PRML笔记: GRAPHICAL MODELS (8-1)-优快云博客

本文探讨了贝叶斯网络及其对应的有向图模型，包括联合分布的分解过程及模型特性。文中详细介绍了多项式回归的贝叶斯模型，并讨论了生成模型的概念及其在图像识别中的应用。此外，还分析了离散变量与线性高斯模型在不同场景下的应用。

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8.1 Bayesian Networks

对应的directed graphical model为

在joint distribution分解过程中，对称性丧失。

Fully connected graph: link between every pair of nodes (not interesting)

Absence of links (interesting): information about the properties of the classes of distributions represented by the graph.

Joint distribution可以分解为:

(1) The key equation expresses the factorization properties of the joint distribution for a directed graphical model.

(2) 限制条件：Directed acyclic graphs (DAGs)

8.1.1 Example: Polynomial regression

Bayesian polynomial regression

Polynormial coefficients: w
Observed data: t
Input data: x
Noise variance: σ²
Hyperparameter: α

其中random variables有w和t,而parameters为x, σ²和α

Joint distribution of the random variables:

Predictive distribution:

8.1.2 Generative models

Ancestral sampling

(1) Sampling from joint distribution;

Sample from lower numbered node to higher numbered node.

(2) Sampling from marginal distribution

Sample from full joint distribution and only keep the values for required variables.

The primary role of latent variables: decompose complicated distribution over observed variables into simpler conditional distributions

例子--图像识别：

其中Object为discrete variable, 而Position和Orientation为continuous variables，最终目的是

The model describes the causal process to generate observed data, therefore, it is called generative model. In polynomial regression model, the observed variable x is not generated by default, so the model is not generative.

8.1.3 Discrete variables

Two K-state discrete variables:

由于constraint , (a)需要 K ² – 1 parameters, (b)需要 2(K – 1) parameters

For fully connected graph of M discrete variables, 有K ^M – 1 parameters. 控制模型的参数数量有如下几种方法：

(a) 选择具有合适connectivity的图结构:

The chain of M discrete variables:

共需要K – 1 + (M – 1) K (K – 1) parameters.

(b) sharing parameters:

μ is shared by all of the conditional distributions p(x_i| x_{i – 1}), 共需要K ² – 1 parameters.

共需要2M + 1 parameters

8.1.4 Linear-Gaussian models

The conditional distribution for each node:

可以求得单个变量的mean和covariance：

从公式中可以看出，mean和covariance都可以表达为recursive form，这有助于推导joint distribution的mean和covariance.

Joint distribution是multivariate Gaussian distribution:

例子：

扩展：

若每个node都是multivariate Gaussian variable,

其joint distribution也为Gaussian

转载于:https://www.cnblogs.com/hh-earlydays/archive/2013/04/03/2997735.html