Analyticity of Entropy Rate of Hidden Markov Chains
Guangyue Han, Brian Marcus
备注：
The title has been changed. The new main theorem now combines the old main theorem and the remark following the old main theorem. A new section is added as an introduction to complex analysis. General principle and an example to determine the domain of analyticity of entropy rate have been added. Relaxed conditions for analyticity of entropy rate and the corresponding examples are added. The section about binary markov chain corrupted by binary symmetric noise is taken out (to be part of another paper)
We prove that under mild positivity assumptions the entropy rate of a hidden Markov chain varies analytically as a function of the underlying Markov chain parameters. A general principle to determine the domain of analyticity is stated. An example is given to estimate the radius of convergence for the entropy rate. We then show that the positivity assumptions can be relaxed, and examples are given for the relaxed conditions. We study a special class of hidden Markov chains in more detail: binary hidden Markov chains with an unambiguous symbol, and we give necessary and sufficient conditions for analyticity of the entropy rate for this case. Finally, we show that under the positivity assumptions the hidden Markov chain {\em itself} varies analytically, in a strong sense, as a function of the underlying Markov chain parameters.
We prove that under certain mild assumptions, the entropy rate of a hidden Markov chain, observed when passing a finite-state stationary Markov chain through a discrete-time continuous-output channel, is jointly analytic as a function of the input Markov chain parameters and the channel parameters. In particular, as consequences of the main theorems, we obtain analyticity for the entropy rate associated with representative channels: Cauchy and Gaussian.
Let $Γ$ be a graph and $P$ be a reversible random walk on $Γ$. From the $L^2$ analyticity of the Markov operator $P$, we deduce that an iterate of odd exponent of $P$ is `lazy', that is there exists an integer $k$ such that the transition probability (for the random walk $P^{2k+1}$) from a vertex $x$ to itself is uniformly bounded from below. The proof does not require the doubling property on $Γ$ but only a polynomial control of the volume.
We consider Hidden Markov Chains obtained by passing a Markov Chain with rare transitions through a noisy memoryless channel. We obtain asymptotic estimates for the entropy of the resulting Hidden Markov Chain as the transition rate is reduced to zero.
A hidden Markov model (HMM) is said to have path-mergeable states if for any two states i,j there exists a word w and state k such that it is possible to transition from both i and j to k while emitting w. We show that for a finite HMM with path-mergeable states the block estimates of the entropy rate converge exponentially fast. We also show that the path-mergeability property is asymptotically typical in the space of HMM topolgies and easily testable.
Consider a hidden Markov chain obtained as the observation process of an ordinary Markov chain corrupted by noise. Zuk, et. al. [13], [14] showed how, in principle, one can explicitly compute the derivatives of the entropy rate of at extreme values of the noise. Namely, they showed that the derivatives of standard upper approximations to the entropy rate actually stabilize at an explicit finite time. We generalize this result to a natural class of hidden Markov chains called ``Black Holes.'' We also discuss in depth special cases of binary Markov chains observed in binary symmetric noise, and …
Consider a filtering process associated to a hidden Markov model with densities for which both the state space and the observation space are complete, separable, metric spaces. If the underlying, hidden Markov chain is strongly ergodic and the filtering process fulfills a certain coupling condition we prove that, in the limit, the distribution of the filtering process is independent of the initial distribution of the hidden Markov chain. If furthermore the hidden Markov chain is uniformly ergodic, then we prove that the filtering process converges in distribution.
We prove what appears to be the first concentration of measure result for hidden Markov processes. Our bound is stated in terms of the contraction coefficients of the underlying Markov process, and strictly generalizes the Markov process concentration results of Marton (1996) and Samson (2000). Somewhat surprisingly, the bound turns out to be the same as for ordinary Markov processes; this property, however, fails for general hidden/observed process pairs.
For the continuous-time and the discrete-time three-state hidden Markov model, the flux of the likelihood function up to 3-dimension of the observed process is shown explicitly. As an application, the sufficient and necessary condition of the reversibility of the observed process is shown.
In this paper we study various properties of finite stochastic systems or hidden Markov chains as they are alternatively called. We discuss their construction following different approaches and we also derive recursive filtering formulas for the different systems that we consider. The key tool is a simple lemma on conditional expectations.
We consider the filtering of continuous-time finite-state hidden Markov models, where the rate and observation matrices depend on unknown time-dependent parameters, for which no prior or stochastic model is available. We quantify and analyze how the induced uncertainty may be propagated through time as we collect new observations, and used to simultaneously provide robust estimates of the hidden signal and to learn the unknown parameters, via techniques based on pathwise filtering and new results on the optimal control of rough differential equations.
We study the convergence in rough path topology of a certain class of discrete processes, the hidden Markov walks, to a Brownian motion with an area anomaly. This area anomaly, which is a new object, keeps track of the time-correlation of the discrete models and brings into light the question of embeddings of discrete processes into continuous time. We also identify an underlying combinatorial structure in the hidden Markov walks, which turns out to be a generalization of the occupation time from the classical ergodic theorem in the spirit of rough paths.
We consider a hidden Markov model with multiplicative noise emerging from studies of software reliability. We show the stability of the optimal filter with respect to general initial conditions in the total variation- and $L^p$-norm and deduce explicit rates. Remarkably, stability turns out to be independent of the ergodic behavior of the signal.
The generalized perturbative approach is an all purpose variant of Stein's method used to obtain rates of normal approximation. Originally developed for functions of independent random variables this method is here extended to functions of the realization of a hidden Markov model. In this dependent setting, rates of convergence are provided in some applications to stochastic geometry, leading, in each instance, to an extra log-factor vis a vis the rate in the independent case.
