Entropy Rate for Hidden Markov Chains with rare transitions

Limit Theorems in Hidden Markov Models

In this paper, under mild assumptions, we derive a law of large numbers, a central limit theorem with an error estimate, an almost sure invariance principle and a variant of Chernoff bound in finite-state hidden Markov models. These limit theorems are of interest in certain ares in statistics and information theory. Particularly, we apply the limit theorems to derive the rate of convergence of the maximum likelihood estimator in finite-state hidden Markov models.

Low and Upper Bound of Approximate Sequence for the Entropy Rate of Binary Hidden Markov Processes

In the paper, the approximate sequence for entropy of some binary hidden Markov models has been found to have two bound sequences, the low bound sequence and the upper bound sequence. The error bias of the approximate sequence is bound by a geometric sequence with a scale factor less than 1 which decreases quickly to zero. It helps to understand the convergence of entropy rate of generic hidden Markov models, and it provides a theoretical base for estimating the entropy rate of some hidden Markov models at any accuracy.

Rényi Divergence in General Hidden Markov Models

In this paper, we examine the existence of the Rényi divergence between two time invariant general hidden Markov models with arbitrary positive initial distributions. By making use of a Markov chain representation of the probability distribution for the general hidden Markov model and eigenvalue for the associated Markovian operator, we obtain, under some regularity conditions, convergence of the Rényi divergence. By using this device, we also characterize the Rényi divergence, and obtain the Kullback-Leibler divergence as α \rightarrow 1 of the Rényi divergence. Several examples, including th…

An efficient algorithm for the entropy rate of a hidden Markov model with unambiguous symbols

We demonstrate an efficient formula to compute the entropy rate $H(μ)$ of a hidden Markov process with $q$ output symbols where at least one symbol is unambiguously received. Using an approximation to $H(μ)$ to the first $N$ terms we give a $O(Nq^3$) algorithm to compute the entropy rate of the hidden Markov model. We use the algorithm to estimate the entropy rate when the parameters of the hidden Markov model are unknown.In the case of $q =2$ the process is the output of the Z-channel and we use this fact to give bounds on the capacity of the Gilbert channel.

Distributed Arithmetic Coding for Sources with Hidden Markov Correlation

Distributed arithmetic coding (DAC) has been shown to be effective for Slepian-Wolf coding, especially for short data blocks. In this letter, we propose to use the DAC to compress momery-correlated sources. More specifically, the correlation between sources is modeled as a hidden Markov process. Experimental results show that the performance is close to the theoretical Slepian-Wolf limit.

Decision Feedback Based Scheme for Slepian-Wolf Coding of sources with Hidden Markov Correlation

We consider the problem of compression of two memoryless binary sources, the correlation between which is defined by a Hidden Markov Model (HMM). We propose a Decision Feedback (DF) based scheme which when used with low density parity check codes results in compression close to the Slepian Wolf limits.

A Decision Feedback Based Scheme for Slepian-Wolf Coding of sources with Hidden Markov Correlation

We consider the problem of compression of two memoryless binary sources, the correlation between which is defined by a Hidden Markov Model (HMM). We propose a Decision Feedback (DF) based scheme which when used with low density parity check codes results in compression close to the Slepian Wolf limits.

Novel Lower Bounds on the Entropy Rate of Binary Hidden Markov Processes

Recently, Samorodnitsky proved a strengthened version of Mrs. Gerber's Lemma, where the output entropy of a binary symmetric channel is bounded in terms of the average entropy of the input projected on a random subset of coordinates. Here, this result is applied for deriving novel lower bounds on the entropy rate of binary hidden Markov processes. For symmetric underlying Markov processes, our bound improves upon the best known bound in the very noisy regime. The nonsymmetric case is also considered, and explicit bounds are derived for Markov processes that satisfy the $(1,\infty)$-RLL constra…

On Hidden Markov Processes with Infinite Excess Entropy

We investigate stationary hidden Markov processes for which mutual information between the past and the future is infinite. It is assumed that the number of observable states is finite and the number of hidden states is countably infinite. Under this assumption, we show that the block mutual information of a hidden Markov process is upper bounded by a power law determined by the tail index of the hidden state distribution. Moreover, we exhibit three examples of processes. The first example, considered previously, is nonergodic and the mutual information between the blocks is bounded by the log…

