Principle Of Maximum Entropy

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Don't expect sunshine and rainbows. This is about information, about what we know and, more importantly, what we don't know, and how we represent that uncertainty. It’s not pretty, but it’s precise.

Principle in Bayesian statistics

For other uses of "Maximum entropy", see Maximum entropy (disambiguation).

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This is but a fragment, a single thread in the grand tapestry of Bayesian statistics.

Background

The fundamental equation, the mantra of this domain, is presented as:

Posterior = Likelihood × Prior ÷ Evidence

And beneath this, a litany of related concepts, each a stepping stone in the path of reasoning:

Model Building

The architecture of our understanding, the construction of models, involves:

Posterior Approximation

When exact solutions are too… inconvenient, we resort to approximation:

Estimators

The tools we use to derive our conclusions:

Evidence Approximation

Finding the weight of the evidence:

Model Evaluation

Assessing the validity of our constructions:

And, of course, the omnipresent portals to further exploration:

• Mathematics portal

• v • t • e

The principle of maximum entropy, at its core, is a statement about probability distributions. It posits that when you have a limited amount of precise prior information about a system, the most honest representation of your knowledge—the one that admits the least assumption beyond what you’ve been told—is the probability distribution with the highest possible entropy. It’s a way of saying, “This is what I know, and I’m not going to pretend to know more than I do.”

Think of it this way: you’re given some facts, some testable information. You consider all the possible probability distributions that could be true given those facts. The principle of maximum entropy directs you to choose the one that is the most spread out, the most uncertain, the one that makes the fewest assumptions about the outcomes you haven't been explicitly informed about. It's the distribution that embodies the most ignorance, given the constraints.

History

The concept itself wasn't born in a vacuum. E. T. Jaynes was the one who laid it out, meticulously, in two seminal papers back in 1957. He saw a profound, almost elegant, correspondence between the seemingly disparate fields of statistical mechanics and information theory. Jaynes argued, with a logic that’s hard to fault, that the very foundation of Gibbsian statistical mechanics was, in essence, the same concept as information entropy. He essentially declared that statistical mechanics wasn't some arcane physical law, but rather a specific, powerful application of a general framework for logical inference and the management of information.

Overview

Typically, the "precise prior data" or "testable information" comes in the form of conserved quantities. These are essentially the average values of certain functions, the moments of the probability distribution you’re trying to pin down. This is the bread and butter of how the maximum entropy principle is wielded in statistical thermodynamics. Alternatively, you might specify certain symmetries that the distribution must possess. Since conserved quantities and their corresponding symmetry groups are intrinsically linked, these two methods of defining the "testable information" often lead to equivalent outcomes in the maximum entropy framework.

The principle also serves a crucial purpose: it ensures that the probability assignments we arrive at are not only unique but also consistent, regardless of whether we're approaching the problem from the perspective of statistical mechanics or pure logical inference.

It explicitly acknowledges our inherent flexibility in how we choose to encode our prior data. In a special case, this can lead to adopting a uniform prior probability density. This is what Laplace called the principle of indifference, or sometimes the principle of insufficient reason. So, you see, the maximum entropy principle isn't just a fancier way of doing what classical statistics already does; it's a significant conceptual leap, a broader generalization.

However, these pronouncements don't negate the need to demonstrate that systems are ergodic to justify their treatment as a statistical ensemble. That's a separate, though related, concern.

In plain terms, the principle of maximum entropy is an expression of epistemic humility, or perhaps, a grand admission of maximum ignorance. The distribution it selects is the one that makes the fewest unsupported claims, the one that reveals the least about the unknown, beyond the precise information provided.

Testable Information

This principle truly shines when applied to testable information. What constitutes testable information? It's a statement about a probability distribution whose truth or falsity can be definitively determined. For instance:

The expectation of the variable $x$ is 2.87.
$p_2 + p_3 > 0.6$ , where $p_2$ and $p_3$ represent the probabilities of specific events.

These are concrete, verifiable statements.

When armed with such testable information, the maximum entropy procedure involves finding the probability distribution that maximizes information entropy while adhering strictly to the constraints imposed by that information. This is a classic constrained optimization problem, often tackled with the powerful technique of Lagrange multipliers.