I describe a new Markov chain method for sampling from the distribution of the state sequences in a non-linear state space model, given the observation sequence. This method updates all states in the sequence simultaneously using an embedded Hidden Markov model (HMM). An update begins with the creation of a ``pool'' of K states at each time, by applying some Markov chain update to the current state. These pools define an embedded HMM whose states are indexes within this pool. Using the forward-backward dynamic programming algorithm, we can then efficiently choose a state sequence at random wit…
In this paper, the forgetting of the initial distribution for a non-ergodic Hidden Markov Models (HMM) is studied. A new set of conditions is proposed to establish the forgetting property of the filter, which significantly extends all the existing results. Both a pathwise-type convergence of the total variation distance of the filter started from two different initial distributions, and a convergence in expectation are considered. The results are illustrated using generic models of non-ergodic HMM and extend all the results known so far.
We consider filtering for a hidden Markov model that evolves with multiple time scales in the hidden states. In particular, we consider the case where one of the states is a scaled Ornstein-Uhlenbeck process with fast reversion to a shifting-mean that is controlled by a continuous time Markov chain modeling regime change. We show that the nonlinear filter for such a process can be approximated by an averaged filter that asymptotically coincides with the true nonlinear filter of the regime-changing Markov chain as the rate of mean reversion approaches infinity. The asymptotics exploit weak conv…
This work is devoted to the almost sure stabilization of adaptive control systems that involve an unknown Markov chain. The control system displays continuous dynamics represented by differential equations and discrete events given by a hidden Markov chain. Different from previous work on stabilization of adaptive controlled systems with a hidden Markov chain, where average criteria were considered, this work focuses on the almost sure stabilization or sample path stabilization of the underlying processes. Under simple conditions, it is shown that as long as the feedback controls have linear g…
We consider the Sequential Probability Ratio Test applied to Hidden Markov Models. Given two Hidden Markov Models and a sequence of observations generated by one of them, the Sequential Probability Ratio Test attempts to decide which model produced the sequence. We show relationships between the execution time of such an algorithm and Lyapunov exponents of random matrix systems. Further, we give complexity results about the execution time taken by the Sequential Probability Ratio Test.
For a wide class of continuous-time Markov processes, including all irreducible hypoelliptic diffusions evolving on an open, connected subset of $\RL^d$, the following are shown to be equivalent: (i) The process satisfies (a slightly weaker version of) the classical Donsker-Varadhan conditions; (ii) The transition semigroup of the process can be approximated by a finite-state hidden Markov model, in a strong sense in terms of an associated operator norm; (iii) The resolvent kernel of the process is `$v$-separable', that is, it can be approximated arbitrarily well in operator norm by finite-ran…
In an effort to aid communication among different fields and perhaps facilitate progress on problems common to all of them, this article discusses hidden Markov processes from several viewpoints, especially that of symbolic dynamics, where they are known as sofic measures, or continuous shift-commuting images of Markov measures. It provides background, describes known tools and methods, surveys some of the literature, and proposes several open problems.
Let $(X, Y) = (X_n, Y_n)_{n \geq 1}$ be the output process generated by a hidden chain $Z = (Z_n)_{n \geq 1}$, where $Z$ is a finite state, aperiodic, time homogeneous, and irreducible Markov chain. Let $LC_n$ be the length of the longest common subsequences of $X_1, \ldots, X_n$ and $Y_1, \ldots, Y_n$. Under a mixing hypothesis, a rate of convergence result is obtained for $\mathbb{E}[LC_n]/n$.
We consider the problem of filtering an unseen Markov chain from noisy observations, in the presence of uncertainty regarding the parameters of the processes involved. Using the theory of nonlinear expectations, we describe the uncertainty in terms of a penalty function, which can be propagated forward in time in the place of the filter.
In this paper we study the problem of learning phylogenies and hidden Markov models. We call a Markov model nonsingular if all transition matrices have determinants bounded away from 0 (and 1). We highlight the role of the nonsingularity condition for the learning problem. Learning hidden Markov models without the nonsingularity condition is at least as hard as learning parity with noise, a well-known learning problem conjectured to be computationally hard. On the other hand, we give a polynomial-time algorithm for learning nonsingular phylogenies and hidden Markov models.
In this paper, we develop methods of nonlinear filtering and prediction of an unobservable Markov chain with a finite set of states. This Markov chain controls coefficients of AR(p) model. Using observations generated by AR(p) model we have to estimate the state of Markov chain in the case of an unknown probability transition matrix. Comparison of proposed non-parametric algorithms with the optimal methods in the case of the known transition matrix is carried out by simulating.
We study finite horizon optimal switching problems for hidden Markov chain models under partially observable Poisson processes. The controller possesses a finite range of strategies and attempts to track the state of the unobserved state variable using Bayesian updates over the discrete observations. Such a model has applications in economic policy making, staffing under variable demand levels and generalized Poisson disorder problems. We show regularity of the value function and explicitly characterize an optimal strategy. We also provide an efficient numerical scheme and illustrate our resul…
To estimate the emission parameters in hidden Markov models one commonly uses the EM algorithm or its variation. Our primary motivation, however, is the Philips speech recognition system wherein the EM algorithm is replaced by the Viterbi training algorithm. Viterbi training is faster and computationally less involved than EM, but it is also biased and need not even be consistent. We propose an alternative to the Viterbi training -- adjusted Viterbi training -- that has the same order of computational complexity as Viterbi training but gives more accurate estimators. Elsewhere, we studied the …
