Continuous-time Bayesian networks is a natural structured representation
language for multicomponent stochastic processes that evolve continuously over
time. Despite the compact representation, inference in such models is
intractable even in relatively simple structured networks. Here we introduce a
mean field variational approximation in which we use a product of inhomogeneous
Markov processes to approximate a distribution over trajectories. This
variational approach leads to a globally consistent distribution, which can be
efficiently queried. Additionally, it provides a lower bound on the probability
of observations, thus making it attractive for learning tasks. We provide the
theoretical foundations for the approximation, an efficient implementation that
exploits the wide range of highly optimized ordinary differential equations
(ODE) solvers, experimentally explore characterizations of processes for which
this approximation is suitable, and show applications to a large-scale
realworld inference problem.

本文介绍了一种基于 Markov 过程的 mean field variational approximation 方法，用于近似描述 Continuous-time Bayesian networks 中的概率分布，并提供了较好的推断和学习效果。

连续时间贝叶斯网络的均场变分近似

Mean Field Variational Approximation for Continuous-Time Bayesian  Networks

The bounded confidence model of opinion dynamics, introduced by Deffuant et
al, is a stochastic model for the evolution of continuous-valued opinions
within a finite group of peers. We prove that, as time goes to infinity, the
opinions evolve globally into a random set of clusters too far apart to
interact, and thereafter all opinions in every cluster converge to their
barycenter. We then prove a mean-field limit result, propagation of chaos: as
the number of peers goes to infinity in adequately started systems and time is
rescaled accordingly, the opinion processes converge to i.i.d. nonlinear Markov
(or McKean-Vlasov) processes; the limit opinion processes evolves as if under
the influence of opinions drawn from its own instantaneous law, which are the
unique solution of a nonlinear integro-differential equation of Kac type. This
implies that the (random) empirical distribution processes converges to this
(deterministic) solution. We then prove that, as time goes to infinity, this
solution converges to a law concentrated on isolated opinions too far apart to
interact, and identify sufficient conditions for the limit not to depend on the
initial condition, and to be concentrated at a single opinion. Finally, we
prove that if the equation has an initial condition with a density, then its
solution has a density at all times, develop a numerical scheme for the
corresponding functional equation, and show numerically that bifurcations may
occur.

本文研究有界信心模型的观点动态演化，证明了当时间趋近无穷大时，各群体内部观点趋于局部集群，随后每个集群内的观点将收敛到集群重心。同时，研究表明，随着同伴数量趋近无穷大，观点处理过程收敛到非线性 Markov（或 McKean-Vlasov）过程。最后，证明了方程具有密度初始条件时具有密度解，并开发了一种数值方案。

意见动态的有界信任模型

The Bounded Confidence Model Of Opinion Dynamics

We prove that for symmetric Markov processes of diffusion type admitting a
"carr\'e du champ", the Poincar\'e inequality is equivalent to the exponential
convergence of the associated semi-group in one (resp. all) $\L^p(\mu)$ spaces
for $p\in (1,+\infty)$. Part of this result extends to the stationary non
necessarily symmetric situation.

对于对称扩散型 Markov 过程，如果存在 carré du champ，则 Poincaré 不等式等价于关联半群在所有 L^p (μ) 空间中的指数收敛性，其中 1<p<+∞。该结果的部分扩展适用于非对称的静止情况。

Poincaré 不等式与半群的 Lp 收敛

Poincaré inequality and the Lp convergence of semi-groups

This paper considers the following stochastic control problem that arises in
opportunistic spectrum access: a system consists of n channels (Gilbert-Elliot
channels)where the state (good or bad) of each channel evolves as independent
and identically distributed Markov processes. A user can select exactly k
channels to sense and access (based on the sensing result) in each time slot. A
reward is obtained whenever the user senses and accesses a good channel. The
objective is to design a channel selection policy that maximizes the expected
discounted total reward accrued over a finite or infinite horizon. In our
previous work we established the optimality of a greedy policy for the special
case of k = 1 (i.e., single channel access) under the condition that the
channel state transitions are positively correlated over time. In this paper we
show under the same condition the greedy policy is optimal for the general case
of k >= 1; the methodology introduced here is thus more general. This problem
may be viewed as a special case of the restless bandit problem, with multiple
plays. We discuss connections between the current problem and existing
literature on this class of problems.

本文研究了随机控制问题在机会式频谱访问中的应用，提出了设计一种通道选择策略，以最大化预期折扣总报酬为目标，将问题与该类问题现有文献进行了讨论。