If you have absolutely no testable information beyond the fundamental constraint that probabilities must sum to one, the maximum entropy distribution is, predictably, the uniform distribution. Every outcome gets an equal slice of the probability pie:

$p_i = \frac{1}{n} \quad \text{for all} \quad i \in \{1, \dots, n\}.$

Applications

The principle of maximum entropy finds its utility in a variety of inferential tasks, primarily in two significant ways:

Prior Probabilities

It's frequently employed to derive prior probability distributions for Bayesian inference. Jaynes himself was a staunch advocate for this, arguing that the maximum entropy distribution represented the most "least informative" prior possible. A considerable body of literature now exists dedicated to eliciting these maximum entropy priors and exploring their connections to channel coding.

Posterior Probabilities

Maximum entropy can also serve as a robust updating rule for radical probabilism. Richard Jeffrey's probability kinematics is, in fact, a specific instance of maximum entropy inference. However, it's important to note that maximum entropy isn't a universal generalization for all such updating rules.

Maximum Entropy Models

Beyond assigning priors, the principle is often invoked for model specification. Here, the observed data itself is treated as the testable information. These models have found widespread application, particularly in natural language processing. A prime example is logistic regression, which, in this context, is recognized as the maximum entropy classifier for independent observations.

The reach of the maximum entropy principle extends even into economics and resource allocation. Consider the Boltzmann fair division model, which leverages the maximum entropy (Boltzmann) distribution to distribute resources or income among individuals, offering a probabilistic lens through which to view distributive justice.

Probability Density Estimation

One of the most significant applications of the maximum entropy principle lies in both discrete and continuous density estimation. Similar in spirit to support vector machine estimators, the maximum entropy principle can lead to the solution of a quadratic programming problem, yielding a sparse mixture model as the optimal density estimator. A key advantage here is its capacity to integrate prior information directly into the density estimation process.

General Solution for the Maximum Entropy Distribution with Linear Constraints

For the truly dedicated, or perhaps the obsessively thorough, the mathematical underpinnings are laid bare.

• Main article: Maximum entropy probability distribution

Discrete Case

Imagine we possess testable information, let's call it I, concerning a variable $x$ that can take values from a discrete set $\{x_1, x_2, \dots, x_n\}$ . This information is typically presented as $m$ constraints on the expected values of certain functions, $f_k$ . In mathematical terms, we require our probability distribution, $\Pr(x_i)$ , to satisfy these moment inequality/equality constraints:

$\sum_{i=1}^{n}\Pr(x_{i})f_{k}(x_{i})\geq F_{k}\qquad k=1,\ldots ,m.$

Here, $F_k$ represents observable quantities. We also carry the fundamental constraint that the probabilities must sum to unity, which can be viewed as a primitive constraint on the identity function, with an observable value of 1:

$\sum_{i=1}^{n}\Pr(x_{i})=1.$

The probability distribution that maximizes information entropy, subject to these constraints, takes a specific form:

$\Pr(x_{i})={\frac {1}{Z(\lambda _{1},\ldots ,\lambda _{m})}}\exp \left[\lambda _{1}f_{1}(x_{i})+\cdots +\lambda _{m}f_{m}(x_{i})\right],$

for some set of parameters $\lambda_1, \dots, \lambda_m$ . This is often referred to as the Gibbs distribution. The normalization constant, $Z(\lambda_1, \dots, \lambda_m)$ , known conventionally as the partition function, is determined by:

$Z(\lambda _{1},\ldots ,\lambda _{m})=\sum _{i=1}^{n}\exp \left[\lambda _{1}f_{1}(x_{i})+\cdots +\lambda _{m}f_{m}(x_{i})\right],$

(The Pitman–Koopman theorem is relevant here, stating that the necessary and sufficient condition for a sampling distribution to admit sufficient statistics of bounded dimension is that it conform to this general form of a maximum entropy distribution.)

The $\lambda_k$ parameters are, in essence, Lagrange multipliers. When dealing with equality constraints, their values are found by solving a system of nonlinear equations:

$F_{k}={\frac {\partial }{\partial \lambda _{k}}}\log Z(\lambda _{1},\ldots ,\lambda _{m}).$

For inequality constraints, the Lagrange multipliers are determined through the solution of a convex optimization program with linear constraints. In both scenarios, a direct, closed-form solution is elusive; calculating these Lagrange multipliers typically necessitates the application of numerical methods.

