# 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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### Shannon entropy and Gini impurity are interchangeable in practice yet different? [closed]

Gini impurity, not to be confused with the Gini coefficient, is also an information theoretic measure and corresponds to Tsallis Entropy with deformation coefficient $q=2$, which in physics is ...
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### If a zero entropy distribution implies high information a priori, what does it mean ex posteriori?

The following counteracts the statements made for the maximum entropy principle case in order to posit a pseudo "minimum entropy principle" case that is simply the polar opposite of the ...
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### Which has minimum concentration: the uniform distribution or the maximum entropy distribution?

For a continuous random variable, the uniform distribution has high entropy because it demonstrates the greatest level uncertainty. However, this conflicts with the maximum entropy principle, which ...
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### Multiplying vector by the covariance matrix only known approximately

(cross-posted on math.SE) For random variable $(x,y)$ in $\mathbb{R}^{2d}$ and vector $v$, I need to perform the following operation on a $d \times d$ covariance matrix $E[xy']$ $$T(v)=E[xy']v$$ The ...
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### Different Maximum Entropy and Cross-Entropy loss?

Hi what's the difference between Maximum Entropy and Cross-Entropy loss for a sentiment classification with 2 labels (positive or negative)?
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### Are discrete maximum entropy distributions uniquely determined by their energy function?

I read this somewhere but can't find a reference, I'd love to be pointed in the right direction. I'm specifically interested in discrete Boltzmann distributions with interactions at different orders, ...
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### How does a distribution's differential entropy correspond to its moments?

The Gaussian distribution maximizes entropy for the following functional constraints $$E(x) = \mu$$ and $$E((x-\mu)^2) = \sigma^2$$ which are just its first and second statistical moments (true ...
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### Does minimizing KL-divergence result in maximum entropy principle?

The Kullback-Leibler divergence (or relative entropy) is a measure of how a probability distribution differs from another reference probability distribution. I want to know what connection it has to ...
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### What does maximizing mutual information do?

In information theory, there is something called the maximum entropy principle. Are other information measures, such as mutual information, also commonly maximized? If mutual information describes the ...
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### How does the maximum entropy principle affect joint entropy, mutual information, and other info measures?

The maximum entropy principle says that we should use the probability distribution of a univariate dataset that has the highest level of entropy because it offers the lowest information. How does the ...
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### Is the maximum entropy probability distribution only determined through comparison?

The maximum entropy probability distribution has entropy at least as great as that of all other members of a specified class of probability distributions (pdf's). Does that mean that the pdf with ...
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### Do all random variables' probability distributions have entropy?

Entropy of probability distributions is the weighted average of the log probabilities of each observation of a random variable. Does this mean that every random variable that has a probability ...
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### Why do we want a maximum entropy distribution, if it has the lowest information?

It is said that the distribution with the largest entropy should be chosen as the least-informative default. That is, we should choose the distribution that maximizes entropy because it has the lowest ...
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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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### 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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### 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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### 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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### 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 ...