Context Tree Estimation in Variable Length Hidden Markov Models

We address the issue of context tree estimation in variable length hidden Markov models. We propose an estimator of the context tree of the hidden Markov process which needs no prior upper bound on the depth of the context tree. We prove that the estimator is strongly consistent. This uses information-theoretic mixture inequalities in the spirit of Finesso and Lorenzo(Consistent estimation of the order for Markov and hidden Markov chains(1990)) and E.Gassiat and S.Boucheron (Optimal error exponents in hidden Markov model order estimation(2003)). We propose an algorithm to efficiently compute t…

Exact Computation of Kullback-Leibler Distance for Hidden Markov Trees and Models

We suggest new recursive formulas to compute the exact value of the Kullback-Leibler distance (KLD) between two general Hidden Markov Trees (HMTs). For homogeneous HMTs with regular topology, such as homogeneous Hidden Markov Models (HMMs), we obtain a closed-form expression for the KLD when no evidence is given. We generalize our recursive formulas to the case of HMMs conditioned on the observable variables. Our proposed formulas are validated through several numerical examples in which we compare the exact KLD value with Monte Carlo estimations.

A Markovian Model for Joint Observations, Bell's Inequality and Hidden States

While the standard approach to quantum systems studies length preserving linear transformations of wave functions, the Markov picture focuses on trace preserving operators on the space of Hermitian (self-adjoint) matrices. The Markov approach extends the standard one and provides a refined analysis of measurements and quantum Markov chains. In particular, Bell's inequality becomes structurally clear. It turns out that hidden state models are natural in the Markov context. In particular, a violation of Bell's inequality is seen to be compatible with the existence of hidden states. The Markov mo…

Improving the Runtime of Algorithmic Polarization of Hidden Markov Models

We improve the runtime of the linear compression scheme for hidden Markov sources presented in a 2018 paper of Guruswami, Nakkiran, and Sudan. Under the previous scheme, compressing a message of length $n$ takes $O(n \log n)$ runtime, and decompressing takes $O(n^{1 + δ})$ runtime for any fixed $δ> 0.$ We present how to improve the runtime of the decoding scheme to $O(n \log n)$ by caching intermediate results to avoid repeating computation.

Efficient tests for equivalence of hidden Markov processes and quantum random walks

While two hidden Markov process (HMP) resp. quantum random walk (QRW) parametrizations can differ from one another, the stochastic processes arising from them can be equivalent. Here a polynomial-time algorithm is presented which can determine equivalence of two HMP parametrizations $\cM_1,\cM_2$ resp. two QRW parametrizations $\cQ_1,\cQ_2$ in time $O(|§|\max(N_1,N_2)^{4})$, where $N_1,N_2$ are the number of hidden states in $\cM_1,\cM_2$ resp. the dimension of the state spaces associated with $\cQ_1,\cQ_2$, and $§$ is the set of output symbols. Previously available algorithms for testing equi…

A note on a complex Hilbert metric with application to domain of analyticity for entropy rate of hidden Markov processes

In this note, we show that small complex perturbations of positive matrices are contractions, with respect to a complex version of the Hilbert metric, on the standard complex simplex. We show that this metric can be used to obtain estimates of the domain of analyticity of entropy rate for a hidden Markov process when the underlying Markov chain has strictly positive transition probabilities.

Analyticity of Entropy Rates of Continuous-State Hidden Markov Models

The analyticity of the entropy and relative entropy rates of continuous-state hidden Markov models is studied here. Using the analytic continuation principle and the stability properties of the optimal filter, the analyticity of these rates is shown for analytically parameterized models. The obtained results hold under relatively mild conditions and cover several classes of hidden Markov models met in practice. These results are relevant for several (theoretically and practically) important problems arising in statistical inference, system identification and information theory.

隐马尔可夫模型中参数估计的信息几何方法

Information Geometry Approach to Parameter Estimation in Hidden Markov Models

本文提出基于信息几何的高效估计量，仅利用 k-记忆数据直方图估计隐马尔可夫过程的转移矩阵，并在切空间上讨论模型等价性。

The Partially Observable Hidden Markov Model and its Application to Keystroke Dynamics

The partially observable hidden Markov model is an extension of the hidden Markov Model in which the hidden state is conditioned on an independent Markov chain. This structure is motivated by the presence of discrete metadata, such as an event type, that may partially reveal the hidden state but itself emanates from a separate process. Such a scenario is encountered in keystroke dynamics whereby a user's typing behavior is dependent on the text that is typed. Under the assumption that the user can be in either an active or passive state of typing, the keyboard key names are event types that pa…

Taylor series expansions for the entropy rate of Hidden Markov Processes

Finding the entropy rate of Hidden Markov Processes is an active research topic, of both theoretical and practical importance. A recently used approach is studying the asymptotic behavior of the entropy rate in various regimes. In this paper we generalize and prove a previous conjecture relating the entropy rate to entropies of finite systems. Building on our new theorems, we establish series expansions for the entropy rate in two different regimes. We also study the radius of convergence of the two series expansions.

