[PGM] Temporal Models

这里的一些东西只是将过去已有的东西用PGM解释了一遍，但优势还是明显的，对整体认识有帮助。

Video: https://www.youtube.com/watch?v=ogs4Oj8KahQ&index=13&list=PL50E6E80E8525B59C

3 - 1 - Overview of Template Models

3 - 2 - Temporal Models 

3 - 3 - Temporal Models - HMMs

3 - 4 - Plate Models (以下主要是此内容)

 

DBM 

动态BM其实就是加入了时间这个变脸，然后随着时间的变化，原有的节点会产生状态转移这样的过程。

 

Nested Plates & Overlapping Plates

  

左：一个courses框框可以包含很多个students框框。

右：如此一来，Courses框框的Difficulty就不是共享模式了。

 

结合后的效果如下：

 

意义在于：

Parameters and structure are reused within a BN and across different BNs.

 

Collective Inference

一个推断思维的例子：

学生选择第二第三课程，分低；但他的第一课程，分高；

那么，第一课程可能真的简单。

 

Plate notation

In Bayesian inference, plate notation is a method of representing variables that repeat in a graphical model.

Instead of drawing each repeated variable individually, a plate or rectangle is used to group variables into a subgraph that repeat together, and a number is drawn on the plate to represent the number of repetitions of the subgraph in the plate.

The assumptions are that

• • the subgraph is duplicated that many times,
  • the variables in the subgraph are indexed by the repetition number, and
  • any links that cross a plate boundary are replicated once for each subgraph repetition.

 

 

In this example, we consider Latent Dirichlet allocation, a Bayesian network that models how documents in a corpus are topically related. There are two variables not in any plate:

• • α is the parameter of the uniform Dirichlet prior on the per-document topic distributions,
  • β is the parameter of the uniform Dirichlet prior on the per-topic word distribution.

The outermost plate represents all the variables related to a specific document, including , the topic distribution for document i.

The M in the corner of the plate indicates that the variables inside are repeated M times, once for each document.

The inner plate represents the variables associated with each of the  words in document i:  is the topic for the jth word in document i, and  is the actual word used.

The N in the corner represents the repetition of the variables in the inner plate  times, once for each word in document i.

• • The circle representing the individual words is shaded, indicating that each  is observable, and
  • the other circles are empty, indicating that the other variables are latent variables.

The directed edges between variables indicate dependencies between the variables: for example, each  depends on  and β.