Bayesian Inference Based On Markov Process

With the continuous development of statistics and computer science,Bayesian inference has been increasingly widely used.Unlike the idea in classical statistics that regards parameters as fixed values,Bayesian inference considers parameters as random variables.We usually need to calculate the posterior distribution,which is the probability distribution of model parameters given observed data.For complex models,this posterior distribution is usually impossible to solve using analytical methods and requires approximation through numerical methods.Markov Chain Monte Carlo(MCMC)is a commonly used stochastic simulation method that can sample the posterior distribution by generating a Markov chain,and estimate statistical properties such as mean and variance of the posterior distribution through large-scale sampling.The advantage of MCMC is that it can handle high-dimensional parameter spaces and complex posterior distributions,and provide a more flexible and comprehensive way for parameter inference and model comparison than traditional methods.By reviewing classical literature,this paper provides a general description of the missing data model in the framework of a typical parameter model.The introduction of Monte Carlo methods has made the calculation of the posterior distribution of parameters in Bayesian inference simple and efficient.By introducing the scope,application methods,and relationships of commonly used stochastic computing methods,we explore the application of EM algorithm,Gibbs sampling,and IBF algorithm and analyze Bayesian inference results under the same known conditions.The innovation of this paper mainly lies in:1.From the perspective of spectral gap analysis,using gene linkage models as an example,the correlation between spectral gap estimation in Gibbs sampling and the convergence rate of the Markov chain is discussed.The impact of transfer matrix correlation on Markov chain convergence is also discussed.The transition probability matrix in the gene linkage model is calculated and discussed.2.In the process of Bayesian statistical inference,especially when discussing the joint posterior distribution,natural Markov processes are obtained based on Gibbs sampling.Using gene linkage models as an example,it is explained how the conjugacy of marginal Markov chains of different parameters leads to mutual constraints and influences.The idea of controlling the convergence rate of Markov chains based on this analysis is proposed.The interrelationship between dimensions is explained from this particular perspective,providing new research angles for future variable transformation and reshaping of parameter relationships,and proposing new algorithms for accelerating algorithms.3.The general framework of Bayesian inference for missing data is summarized,and a comprehensive discussion of gene linkage models is conducted within this framework.Analytical expressions for the marginal posterior distribution p(θ|Y),p(Z|Y),and the complete posterior distribution p(θ,Z|Y)of parameters are given under the condition of beta distribution prior,as well as common Bayesian point estimation methods.The changes in posterior distribution after introducing latent variables are discussed,and various estimation methods for latent variable Z are given.After introducing prior information,the advantages and disadvantages of each method are compared through the comparison of EM algorithm,Gibbs sampling,and IBF algorithm.Confidence intervals or credible regions of parameters are calculated in multiple ways.Through a complete theoretical framework and simulation calculations,the research provides important theoretical and application value for researchers using Bayesian inference for statistical analysis and exploration.4.The application of importance sampling method to calculate simplicial depth is emphasized,and the advantages of this algorithm over other Monte Carlo algorithms and exact algorithms are demonstrated through simulation and actual data examples.The simplicial depth is extended to regression analysis,obtaining more robust estimation results than traditional least squares methods.As an effective variance reduction technique,importance sampling method can be applied to the computation of various statistical estimation problems.We verified through a specific physical data example that the SD method based on regression analysis has better robustness compared to least squares estimation,thus having good application prospects.The main research content of this paper is as follows:Chapter 1 introduces the framework of Bayesian inference and the posterior distribution of missing data models,along with commonly used statistical computing methods for statistical inference on the posterior distribution.The convergence properties of Markov chains in transition probability density functions/matrices are discussed,and it is pointed out that estimates of convergence speed can be obtained by calculating the second largest eigenvalue.Chapter 2 uses the simple model of a genetic linkage model and the normal distribution to conduct calculations and discussions on the aforementioned issues.Special attention is given to changes in the posterior distribution of parameters and inference processes when prior information and latent variables are introduced.By decomposing the model completely and using analytical expressions,results from EM algorithm,Gibbs sampling,and IBF algorithm are compared.The estimation and analysis of the convergence speed of the Markov process involved in Gibbs sampling stochastic simulation is also discussed.Chapter 3 uses the normal distribution as an example to illustrate that,unlike classical statistical theory,the introduction of a prior distribution may change the independent relationships between parameters during Bayesian inference.Under convenient calculation conditions,it is suggested that the estimation problem of parameters should prioritize the estimation of their joint posterior distribution,which can provide richer correlation relationships between parameters.The spectral gap estimation of the transition probability density function of parameters in Gibbs sampling is also discussed,and an algorithm for solving the discretized continuous smooth density function is proposed.Chapter 4 discusses the calculation method of simplicial depth from the Bayesian perspective.Simulations using importance sampling are conducted,and the advantages of this method over other Monte Carlo algorithms and exact methods are demonstrated through simulations and real-world examples.simplicial depth is also extended to regression analysis.Chapter 5 provides a summary and outlook for future research based on the main conclusions drawn in the previous chapters.The limitations of this study are discussed,and potential areas for continued research are suggested.