Markov Chain
A class of stochastic processes whose core property is that the conditional distribution of the current state depends only on a finite number of past states. A k-th order Markov chain depends on the most recent k states; a first-order chain satisfies the memoryless property that, given the current state, the future is independent of the past.
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A research paper analyzes the convergence of Markov chains under the framework of Hilbert space embedding geometry induced by positive definite kernels, namely Maximum Mean Discrepancy (MMD).
For a reversible Markov chain, the largest eigenvalue of the transition operator is 1, the absolute value of the second largest eigenvalue is denoted λ₂, and the spectral gap is defined as 1−|λ₂|.
This paper extends the classical least squares framework of source conditions and capacity conditions to the setting of data-dependent (Markov chain trajectory) sampling.
The paper proves that for finite-order Markov chains, the minimum memory length is upper-bounded by the Markov order.
A k-th order Markov chain is a stochastic process in which the conditional distribution of the current state depends only on the most recent k historical states and is independent of earlier history.
Markov chains and their variants remain continuously active as foundational tools for probabilistic inference, and have regained attention in the theoretical analysis of diffusion models.
All Facts (6)
A research paper analyzes the convergence of Markov chains under the framework of Hilbert space embedding geometry induced by positive definite kernels, namely Maximum Mean Discrepancy (MMD).
50%UnverifiedFor a reversible Markov chain, the largest eigenvalue of the transition operator is 1, the absolute value of the second largest eigenvalue is denoted λ₂, and the spectral gap is defined as 1−|λ₂|.
50%UnverifiedThis paper extends the classical least squares framework of source conditions and capacity conditions to the setting of data-dependent (Markov chain trajectory) sampling.
50%UnverifiedThe paper proves that for finite-order Markov chains, the minimum memory length is upper-bounded by the Markov order.
50%UnverifiedA k-th order Markov chain is a stochastic process in which the conditional distribution of the current state depends only on the most recent k historical states and is independent of earlier history.
50%UnverifiedMarkov chains and their variants remain continuously active as foundational tools for probabilistic inference, and have regained attention in the theoretical analysis of diffusion models.
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