Shannon Entropy Rate of Hidden Markov Processes

Hidden Markov chains are widely applied statistical models of stochastic processes, from fundamental physics and chemistry to finance, health, and artificial intelligence. The hidden Markov processes they generate are notoriously complicated, however, even if the chain is finite state: no finite expression for their Shannon entropy rate exists, as the set of their predictive features is generically infinite. As such, to date one cannot make general statements about how random they are nor how structured. Here, we address the first part of this challenge by showing how to efficiently and accurately calculate their entropy rates. We also show how this method gives the minimal set of infinite predictive features. A sequel addresses the challenge’s second part on structure.

Coupon Collecting with Quotas

We analyze a variant of the coupon collector's problem, in which the probabilities of obtaining coupons and the numbers of coupons in a collection may be non-uniform. We obtain a finite expression for the generating function of the probabilities to complete a collection and show how this generalizes several previous results about the coupon collector's problem. Also, we provide applications about computational complexity and approximation.

Is Creation Eternal?

[The author examines the integral relationship between the Trinity and creation based on the theology of St. Bonaventure. Divine action is considered not as episodic intervention but within the context of relationship. Because creation is a finite expression of the infinite Word of God, it is intrinsically related to the primacy of Christ. An eternal creation rests on God's eternal resolve to love a finite order, and Christ is first in God's intention to love. It is argued that the eternal act of creation lies in the eternal nature of God's love.]

XXXVI. On the comparison of transcendents, with certain applications to the theory of definite integrals

1. The following objects are contemplated in this paper:— 1st. The demonstration of a fundamental theorem for the summation of integrals whose limits are determined by the roots of an algebraic equation. 2ndly. The application of that theorem to the problem of the comparison of algebraic transcendents. The immediate object of this application will in each case be the finite expression of the value of the sum of a series of integrals, Ʃ∫X dx , the differential coefficient, X, being an algebraic function, and the values of x at the limits being determined by the roots of an algebraic equation.