Questions tagged [maximum-entropy]

maximum entropy or maxent is a statistical principle derived from information theory. Distributions maximizing entropy (under some constraints) are thought to be "maximally uninformative" given the constraints. Maximum entropy can be used for multiple purposes, like choice of prior, choice of sampling model, or design of experiments.

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Am I understanding correctly the Maximum Entropy concept using a sentence?

In the sentence "The house is white", each word carries a different amount of information. If I remove the "The" from the sentence, almost nothing happens: you are still able to ...
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Continuous Entropy and Maximum Entropy Solution

This is a problem that I have been working on and the mathematics of it have me fairly stumped. I am given the continuous entropy for a density $p(x)$. It is $H(X)=-\int_{0}^{\infty}p(x)\text{log}\: ...
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Is this implementation detail for solving maximum entropy on a computer correct?

I am currently looking at a paper by Mattos and Veiga, who describe an approach to solving the maximum entropy problem subject to linear constraints: $$\begin{aligned} \max_{p_i} -\sum_{i=1}^N p_i \...
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When is uniform distribution have maximum entropy instead of normal distribution?

As far as I know, when we have just data and no constraints (other than probabilities must add up to 1), the distribution that gives maximum entropy is uniform distribution. But when we know mean and ...
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Why are $\mathbb{E}( \ln(x))$ and $\mathbb{E} ( \ln(1 - x))$ reasonable descriptions of knowledge about a beta distribution?

The max entropy philosophy states that given some constraints on the prior, we should choose the prior that is maximum entropy subject to those constraints. I know that the Beta($\alpha, \beta$) is ...
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How do you use MaxEnt for Bayesan updating given new data on a toy problem about a biased coin

The article "Updating, supposing, and maxent" (paywalled) says, Bayes' rule is a special case of MAXENT. Let the random variable $I_c$ be the indicator function which takes the value 1 in the set $...
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Is there a connection between the binomial pmf and the formula for entropy? [duplicate]

Many times when two formulas "look" the same, there is some interesting mathematical result linking them. Both the log binomial likelihood and the entropy formula kind of "look" the same, in that they ...
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Do Lévy α-stable distributions maximize entropy subject to a simple constraint?

Is there a simple constraint on real-valued distributions such that the maximum entropy distribution is Lévy α-stable? Special cases include the Normal and Cauchy distributions for which the answer is ...
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Four different ways to deal with the log-likelihood of a probability density function (Python code included)

This is not really a question but more of a discussion. Please correct me where wrong and share your thoughts and past experience with regards to computing the likelihoods for continuous data models. ...
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How do I prove conditional entropy is a good measure of information?

This question is a follow-up of Does "expected entropy" make sense?, which you don't have to read as I'll reproduce the relevant parts. Let's begin with the statement of the problem A student has ...
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Are there nonparametric generative models for datasets?

Typically when I see generative models, e.g., Latent Dirichlet Allocation (JMLR) or Linear/Quadratic Discriminant Analysis (wikipedia LDA), they are probabilistic models that belong to the exponential ...
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Intuition for the uniform distribution having the maximum entropy

I saw the following explanation for Entropy in probability: (Entropy). The surprise of learning that an event with probability $p$ happened is defined as $\log_2(1/p)$, measured in a unit called ...
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MaxEnt model vs cross entropy loss

Pardon my ignorance. I am still learning. We try to minimize the cross-entropy loss for best results. However, why should the entropy be high for a MaxEnt model for the model to be good? My ...
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Maximum entropy function, with f(0)=0?

I want to derive the Maximum Entropy distribution (f(x)) with the following constraints: 1. non-negative 2. specified mean 3. specified variance 4. f(0)=0 I know how to derive the MaxEnt distro with ...
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Maximum entropy prior for dichotomous variables [closed]

I have a set of dichotomous variables $A, B, C,$... and I know their probabilities $P(A), P(B), P(C),$... as well es their pairwise dependencies $P(A \cap B), P(A \cap C), P(B \cap C),$... . Or in ...
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Why generative models in Machine Learning are Boltzmann distribution-backed?

