22 problems

Minimality conjecture. If every mixed state in $\boldsymbol{\mathcal{R}}$ corresponds to a unique probability distribution over symbols, then the corresponding mixed-state presenta…

Let $(X_n)_{n\ge 0}$ be the underlying Markov chain of a hidden Markov model, and suppose that it is irreducible. The associated filtering process is the measure-valued Markov proc…

Let $C_N$ be the upper-bound entropy for a chain of length $N$, with expansion … Here $C_N^{(k)}$ denotes the coefficient of order $k$, and $C^{(k)}$ is its value once the coeffici…

Let $cmathcal{E}_cperp$ be the natural orthogonal-complement effective subspace defined above, and let $coverline{\bm p}$ be the vector defined in Corollary corresponding to…

Consider a hidden Markov model with trends, with true transition matrix $Q^*$ and maximum likelihood estimator $Q^{\hat{\theta}_n}$ based on the first $n$ observations. Let…

Let $H_5$ be the graph described in the source, and let $P$ be its corresponding Markov-chain transition matrix. The observed sites are vertices of the graph, and identification me…

Consider a combinatorially symmetric Markov chain with underlying graph $H_5$, whose forcing number is $3$ and whose maximum nullity is $2$. Smaller-than-zero-forcing identificatio…

Let $P$ be the transition matrix of the Markov chain whose underlying directed graph is a directed cycle on $S$ vertices. Observing a set of vertices means using the corresponding…

Let $\mathcal{Z}$ be the set of precise models under consideration, and define the lower updated expectation by … whenever the set over which the infimum is taken is nonempty. Limi…

Consider the joint posterior density ${\rm P}_n(\epsilon^{21},\epsilon^{12})$ for the two transition rates of a two-state model with noisy measurements, and let the true rate value…

Let ${\rm P}_n(\epsilon)$ denote the posterior distribution of the transition rate $\epsilon$ after $n$ observations, where the measurements are noisy. Consistency conjecture. The…

The decay rate $\rho$ in the exponential variation bound … for the $g$-function associated with the binary symmetric channel should admit an improved estimate that incorporates dep…

Let $(X_k,Y_k)_{k\,in\mathbb{Z}}$ be a stationary infinite-dimensional hidden Markov model as in Section 3, with $X_k\in\{-1,1\}^{\mathbb{Z}}$ and $Y_k\in\{-1,1\}^{\mathbb{Z}}$. Su…

Let the hidden Markov chain have transition probability matrix (tr.pr.m.) and let the associated filtering process be the Markov chain of conditional distributions of the hidden st…

Threshold-policy entropy optimality conjecture. In the special case, the infimum of entropy attainable under threshold policies is the same as that under general policies.

Ergodicity conjecture. The conditions of that theorem can be weakened to the case when the underlying chain is only ergodic.

Kunita's exchange conjecture. If the hidden process is purely nondeterministic and the observations are nondegenerate, then $$ holds true.

Piecewise computability conjecture. The dynamic programming algorithm for finding the minimizer of the ${\bar R}_C$-risk, together with exponential smoothing, could be used to find…

Let $\widetilde{M}^+$ be the time-reversed presentation and let $\mathcal{U}$ denote the mixed-state presentation operation. An \text{epsilon-machine} is the minimal unifilar p…

Let $\mathbf{f}_{n,l}$ be the unconstrained hidden Markov model with $l$ hidden states and strings of length $n$, and let $d(n,l)$ denote the maximum degree of its invariants. Mono…

Let $\Sigma$ be the alphabet, let $\mathbf{g}_{n,d}$ denote the parameterization of the relevant hidden Markov model, and let $p=(p(v))_{v\in\Sigma^n}$ be a string function. For wo…

In a hidden Markov model whose states at sample size $n$ include count vectors $(0,n),(1,n-1),\ldots,(n,0)$, suppose that states can be grouped into ranges and that the transition…