arXiv subjects
Publications and source records attributed to M. Grendar, Jr..
(Jaynes') Method of (Shannon-Kullback's) Relative Entropy Maximization (REM or MaxEnt) can be - at least in the discrete case - according to the Maximum Probability Theorem (MPT) viewed as an asymptotic instance of the Maximum Probability method (MaxProb). A simple bayesian interpretation of MaxProb is given here. MPT carries the interpretation over into REM.
Traditionally, the Method of (Shannon-Kullback's) Relative Entropy Maximization (REM) is considered with linear moment constraints. In this work, the method is studied under frequency moment constraints which are non-linear in probabilities. The constraints challenge some justifications of REM since a) axiomatic systems are developed for classical linear moment constraints, b) the feasible set of distributions which is defined by frequency moment constraints admits several entropy maximizing distributions (I-projections), hence probabilistic justification of REM via Conditioned Weak Law of Large Numbers cannot be invoked. However, REM is not left completely unjustified in this setting, since Entropy Concentration Theorem and Maximum Probability Theorem can be applied. Maximum Renyi-Tsallis' entropy method (maxTent) enters this work because of non-linearity of X-frequency moment constraints which are used in Non-extensive Thermodynamics. It is shown here that under X-frequency moment constraints maxTent distribution can be unique and different than the I-projection. This implies that maxTent does not choose the most probable distribution and that the maxTent distribution is asymptotically conditionally improbable. What are adherents of maxTent accomplishing when they maximize Renyi's or Tsallis' entropy?
Chernoff's bound binds a tail probability (ie. $Pr(X \ge a)$, where $a \ge EX$). Assuming that the distribution of $X$ is $Q$, the logarithm of the bound is known to be equal to the value of relative entropy (or minus Kullback-Leibler distance) for $I$-projection $\hat P$ of $Q$ on a set $\mathcal{H} \triangleq \{P: E_PX = a\}$. Here, Chernoff's bound is related to Maximum Likelihood on exponential form and consequently implications for the notion of complementarity are discussed. Moreover, a novel form of the bound is proposed, which expresses the value of the Chernoff's bound directly in terms of the $I$-projection (or generalized $I$-projection).
Ill-posed inverse problems of the form y = X p where y is J-dimensional vector of a data, p is m-dimensional probability vector which cannot be measured directly and matrix X of observable variables is a known J,m matrix, J < m, are frequently solved by Shannon's entropy maximization (MaxEnt). Several axiomatizations were proposed to justify the MaxEnt method (also) in this context. The main aim of the presented work is two-fold: 1) to view the concept of complementarity of MaxEnt and Maximum Likelihood (ML) tasks from a geometric perspective, and consequently 2) to provide an intuitive and non-axiomatic answer to the 'Why MaxEnt?' question.
Randomness is viewed through an analogy between a physical quantity, density of gas, and a mathematical construct -- probability density. Boltzmann's deduction of equilibrium distribution of ideal gas placed in an external potential field than provides a way of viewing probability density from a perspective of forces/potentials, hidden behind it.