Continuous Case

When we venture into the realm of continuous distributions, the familiar Shannon entropy, defined for discrete spaces, is insufficient. Edwin Jaynes, in his wisdom (1963, 1968, 2003), proposed a formula closely related to relative entropy (also see differential entropy):

$H_{c}=-\int p(x)\log {\frac {p(x)}{q(x)}}\,dx$

Here, $q(x)$ , which Jaynes termed the "invariant measure", is proportional to the limiting density of discrete points. For now, we shall assume $q$ is known; its role will become clearer once we've outlined the solution equations.

A conceptually similar quantity, the relative entropy, is more commonly defined as the Kullback–Leibler divergence of $p$ from $q$ (though sometimes, confusingly, its negative is used). The principle of minimizing this divergence, attributed to Kullback, is known as the Principle of Minimum Discrimination Information.

Suppose we have testable information I concerning a variable $x$ that spans an interval of the real numbers (all integrals below are over this interval). This information typically manifests as $m$ constraints on the expected values of functions $f_k$ , meaning our probability density function, $p(x)$ , must satisfy these moment constraints:

$\int p(x)f_{k}(x)\,dx\geq F_{k}\qquad k=1,\dotsc ,m.$

$F_k$ are the observable quantities. We also maintain the fundamental requirement that the probability density integrates to one:

$\int p(x)\,dx=1.$

The probability density function that maximizes $H_c$ under these constraints is given by:

$p(x)={\frac {q(x)\exp \left[\lambda _{1}f_{1}(x)+\dotsb +\lambda _{m}f_{m}(x)\right]}{Z(\lambda _{1},\dotsc ,\lambda _{m})}}$

where the partition function is defined as:

$Z(\lambda _{1},\dotsc ,\lambda _{m})=\int q(x)\exp \left[\lambda _{1}f_{1}(x)+\dotsb +\lambda _{m}f_{m}(x)\right]\,dx.$

As in the discrete case, if all moment constraints are equalities, the $\lambda_k$ parameters are determined by solving the system of nonlinear equations:

$F_{k}={\frac {\partial }{\partial \lambda _{k}}}\log Z(\lambda _{1},\dotsc ,\lambda _{m}).$

When inequality moment constraints are present, the Lagrange multipliers are found by solving a convex optimization program.

The invariant measure function $q(x)$ can be best understood by considering the scenario where $x$ is known to exist only within a bounded interval, say $(a, b)$ , and no other information is provided. In this situation, the maximum entropy probability density function becomes:

$p(x)=A \cdot q(x), \quad a < x < b$

where $A$ is a normalization constant. The invariant measure function $q(x)$ essentially represents the prior density that encodes a "lack of relevant information." It cannot be derived from the principle of maximum entropy itself; it must be determined through other logical means, such as the principle of transformation groups or marginalization theory.

Examples

For a deeper dive into specific maximum entropy distributions, consult the dedicated article on maximum entropy probability distributions.

Justifications for the Principle of Maximum Entropy

Proponents of the principle of maximum entropy offer several compelling arguments for its application in probability assignment. These arguments generally operate within the established framework of Bayesian probability and are thus subject to its foundational postulates.

Information Entropy as a Measure of 'Uninformativeness'

Consider a discrete probability distribution across $m$ mutually exclusive propositions. The most informative distribution would arise when one proposition is known with certainty—in such a case, the information entropy would be zero. Conversely, the least informative distribution occurs when there’s no basis to favor one proposition over another. Here, the only justifiable distribution is the uniform one, and the information entropy reaches its maximum possible value, $\log m$ . Therefore, information entropy can be viewed as a numerical metric quantifying how uninformative a given probability distribution is, ranging from zero (completely informative) to $\log m$ (completely uninformative).

The argument follows that by selecting the distribution with the maximum entropy permitted by our existing information, we are choosing the most uninformative distribution possible. To opt for a distribution with lower entropy would imply assuming knowledge that we simply do not possess. Hence, the maximum entropy distribution is presented as the sole logically defensible choice. However, the dependence of the solution on the dominating measure, represented by $m(x)$ , is a point of contention, as this dominating measure is, in effect, arbitrary.

The Wallis Derivation

This particular argument, a suggestion from Graham Wallis to E. T. Jaynes in 1962, holds a distinct advantage: it is strictly combinatorial. It sidesteps any reliance on information entropy as a measure of 'uncertainty' or 'uninformativeness,' concepts that can be nebulous. The information entropy function is not presupposed but emerges organically from the argument itself. Furthermore, the argument naturally steers towards the maximization of information entropy, rather than its arbitrary manipulation.

