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扩散过程的非参数估计：微分方程方法AI 解读

We consider estimation of scalar functions that determine the dynamics of diffusion processes. It has been recently shown that nonparametric maximum likelihood estimation is ill-posed in this context. We adopt a probabilistic approach to regularize the problem by the adoption of a prior distribution for the unknown functional. A Gaussian prior measure is chosen in the function space by specifying its precision operator as an appropriate differential operator. We establish…

Summary We propose a flexible non-parametric specification of the emission distribution in hidden Markov models and we introduce a novel methodology for carrying out the computations. Whereas current approaches use a finite mixture model, we argue in favour of an infinite mixture model given by a mixture of Dirichlet processes. The computational framework is based on auxiliary variable representations of the Dirichlet process and consists of a forward–backward Gibbs sampling…

狄利克雷过程层级模型的回溯式马尔可夫链蒙特卡洛方法AI 解读

Inference for Dirichlet process hierarchical models is typically performed using Markov chain Monte Carlo methods, which can be roughly categorized into marginal and conditional methods. The former integrate out analytically the infinite-dimensional component of the hierarchical model and sample from the marginal distribution of the remaining variables using the Gibbs sampler. Conditional methods impute the Dirichlet process and update it as a component of the Gibbs sampler.…

可逆跳跃 MCMC 提议分布的高效构造AI 解读

Summary The major implementational problem for reversible jump Markov chain Monte Carlo methods is that there is commonly no natural way to choose jump proposals since there is no Euclidean structure in the parameter space to guide our choice. We consider mechanisms for guiding the choice of proposal. The first group of methods is based on an analysis of acceptance probabilities for jumps. Essentially, these methods involve a Taylor series expansion of the acceptance…

利用Metropolis–Hastings算法推断部分观测非线性扩散模型AI 解读

In this paper, we introduce a new Markov chain Monte Carlo approach to Bayesian analysis of discretely observed diffusion processes. We treat the paths between any two data points as missing data. As such, we show that, because of full dependence between the missing paths and the volatility of the diffusion, the rate of convergence of basic algorithms can be arbitrarily slow if the amount of the augmentation is large. We offer a transformation of the diffusion which breaks…

Gibbs 采样器的更新方案、相关结构、分块与参数化AI 解读

Summary In this paper many convergence issues concerning the implementation of the Gibbs sampler are investigated. Exact computable rates of convergence for Gaussian target distributions are obtained. Different random and non-random updating strategies and blocking combinations are compared using the rates. The effect of dimensionality and correlation structure on the convergence rates are studied. Some examples are considered to demonstrate the results. For a Gaussian image…

扩散过程离散观测的贝叶斯因子AI 解读

We present an approach to model selection for a time series of data on a fine time scale. The underlying process generating the data is modelled as a continuous time stochastic process. The underlying continuous processes are assumed to be diffusions with time varying drift and diffusion coefficient. Several approaches to modelling the diffusion coefficient are described. To perform model selection, we propose an approximation to the Bayes factor that uses only the discrete…

SUMMARY The use of the Gibbs sampler for Bayesian computation is reviewed and illustrated in the context of some canonical examples. Other Markov chain Monte Carlo simulation methods are also briefly described, and comments are made on the advantages of sample-based approaches for Bayesian inference summaries.