CS5340-Uncertainty Modeling in AI Solved

Your goal is to implement the sum-product algorithm for inference in directed trees with nodes representing discrete random variables.

Setup Description
The main.py script accepts a list of json files and outputs the number of marginals that were computed correctly by your program.

Example usage: python main.py problem1.json.

The main script first constructs a tree.Tree object based on the description in the JSON file and then calls the inference.sum product function to compute the marginals. Finally, it compares the marginals computed by the inference.sum  product function to the correct marginals present in the JSON file.

Your task is to only modify the functions in inference.py. Your code will only be tested on tree-like Bayesian networks (no poly-trees). Refer to the comments in the code for details.

 

Figure 1: An example tree.

Structure of the JSON file

Let’s consider the tree shown in Fig. 1 with the corresponding probability tables shown in Fig. 2. The following json file represents this tree.

The nodes key holds the number of nodes in the tree and their corresponding dimension where dimension implies the number of categories in the support of the random variable. For example, a dimension of 4 would mean that the random variable can take one of four possible values from {0,1,2,3}. In the case of Fig. 1, nodes 0 & 1 represent binary random variables and node 2 represents a ternary random variable.

The factors key holds all the pair factors (or conditional distributions of the form p(x|y)); e.g., the 0 key holds the two factors between (0, 1) and (0, 2) represented by the probability tables in Fig. 2 (b) and (c) respectively.

The observations key holds the observations, if any.

The p  0 key holds p(x0) (Fig. 2 (a)).

The marginals key holds the correct marginals obtained after running the sum-product algorithm. The marginals generated by your implementation should match these values.

 

  Figure 2: Probability tables.

 

1                        {"nodes":

2                        {"0": {"dim": 2},

3                        "1": {"dim": 2}, 4                     "2": {"dim": 3}}, 5        "factors":

6                                                                         {"0": {"1": [[0.3, 0.7],

7                                                                         [0.2, 0.8]],

8                                                                         "2": [[0.2, 0.3, 0.5],

9                                                                         [0.6, 0.1, 0.3]]}

10                                                                       },

11                                                                       "observations": {}, 12       "p_0": [[0.1, 0.9]], 13             "marginals":

14                            {"1": [[0.21, 0.79]],

15                            "2": [[0.56, 0.12, 0.32]],

16                            "0": [[0.1, 0.9]]

17                            }

18                            }

 

The following json file contains the case when the variable x2 has been observed to be 2.

 

1                        {"nodes":

2                        {"0": {"dim": 2},

3                        "1": {"dim": 2}, 4                     "2": {"dim": 3}}, 5        "factors":

6                                                                         {"0": {"1": [[0.3, 0.7],

7                                                                         [0.2, 0.8]],

8                                                                         "2": [[0.2, 0.3, 0.5],

9                                                                         [0.6, 0.1, 0.3]]}

10                                                                       },

11                                                                       "observations": {"2": 2},

12                                                                       "p_0": [[0.1, 0.9]], 13        "marginals":

14                            {"1": [[0.215625, 0.784375]],

15                            "2": [[0.0, 0.0, 1.0]],

16                            "0": [[0.15625, 0.84375]]

17                            }

18                            }

 

Some example JSON files have been included with the code for you to test your implementation. You can follow the structure of the JSON, as described above, and construct more problem files; the correct marginals can be obtained using a tool such as Belief and Decision Networks[1].