The Entropy of a Binary Hidden Markov Process

The entropy of a binary symmetric Hidden Markov Process is calculated as an expansion in the noise parameter epsilon. We map the problem onto a one-dimensional Ising model in a large field of random signs and calculate the expansion coefficients up to second order in epsilon. Using a conjecture we extend the calculation to 11th order and discuss the convergence of the resulting series.

A hybrid scheme for encoding audio signal using hidden Markov models of waveforms

This paper reports on recent results related to audiophonic signals encoding using time-scale and time-frequency transform. More precisely, non-linear, structured approximations for tonal and transient components using local cosine and wavelet bases will be described, yielding expansions of audio signals in the form tonal + transient + residual. We describe a general formulation involving hidden Markov models, together with corresponding rate estimates. Estimators for the balance transient/tonal are also discussed.

具有马尔可夫或隐马尔可夫信号先验的线性模型的复本分析

Replica Analysis of the Linear Model with Markov or Hidden Markov Signal Priors

本文采用统计物理中的复本方法，估计了信源为马尔可夫链或隐马尔可夫模型的线性模型的自由能、平均互信息与MMSE，证明其可解耦为带状态信息的单输入AWGN信道，且数值结果与现有算法结果高度吻合。

Exactly Optimal Bayesian Quickest Change Detection for Hidden Markov Models

This paper considers the quickest detection problem for hidden Markov models (HMMs) in a Bayesian setting. We construct an augmented HMM representation of the problem that allows the application of a dynamic programming approach to prove that Shiryaev's rule is an (exact) optimal solution. This augmented representation highlights the problem's fundamental information structure and suggests possible relaxations to more exotic change event priors not appearing in the literature. Finally, this augmented representation also allows us to present an efficient computational method for implementing th…

Learning Hidden Markov Models using Non-Negative Matrix Factorization

The Baum-Welsh algorithm together with its derivatives and variations has been the main technique for learning Hidden Markov Models (HMM) from observational data. We present an HMM learning algorithm based on the non-negative matrix factorization (NMF) of higher order Markovian statistics that is structurally different from the Baum-Welsh and its associated approaches. The described algorithm supports estimation of the number of recurrent states of an HMM and iterates the non-negative matrix factorization (NMF) algorithm to improve the learned HMM parameters. Numerical examples are provided as…

From finite-system entropy to entropy rate for a Hidden Markov Process

A recent result presented the expansion for the entropy rate of a Hidden Markov Process (HMP) as a power series in the noise variable $\eps$. The coefficients of the expansion around the noiseless ($\eps = 0$) limit were calculated up to 11th order, using a conjecture that relates the entropy rate of a HMP to the entropy of a process of finite length (which is calculated analytically). In this communication we generalize and prove the validity of the conjecture, and discuss the theoretical and practical consequences of our new theorem.

Informational and Causal Architecture of Continuous-time Renewal and Hidden Semi-Markov Processes

We introduce the minimal maximally predictive models (ε-machines) of processes generated by certain hidden semi-Markov models. Their causal states are either hybrid discrete-continuous or continuous random variables and causal-state transitions are described by partial differential equations. Closed-form expressions are given for statistical complexities, excess entropies, and differential information anatomy rates. We present a complete analysis of the ε-machines of continuous-time renewal processes and, then, extend this to processes generated by unifilar hidden semi-Markov models and semi-M…

The Markov Memory for Generating Rare Events

We classify the rare events of structured, memoryful stochastic processes and use this to analyze sequential and parallel generators for these events. Given a stochastic process, we introduce a method to construct a new process whose typical realizations are a given process' rare events. This leads to an expression for the minimum memory required to generate rare events. We then show that the recently discovered classical-quantum ambiguity of simplicity also occurs when comparing the structure of process fluctuations.

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 accura…