[SOLVED] CSC384 Assignment 4 variable elimination for Bayes Nets

In this assignment you will implement variable elimination for Bayes Nets.What is supplied: Python code that implements Variable, Factor, and BN objects. The file bnetbase.pycontains the class definitions for these objects. The code supports representing factors as tables of valuesindexed by various settings of the variables in the factors scope.The template file bnetbase.py also contains function prototypes for the functions you must implement.Question 1. Implement Variable Elimination (worth 60/100 marks)Implement the following functions that operate on Factor objects and then use these functions to implementVE (variable elimination): multiply factors (worth 10 points). This function takes as input a list of Factor objects; it createsand returns a new factor that is equal to the product of the factors in the list. Do not modify any ofthe input factors. restrict factor (worth 10 points). This function takes as input a single factor, a variable V and avalue d from the domain of that variable. It creates and returns a new factor that is the restriction ofthe input factor to the assignment V = d. Do not modify the input factor. sum out variable (worth 10 points). This function takes as input a single factor, and a variable V;it creates and returns a new factor that is the result of summing V out of the input factor. Do notmodify the input factor. VE (worth 30 points). This function takes as input a Bayes Net object (object of class BN), a variablethat is the query variable Q, and a list of variables E that are the evidence variables (all of whichhave had some value set as evidence using the variables set evidence interface). Compute theprobability of every possible assignment to Q given the evidence specified by the evidence settingsof the evidence variables. Return these probabilities as a list, where every number corresponds theprobability of one of Qs possible values. Do not modify any factor of the input Bayes net.Assignment 4, University of Toronto, CSC384 Intro to AI, Winter 2020 3Question 2: Problem Solving with your VE Implementaion (worth 40/100marks)For the following questions, you will submit your answers using the Google form that is located at https://forms.gle/suJ2eVbusTimJb359.1. Use your implementation to answer questions regarding the following Bayesian network. All variables are binary, with values Dom(A) = {a,a}, Dom(B) = {b,b}, etc.The probability table values are as follows (probabilities add to one so you can compute the remaining probability values):P(a) = 0.9P(b|ah) = 1.0;P(b|ah) = 0.0;P(b|a;h) = 0.5;P(b|ah) = 0.6P(c|bg) = 0.9;P(c|bg) = 0.9;P(c|b;g) = 0.1;P(c|bg) = 1.0P(d|c f) = 0.0;P(d|c f) = 1.0;P(d|c; f) = 0.7;P(d|cf) = 0.2P(e|c) = 0.2;P(e|c) = 0.4P(f) = 0.1P(g) = 1.0P(h) = 0.5P(i|b) = 0.3;P(i|b) = 0.9Using your Variable Elimination implementation (or by hand!), compute the following probabilitiesand post these to the Google Form (worth 10 points):(a) P(b|a)(b) P(c|a)(c) P(c|ae)(d) P(c|a f)Assignment 4, University of Toronto, CSC384 Intro to AI, Winter 2020 42. Next, examine the file carDiagnosis.py. This specifies a Bayes Net for diagnosing various reasonswhy a car might not start. The layout of this Bayes Net is shown below, and the various CPTs for theNet are specified in carDiagnosis.py as Factors:Each variable of the Net is shown in a square box along with the values that the variable can take.For example, The variable BatteryVoltage (which is the variable bv in the file carDiagnosis.py)can take on one of three different values: strong, weak and dead.The numbers and bars show the unconditional probabilities of the variables taking on their different values. For example, in the Net we have P(BatteryVoltage = dead) = 0.41. For the variousCPTs for the Net, see carDiagnosis.py.Using your Variable Elimination implementation (or based on inspection!), answer the followingquestions and post your answers to the Google Form:(a) (worth 5 points) Show a case of conditional independence in the Net where knowing someevidence item V1 = d1 makes another evidence item V2 = d2 irrelevant to the probability ofsome third variable V3. (Note that conditional independence requires that the independenceholds for all values of V3).(b) (worth 5 points) Show a case of conditional independence in the Net where two variables areindependent given NO evidence at a third variable yet dependent given evidence at the thirdvariable.Assignment 4, University of Toronto, CSC384 Intro to AI, Winter 2020 5(c) (worth 10 points) Show a sequence of accumulated evidence items V1 = d1,,V k = dk (i.e.,each evidence item in the sequence is added to the previous evidence items) such that eachadditional evidence item increases the probability that some variable V has the value d. (Thatis, the probability of V = d increases monotonically as we add evidence items). What is P(V =dV1 = d1,,Vk = dk)?(d) (worth 10 points) Show a sequence of accumulated evidence items V1 = d1,,V k = dk (i.e.,each evidence item in the sequence is added to the previous evidence items) such that eachadditional evidence item decreases the probability that some variable V has the value d. (Thatis, the probability of V = d decreases monotonically as we add evidence items). What is P(V =dV1 = d1,,Vk = dk)?