I learned from this review paper that MaxEnt models naturally display a Boltzmann distribution for the data samples, it comes from the Principle of Maximum Entropy. But I could not understand why this ...
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How can a probability densitiy be estimated based on the maximum entropy principle, with constraints in the order statistics?

Let's say we are given a sample $\{ z_i \}$, $i=1,2,\ldots,N$, such that each value $z_i$ corresponds to the minimum of $n$ random variables $x$, i.e., $z = \min \{ x_1, x_2,\ldots,x_n \}$. The ...
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How to minimise and expectation with respect to a parameter?

Suppose that $X$ is a random variable with distribution $G$. Let $H(X;\theta)$ be a parametric function with $\theta \in \Theta \subset {\mathbb R}^p$. I want to maximize the function $$\varphi(\...
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Maximum entropy prior for r.v. supported on real line with no other constraints?

What would be a suitable maximum entropy prior for a random variable supported on the real line with no other constraints (i.e. unknown mean, unknown variance, unknown bounds)? All kinds of answers (...
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how to calculate entropy on matrix of words, topics

I have been digging in the concept of entropy for a while, now it comes to the implementation part I feel I am confused. Imagine that we have a matrix 20 * 3 standing for 20 words 3 topics (by 20 ...
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Determining a probability distribution from constraints on where its mass is

Let $X$ be a random variable over the real line. Suppose that we know that $X$ is a Pearson distribution. Furthermore, suppose we know how the mass of $X$ is distributed into 6 intervals, so that if $...
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Maximum entropy probability distribution over non-negative support and finite mean?

I'm trying to derive which univariate probability distribution maximizes entropy, assuming finite mean $\mu$ and non-negative support $[0, \infty)$. I know that the answer is the exponential ...
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Proving that Shannon entropy is maximised for the uniform distribution

I know that Shannon entropy is defined as $-\sum_{i=1}^kp_i\log(p_i)$. For the uniform distribution, $p_i=\frac{1}{k}$, so this becomes $-\sum_{i=1}^k\frac{1}{k}\log\left(\frac{1}{k}\right)$. Further ...
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Estimating maximum entropy distribution given first n moments

Is there a good way to estimate the pdf, pdf up to a constant multiple, cdf, or quantile function of a distribution given the first n moments? A closed form for one of those functions in terms of the ...
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78 views

Is the marginal distribution of a maximum entropy distribution also a maximum entropy distribution?

I am considering under what circumstances maximum entropy distributions are closed under marginalization. The main case I am interested is the following setting: a finite set of discrete random ...
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Maximum entropy distribution on the hypercube

Given the first two moments, the maximum entropy distribution over $\mathbb{R}$ is known the be the normal distribution. What is the analogue for a distribution over $[0,1]$ given either only the ...
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115 views

Same maximum entropy and measures [closed]

As the author Christian Robert asks about reference measures that are absolutely continuous to one another, and from what I can gather this just means they have the same null set? But would that not ...
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Normal max entropy

In the following link, at the bottom of the page. https://www.dsprelated.com/freebooks/sasp/Maximum_Entropy_Property_Gaussian.html The Gaussian maximum entropy is derived. I have a question about ...
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Some Questions about reference measures and maximum entropy priors (from The Bayesian Choice)

I am relatively new to statistics and Bayesian theory but I am trying to understand it by working through a few books. There are some things I am confused about. ( As I believe my analysis and such is ...
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Where is my misunderstanding/mistake in calculation about maximum entropy

I am looking through the book Philosophy of Science : Entropy and Uncertainty by Teddy Seidenfeld and have some questions. He talks about the basic dice example, where by you have a fair six sided ...
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229 views

Maximum Entropy with bounded constraints

Assume we have the problem of estimating the probabilities $\{p_1,p_2,p_3\}$ subject to: $$0 \le p_1 \le .5$$ $$0.2 \le p_2 \le .6$$ $$0.3 \le p_3 \le .4$$ with only the natural constraint of $p_1+...
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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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Incorporating knowledge of aggregate outcomes to constrain predictions on finer scales

I've got county-level longitudinal data on the timing of an event between the years 1998 and 2012, and I want to use it to form a predictive model for the time that that event will occur in future ...
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What is the maximum entropy distribution given the median (instead of the mean)?