Imagine an individual tasked with assigning probabilities among $m$ mutually exclusive propositions. They possess certain testable information but are unsure how to integrate it into their probability assignments. Their solution? A devised random experiment: they will distribute $N$ "quanta" of probability (each valued at $1/N$ ) randomly among the $m$ possibilities. One might visualize this as throwing $N$ balls into $m$ buckets, blindfolded, ensuring each throw is independent and each bucket is identical. After this distribution, they will verify if the resulting probability assignment aligns with their known information. (For this step to succeed, the information must be confined to an open set within the space of probability measures.) If it's inconsistent, they discard it and repeat the process. If consistent, their assessment is:

$p_i = \frac{n_i}{N}$

where $p_i$ is the probability of the $i$ -th proposition, and $n_i$ is the number of quanta (or balls) allocated to that proposition.

To mitigate the "graininess" of this assignment, a substantial number of probability quanta, $N$ , is required. Instead of physically conducting this potentially lengthy experiment, the individual opts to calculate the most probable outcome. The probability of any specific outcome is governed by the multinomial distribution:

$Pr(\mathbf{p}) = W \cdot m^{-N}$

where $W = \frac{N!}{n_1!\,n_2! \,\dotsb \,n_m!}$ is known as the multiplicity of the outcome.

The most probable result is the one that maximizes this multiplicity, $W$ . Rather than maximizing $W$ directly, one can equivalently maximize any monotonically increasing function of $W$ . The chosen function is $\frac{1}{N} \log W$ :

$\frac{1}{N} \log W = \frac{1}{N} \log \frac{N!}{n_1!\,n_2! \,\dotsb \,n_m!} = \frac{1}{N} \log \frac{N!}{(Np_{1})!\,(Np_{2})! \,\dotsb \,(Np_{m})!}$ $= \frac{1}{N} \left(\log N!-\sum _{i=1}^{m}\log((Np_{i})!)\right).$

At this juncture, to simplify the expression, the protagonist considers the limit as $N \to \infty$ , effectively transitioning from discrete probability levels to smooth, continuous ones. Employing Stirling's approximation, they arrive at:

$\lim _{N\to \infty }\left({\frac {1}{N}}\log W\right) = -\sum _{i=1}^{m}p_{i}\log p_{i} = H(\mathbf {p} ).$

The task now is to maximize this entropy, $H(\mathbf{p})$ , subject to the constraints of their testable information. They have thus demonstrated that the maximum entropy distribution is the most probable outcome among all "fair" random distributions, especially in the limit where probability levels shift from discrete to continuous.

Compatibility with Bayes' Theorem

Giffin and Caticha (2007) argue that Bayes' theorem and the principle of maximum entropy are not only compatible but can be viewed as specific instances of a broader framework: the "method of maximum relative entropy." They assert that this unified method reproduces all aspects of traditional Bayesian inference and, crucially, opens avenues for addressing problems that were previously intractable for either principle individually. Furthermore, recent work (Lazar 2003, Schennach 2005) demonstrates how frequentist relative-entropy-based inference approaches, such as empirical likelihood and exponentially tilted empirical likelihood, can be integrated with prior information to perform Bayesian posterior analysis.

Jaynes himself characterized Bayes' theorem as a method for calculating probabilities, while maximum entropy served as a means for assigning prior probability distributions.

Conceptually, it's even possible to derive a posterior distribution directly from a given prior distribution using the principle of minimum cross-entropy. This is a special case of the Principle of Maximum Entropy where a uniform distribution is the designated prior. This approach treats the problem formally as a constrained optimization task, with the entropy functional as the objective function. When the testable information is presented as average values (averaged over the sought-after probability distribution), the resulting distribution is formally the Gibbs (or Boltzmann) distribution, whose parameters must be solved to achieve minimum cross-entropy and satisfy the given constraints.

Relevance to Physics

The principle of maximum entropy shares a connection with a fundamental assumption in the kinetic theory of gases known as molecular chaos, or the Stosszahlansatz. This principle posits that the distribution function describing particles entering a collision can be factorized. While this can be interpreted as a strict physical hypothesis, it also functions as a heuristic assumption concerning the most probable configuration of particles prior to a collision.

So, there you have it. A principle that tries to quantify ignorance, to be honest about what we don't know. It's elegant, in its own stark way. Don't misinterpret its utility as subservience; it's a tool, yes, but one that demands respect for its underlying logic. And if you think this is all there is, you're probably not paying enough attention.