﻿ 基于Gibbs抽样的分层贝叶斯模型在火灾发生次数统计推断中的应用 Application of Hierarchical Bayesian Model Based on Gibbs Sampling in Statistical Inference of Fire Occurrences

Statistics and Application
Vol.07 No.02(2018), Article ID:24681,9 pages
10.12677/SA.2018.72029

Application of Hierarchical Bayesian Model Based on Gibbs Sampling in Statistical Inference of Fire Occurrences

Kang Cao

Shanghai Maritime University, Shanghai

Received: Apr. 3rd, 2018; accepted: Apr. 21st, 2018; published: Apr. 28th, 2018

ABSTRACT

Gibbs sampling method is the most widely used method in MCMC algorithm. The basic idea of Gibbs sampling is to construct the Markov chain by the conditional distribution family of the components of the parameter vector when the high-dimensional parameters are posteriorly inferred, so that its invariant distribution is the target distribution. This topic is based on the method to determine the parameters of the model, which can be based on existing information to estimate the number of years, the number of fire occurred in the region and the estimated confidence interval of the parameters.

Keywords:Gibbs Sampling, Hierarchical Bayesian Model, Markov Chain, Fire

Gibbs抽样是MCMC抽样算法中应用最广泛的方法之一，其核心思想是对高维参数进行后验推断时，通过参数向量的分量的条件分布族来构造Markov链，使其不变分布为目标分布。本文利用Gibbs抽样方法结合分层贝叶斯模型，对我国各地区火灾发生次数进行了回测，结果显示，相比于传统的poisson分布刻画方法，基于Gibbs抽样的分层贝叶斯方法充分利用了历史信息使结果更具可信度。

1. 引言

2. 分层贝叶斯模型

1) 写出联合后验密度p(θ,φ|y)，其非正规化的形式是超先验分布p(φ)、总体分布p(θ|φ)和似然函数p(y|θ)的乘积。

2) 在给定超参数φ的情况下，确定θ的条件后验密度，固定观测值y的情况下，它是φ的函数p(θ|φ, y)。

3) 使用贝叶斯分析范例估计φ，也就是要获取边缘后验分布p(φ|y)。

3. Gibbs抽样

3.1. Gibbs抽样原理

Gibbs抽样 [7] 简单、应用最广泛的MCMC抽样方法之一，应用该抽样方法的前提是要分布π(x)的满条件分布已知，即对于任意i，在已知x的第i个分量以外其他分量值的条件下，第i个分量的条件分布

(1) 生成,

···

(i) 生成,

···

(p) 生成

3.2. Gibbs抽样的具体实现方法

···

4. 应用实例

4.1. 案例介绍

4.2. 模型建立

Table 1. Number of fire occurrences by region in 2012

4.3. 算法的实现与结果

Figure 1. Some parameters posterior histogram and density curve

Figure 2. Iterate over 10,000 times the dynamic average of some parameters

Figure 3. Remove the first 2000 posterior histograms and density plots

Table 2. Parameter confidence interval

5. 结论

Application of Hierarchical Bayesian Model Based on Gibbs Sampling in Statistical Inference of Fire Occurrences[J]. 统计学与应用, 2018, 07(02): 247-255. https://doi.org/10.12677/SA.2018.72029

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