Given that the median seems to be a more robust statistic than the mean/average, I was wondering if there is a solution of the maximum entropy distribution given the median (or the median and some ...
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Maximum Entropy: another name for Maximum Likelihood or a legit Bayes procedure?

In some more recent works (like this and this for instance), MaxEnt is unequivocally bound to Maximum Likelihood (ergo Classical Inference). In some other older works (like Jaynes article), MaxEnt ...
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164 views

Computational complexity of MaxiEnt classifier

I know that the time complexity of logistic regression can be as low as linear when the optimizer/solver is assumed to be linear, such as L-BFGS (this link) I know that multinomial logistic regression ...
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459 views

Maximum Entropy Bootstrap - how to implement?

I have a question about the Maximum Entropy Bootstrap algorithm for time series. I'm confused about how to implement step 5, where you compute the sample quantiles of the Maximum Entropy ...
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Thompson sampling with adaptive kernel density estimation

This is an extension to this question, which is about handling arbitrary (potentially unbounded) reward distributions for the multi-armed bandit problem. Given a sequence of observed rewards $r_t \in \...
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Why do we care about maximum entropy? [duplicate]

One justification for the ubiquity of the (multivariate) normal distribution in statistical/machine learning modeling is that it maximizes entropy among distributions with mean $\mu$ and variance ...
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How to express tail index as an expectation (for MaxEnt procedure)

I'm trying to construct a prior probability density function, $f_X(x)$, for a fat-tailed distribution using the maximum entropy (MaxEnt) method. For my known "testable" information, I have the ...
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80 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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259 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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Support vector machines (SVMs) are the zero temperature limit of logistic regression?

I was had a quick discussion recently with a knowledgeable friend who mentioned that SVMs are the zero temperature limit of logistic regression. The rationale involved marginal polytopes and fenchel ...
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Understanding the composition law of maximum entropy

Shannon derived the principle of maximum entropy from 3 assumptions. One of those assumptions is the composition law. In other words, he derived that maximizing the probability weighted log of ...
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Is cross-entropy loss for MaxEnt models convex? If so, how do I prove it?

I am trying to prove if cross-entropy loss for MaxEnt models is convex. My first attempt at approaching this problem is to compute the Hessian matrix of second-order partial derivatives and showing ...
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Bimodal MaxEnt distributions?

What kind of constrains give rise to bimodal distributions in the Maximum Entropy formalism? Are there any known results in this topic?
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Maximum Entropy modelling - likelihood equation

I am trying to understand maximum entropy modelling and I came across log likelihood equation of the empirical distribution, which I did not quite understand, which also eventually turns out to be ...
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Measure non-randomness of numeric license plates

License plates for cars in Switzerland have the 2-letter abbreviation from the Canton and then between 1 and 6 numeric digits. There are no alphabet characters in the license plate, and therefore no ...
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Sampling MaxEnt Distribution with PyMC3

I’m trying to use PyMC3 to sample from a maximum entropy distribution over binary patterns of length $N$, constrained by means and pairwise correlations. The ...
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Distribution (CDF) estimation for strictly increasing, continuous distribution with compact support

For all $t\in 1,\dots,T$, suppose $x_t\in [0,1]$ is a draw from a distribution with unknown CDF $F:[0,1]\rightarrow [0,1]$. For future use, define $\tilde{x}\in [0,1]^T$ to be a vector containing $x_1,...