Pathwise guessing in categorical time series ith unbounded alphabets

The following learning problem arises naturally in various applications: Given a finite sample from a categorical or count time series, can we learn a function of the sample that (nearly) maximizes the probability of correctly guessing the values of a given portion of the data using the values from the remaining parts? Unlike the classical task of estimating conditional probabilities in a stochastic process, our approach avoids explicitly estimating these probabilities. We propose a non-parametric guessing function with a learning rate that is independent of the alphabet size. Our analysis focuses on a broad class of time series models that encompasses finite-order Markov chains, some hidden Markov chains, Poisson regression for count process, and one-dimensional Gibbs measures. Additionally, we establish a minimax lower bound for the rate of convergence of the risk associated with our guessing problem. This lower bound matches the upper bound achieved by our estimator up to a logarithmic factor, demonstrating its near-optimality.