am trying to use Information Gain to select features when classifying text with a Support Vector Machine. For each word in our training data, we computed its information gain. Then, we should keep ...
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1answer
36 views

Differential Entropy of Gaussian Process

I have $N$ datapoints that have $d$ features in a GP and their covariance matrix $K$ and I want to calculate the differential entropy of that GP. Is this formula right? $E(I)= \frac{1}{2} ...
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1answer
80 views

Information gain with numerical data

I'm making a random forest classifier. In every tutorial, there is a very simple example of how to calculate entropy with Boolean attributes. In my problem I have attribute values that are calculated ...
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1answer
78 views

Relationship between entropy and information gain

Based on papers :1. Deniz Erdogmus, Member, IEEE, and Jose C. Principe, An Error-Entropy Minimization Algorithm for Supervised Training of Nonlinear Adaptive Systems J. Principe, D. Xu, and J. ...
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44 views

Entropy and information content

I am curious to know about the relation between Entropy and information content of a signal or trajectory of time series. When a system is at equilibrium, then it has maximum entropy. Does entropy ...
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39 views

Combining values of Shannon Entropy

I have a set of n variables. Each of these variables an be one of a finite number, m, of values. The number of values is different for each variable and independent. From this definition I can ...
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125 views

Estimating entropy of multidimensional variable through dimension reduction

Is there anything inherently wrong with trying to estimate the entropy of a multidimensional random variable by first transforming it (by some method) into a single-dimensional variable?
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47 views

$I(X:Y|Z=z) \leq I(X:Y|Z)$?

$X,Y$ and $Z$ are discrete random variables. $I$ is the mutual information. The question is in the title. If the inequality is true, how would you show it? Thanks. My intuition: having more ...
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41 views

Entropy of Cauchy (Lorentz) Distribution

Entropy is defined as $H$ = -$\int p(x)$ $log$ $p(x)$ $dx$ The Cauchy Distribution is defined as $f(x)$ = $\frac{\gamma}{\pi}$ $\frac{1}{\gamma^2 + x^2} $ I kindly ask to show the steps to ...
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93 views

kozachenko-leonenko entropy estimation

I'm trying to implement the entropy estimation based on the closest neighbour from Kozachenko and Leonenko but I'm facing a problem I can't solve. The idea is to work in a new set ...
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40 views

Using similarity matrix to measure diversity of a group

I wish to measure the "diversity" of a group of objects. Right now I'm using Euclidean distance to compute the similarity matrix between all the objects in the group. I'm searching for a measure of ...
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3answers
49 views

$\phi$-divergence?

I am frustrated of looking for a simple explanation of this term $\phi$-divergence, but I cannot find any. Therefore I would be really grateful if somebody could introduce a reference or write a ...
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179 views

Why am I getting information entropy greater than 1?

I implemented the following function to calculate entropy: ...
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97 views

Why can we use entropy to measure the quality of a language model?

I am reading the < Foundations of Statistical Natural Language Processing >. It has the following statement about the relationship between information entropy and language model: ...The ...
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1answer
128 views

Entropy estimation for a symbol sequence

I am looking for an R-implementation of the Lempel-Ziv data compression algorithm, to estimate the source entropy of a time-series consisting of a sequence of symbols. Rather than simply measuring ...
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28 views

Confidence interval on entropy estimation

Over a sample (of about a hundred observations), I have estimated the "empirical" entropy of a sequence of identically independently distributed random variables which have 5 possible outcomes. I ...
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41 views

How to compute the total entropy of a system of binary outcomes

This is nowhere near my field of expertise, so if my question is poorly formatted for the context please feel free to edit. My question is at the end of the text below. Let's assume I flip a coin 5 ...
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47 views

Entropy of generalized distributions?

What's the entropy of the following generalized probability distributions? $P_1(x) = \delta(x)$ $P_2(x,y) = \delta(x+y)$, for $0\le x\le 1$, and $P_2(x,y)=0$ otherwise. Integrals of the type $-\int ...
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137 views

Compute Shannon entropy between every row of a large, sparse matrix

I have a sparse, binary matrix of user (rows) and items (columns). Each element of this matrix is either 0 or 1: ...
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62 views

Better markov chain rank aggregation using an entropy-based approach

Background: Dwork et al. in Rank Aggregation Methods for the Web have proposed a few markov chain-based methods to perform rank aggregation (finding an aggregated ranking of items from a set of N ...
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1answer
108 views

LLR with Positive and Negative Values vs. Dunning method with Entropy-based Calculation

Ted Dunning has a blog post about calculating G2 (aka LLR) using Entropy calculations as components. I found this really intriguing. Ted's original post: ...
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What is the role of the logarithm in Shannon's entropy?

Shannon's entropy is the negative of the sum of the probabilities of each outcome multiplied by the logarithm of probabilities for each outcome. What purpose does the logarithm serve in this equation? ...
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1answer
56 views

When and why does the “brittleness” of mutual information cause overfitting?

I have frequently heard concern over "brittleness" of entropy and mutual information as performance metrics for a statistical fitting and the fact that it leads to overfitting. You can see an example ...
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2answers
51 views

Multidimensional Differential Entropy

I am looking for a measure of entropy over multiple random variables, each with values between 0 and 1. Intuitively, it seems possible to talk about the expected value of information of several ...
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1answer
120 views

Jensen-Shannon divergence for finite samples

I have two finite samples $s_1$ and $s_2$ and two distributions $p_1(s_1)$ and $p_2(s_2)$ that are associated to these samples. I'm essentially interested to measure the distance or similarity between ...
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1answer
80 views

Difference between different kinds of entropy

The following concepts are baffling and would be obliged for a constructive explanation. (Q1) What is the conceptual difference between (a) Kolmogorov-Sinai entropy, (b) Shannon entropy, (c) Source ...
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24 views

Modeling a list with a tunable degree of disorder/shuffling

Imagine we have a list of ordered numbers $L = (1, 2,\dots, N)$. I want to add an arbitrary amount of "disorder" to that list. For instance: Adding a little bit of disorder would permute a few ...