A new continuous-time Markov chain rate change formula is proven. This theorem is used to derive existence and uniqueness of novel filtering equations akin to the Duncan–Mortensen–Zakai equation and the Fujisaki–Kallianpur–Kunita equation but for Markov signals with general continuous-time Markov chain observations. The equations in this second theorem have the unique feature of being driven by both the observations and the process counting the observation transitions. A direct method of solving these filtering equations is also derived. Most results apply as special cases to the continuous-time Hidden Markov Models (CTHMM), which have become important in applications like disease progression tracking, The corresponding CTHMM results are stated as corollaries. Finally, application of our general theorems to Markov chain importance sampling, rejection sampling and branching particle filtering algorithms is also explained and these are illustrated by way of disease tracking simulations.

Herein, we propose generating CAPTCHAs through random field simulation and give a novel, effective and efficient algorithm to do so. Indeed, we demonstrate that sufficient information about word tests for easy human recognition is contained in the site marginal probabilities and the site-to-nearby-site covariances and that these quantities can be embedded directly into certain conditional probabilities, designed for effective simulation. The CAPTCHAs are then partial random realizations of the random CAPTCHA word. We start with an initial random field (e.g., randomly scattered letter pieces) and use Gibbs resampling to re-simulate portions of the field repeatedly using these conditional probabilities until the word becomes human-readable. The residual randomness from the initial random field together with the random implementation of the CAPTCHA word provide significant resistance to attack. This results in a CAPTCHA, which is unrecognizable to modern optical character recognition but is recognized about 95% of the time in a human readability study.

Many stochastic approximation procedures result in a stochastic algorithm of the form h/sub k+1/=h/sub k/+1/k(b/sub k/-A/sub k/h/sub k/), for all k=1,2,3,... . Here, {b/sub k/,k=1,2,3...} is a R/sup d/-valued process, {A/sub k/,k=1,2,3,...} is a symmetric, positive semidefinite Re/sup d/spl times/d/-valued process, and {h/sub k/,k=1,2,3,...} is a sequence of stochastic estimates which hopefully converges to h/sup /spl Delta//=[lim/sub N/spl rarr//spl infin//1/N/spl Sigma//sub k=1//sup N/EA/sub k/]/sup -1/ {lim/sub N/spl rarr//spl infin//1/N/spl Sigma//sub k=1//sup N/Eb/sub k/} (assuming everything here is well defined). We give an elementary proof which relates the almost sure convergence of {h/sub k/,k=1,2,3,...} to strong laws of large numbers for {b/sub k/,k=1,2,3,...} and {A/sub k/,k=1,2,3,...}.

Almost-sure (a.s.) bounds for linear, constant-gain, adaptive filtering algorithms are investigated. For instance, under general pseudo-stationarity and dependence conditions on the driving data {/spl psi/k,k=1,2,3,...}, {Y/sub k/,k=0,1,2,...} a.s. convergence and rates of a.s. convergence (as the algorithm gain /spl epsiv//spl rarr/0) are established for the LMS algorithm h/sub k+1//sup /spl epsiv//=h/sub k//sup /spl epsiv//+/spl epsiv/Y/sub k/(/spl psi//sub k+1/-Y/sub k//sup T/h/sub k//sup /spl epsiv//) subject to some nonrandom initial condition h/sub 0//sup /spl epsiv//=h/sub 0/. In particular, defining {g/sub k//sup /spl epsiv//}/sub k=0//sup /spl infin// by g/sub 0//sup /spl epsiv//=h/sub 0/ and g/sub k+1//sup /spl epsiv//=g/sub k//sup /spl epsiv//+/spl epsiv/(E[Y/sub k//spl psi//sub k+1/]-E[Y/sub k/Y/sub k//sup T/]g/sub k//sup /spl epsiv//) for k=0,1,2,..., the author shows that for any /spl gamma/>0 max/sub 0/spl les/k/spl les//spl gamma//spl epsiv/(-1/)|h/sub k//sup /spl epsiv//-g/sub k//sup /spl epsiv//|/spl rarr/0 as /spl epsiv//spl rarr/0 a.s. and under a stronger dependency condition, the author shows that for any 00, max/sub 0/spl les/k/spl les//spl gamma//spl epsiv/(-/spl zeta//)|h/sub k//sup /spl epsiv//-g/sub k//sup /spl epsiv//| converges (as /spl epsiv//spl rarr/0) a.s. At a rate marginally slower than O((/spl epsiv//sup 2-/spl zeta//log log(/spl epsiv//sup -/spl zeta//))/sup 1/2 /). Then, under a stronger pseudostationarity assumption it is shown that similar results hold if the sequences {g/sub k//sup /spl epsiv//}/sub k=0//sup /spl infin//,/spl epsiv/>0 in the above results are replaced with the solution g/sup 0/(/spl middot/) of a nonrandom linear ordinary differential equation, i.e. one has, max/sub 0/spl les/k/spl les/[/spl gamma//spl epsiv/(-/spl zeta//])|h/sub k//sup /spl epsiv//-g/sup 0/(/spl epsiv/k)|/spl rarr/0 as /spl epsiv//spl rarr/0 a.s., where one can attach a rate to this convergence under the stronger dependency condition. The almost-sure bounds contained in the paper complement previously developed weak convergence results in Kushner and Shwartz [IEEE Trans. Information Theory, IT-30(2), 177-182, 1984] and, as are "near optimal". Moreover, the proofs used to establish these bounds are quite elementary.>