[PGM] Stanford probability graph model (Probabilistic graphical model)-Lecture 2: template models and structured CPDs

The probabilistic graphical model series is explained by Daphne Koller In the probabilistic graphical model of the Stanford open course. Https://class.coursera.org/pgm-2012-002/class/index)

Main contents include (reprinted please indicate the original source http://blog.csdn.net/yangliuy)

1. probabilistic Graph Model Representation and deformation of Bayesian Networks and Markov networks.

2. Reasoning and inference methods, including Exact Inference (variable elimination, clique trees) and approximate inference (belief propagation message passing, Markov Chain Monte Carlo methods ).

3. Learning Methods for parameters and structures in the probability graph model.

4. Use the probability graph model for Statistical Decision modeling.

Lecture 2. template models and structured CPPS.

1 template models

The graphic layout model is a more compact description of the graphic model. Template variables are repeated variables in the graph model multiple times, such as the IQ of multiple students and the difficulty of multiple courses. The template graph model describes how the template variables inherit Dependencies from the template. Typical templatemodels include the dynamic Bayesian Model DBN, Hidden Markov Model hmm, and platemodels. The Dynamic Bayesian model mainly introduces Markov hypothesis and time immutability in Bayesian Networks. These models will be further introduced in the next few lectures. Let's take a look at the exercises of templatemodels.

2 CPD

CPD is the conditional probability distribution. The table method is not suitable for CPD. If a node has K parents, the table has O (2exp (k) rows. The general CPD definition is given below

It mainly includes the following models:

The deterministic CPD is also called context specific independence, as shown in the following formula:

Given the value of variable C, the random variable X is independent from the random variable Y condition, given Z.

3 treestructured CPD

CPD in the tree structure, different path selection in the tree structure will lead to different conditional independence. In the following example, given J and C1, that is, if someone is known to have selected recommendation letter 1, whether or not the work can be obtained depends only on the Validity probability distribution of recommendation letter L1, then the L2 → J Path becomes spurious, therefore, there is no active trails between L1 and L2. L1 and L2 are independent of each other. We call L1 and L2 belong to D-separated, and the conditional independence relationship given J and context c = C1. is also called CSI-separation, that is, context-related conditions are independent.

 

4 noisy or CPD

As shown in, the y node has a parent node Z1 to ZK, or relationship, that is, only Z1 to ZK do not occur, then y = 0, whether Zi is affected by the random variable XI. If we regard y as the Professor's recommendation letter, then X1 to XK can be seen as a student's performance, the probability of being admired by the professor when the student's item I is excellent is λ I. This figure means that if the professor appreciates at least one of the students' excellent performances, there will be a letter of recommendation. So what is z0? This probability is called leak probability. It can be considered that the professor is in a good mood one day. Even if every performance of a student is messy, he also gives a letter of recommendation, that is, this variable is not affected by the student's performance Xi, it depends entirely on the professor. So the probability of no recommendation letter is that the probability that each item of the student performs poorly multiplied by the probability that the professor is in a bad mood. Therefore, the conditional probability formula shown below is shown.

5 sigmoid CPD

Clearly understand the sigmoid function. This conditional probability is well understood, for example:

 

Exercise questions

6. Linear Gaussian Model

Y follows the Gaussian distribution, and its mean value is a linear combination of the father node XI. The variance is irrelevant to the father node.

In a linear Gaussian Model, if the coefficients in the linear combination of mean values depend on the random variable A, the variance and the random variable A are called conditional linear Gaussian models.