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We introduce a new nonparametric density estimator inspired by Markov Chains, and generalizing the well-known Kernel Density Estimator (KDE). Our estimator presents several benefits with respect to the usual ones and can be used straightforwardly as a foundation in all density-based algorithms. We prove the consistency of our estimator and we find it typically outperforms KDE in situations of large sample size and high dimensionality. We also employ our density estimator to build a local outlier detector, showing very promising results when applied to some realistic datasets.
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Conditional density estimation generalizes regression by modeling a full density f(yjx) rather than only the expected value E(yjx). This is important for many tasks, including handling multi-modality and generating prediction intervals. Though fundamental and widely applicable, nonparametric conditional density estimators have received relatively little attention from statisticians and little or none from the machine learning community. None of that work has been applied to greater than bivariate data, presumably due to the computational difficulty of data-driven bandwidth selection. We describe the double kernel conditional density estimator and derive fast dual-tree-based algorithms for bandwidth selection using a maximum likelihood criterion. These techniques give speedups of up to 3.8 million in our experiments, and enable the first applications to previously intractable large multivariate datasets, including a redshift prediction problem from the Sloan Digital Sky Survey.
We consider the problem of flexible modeling of higher order Markov chains when an upper bound on the order of the chain is known but the true order and nature of the serial dependence are unknown. We propose Bayesian nonparametric methodology based on conditional tensor factorizations, which can characterize any transition probability with a specified maximal order. The methodology selects the important lags and captures higher order interactions among the lags, while also facilitating calculation of Bayes factors for a variety of hypotheses of interest. We design efficient Markov chain Monte Carlo algorithms for posterior computation, allowing for uncertainty in the set of important lags to be included and in the nature and order of the serial dependence. The methods are illustrated using simulation experiments and real world applications.
The spectral gap $gamma$ of a finite, ergodic, and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $P$ may be unknown, yet one sample of the chain up to a fixed time $n$ may be observed. We consider here the problem of estimating $gamma$ from this data. Let $pi$ be the stationary distribution of $P$, and $pi_star = min_x pi(x)$. We show that if $n = tilde{O}bigl(frac{1}{gamma pi_star}bigr)$, then $gamma$ can be estimated to within multiplicative constants with high probability. When $pi$ is uniform on $d$ states, this matches (up to logarithmic correction) a lower bound of $tilde{Omega}bigl(frac{d}{gamma}bigr)$ steps required for precise estimation of $gamma$. Moreover, we provide the first procedure for computing a fully data-dependent interval, from a single finite-length trajectory of the chain, that traps the mixing time $t_{text{mix}}$ of the chain at a prescribed confidence level. The interval does not require the knowledge of any parameters of the chain. This stands in contrast to previous approaches, which either only provide point estimates, or require a reset mechanism, or additional prior knowledge. The interval is constructed around the relaxation time $t_{text{relax}} = 1/gamma$, which is strongly related to the mixing time, and the width of the interval converges to zero roughly at a $1/sqrt{n}$ rate, where $n$ is the length of the sample path.
We describe estimators $chi_n(X_0,X_1,...,X_n)$, which when applied to an unknown stationary process taking values from a countable alphabet ${cal X}$, converge almost surely to $k$ in case the process is a $k$-th order Markov chain and to infinity otherwise.
We consider state-aggregation schemes for Markov chains from an information-theoretic perspective. Specifically, we consider aggregating the states of a Markov chain such that the mutual information of the aggregated states separated by T time steps is maximized. We show that for T = 1 this approach recovers the maximum-likelihood estimator of the degree-corrected stochastic block model as a particular case, thereby enabling us to explain certain features of the likelihood landscape of this popular generative network model from a dynamical lens. We further highlight how we can uncover coherent, long-range dynamical modules for which considering a time-scale T >> 1 is essential, using synthetic flows and real-world ocean currents, where we are able to recover the fundamental features of the surface currents of the oceans.
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