人工智能代写｜COMS W4701: Artificial Intelligence Homework 5

这是一篇美国的人工智能cs代写

Instructions: Compile all solutions to the written problems on this assignment in a single PDF file. Show your work by writing down relevant equations or expressions, and/or by explaining any logic that you are using to bypass known equations. When ready, follow the submission instructions to submit all files to Gradescope. Please be mindful of the deadline, as late submissions are not accepted, as well as our course policies on academic honesty.

Problem 1: Descendants of Effects (20 points)

We will investigate the absence of conditional independence guarantees between two random variables when an arbitrary descendant of a common effect is observed. We will consider the simple case of a causal chain of descendants:

Suppose that all random variables are binary. The marginal distributions of A and B are both uniform (0.5, 0.5), and the CPTs of the common effect D0 and its descendants are as follows:

(a) Give an analytical expression for the joint distribution Pr(D0, D1, · · · , Dn). Your expression should only contain CPTs from the Bayes net parameters. What is the size of the full joint distribution, and how many entries are nonzero?

(b) Suppose we observe Dn = +dn. Numerically compute the CPT Pr(+dn|D0). Please show how you can solve for it using the joint distribution in (a), even if you do not actually use it.

(c) Let’s turn our attention to A and B. Give a minimal analytical expression for Pr(A, B, D0, +dn).Your expression should only contain CPTs from the Bayes net parameters or the CPT you found in part (b) above.

(d) Lastly, compute Pr(A, B | +dn). Show that A and B are not independent conditioned on Dn.

Problem 2: Bayes Net 1 (20 points)

The following Bayes net is the “Fire Alarm Belief Network” from the Sample Problems of the Belief and Decision Networks tool on AIspace. All variables are binary.

(a) Which pair(s) of nodes are guaranteed to be independent given no observations in the Bayes net? Now suppose Alarm is observed. Identify and briefly explain the nodes whose conditional independence guarantees, given Alarm, are different from their independence guarantees, given no observations.

(b) We are interested in computing the conditional distribution Pr(Smoke | report). Give an analytical expression in terms of the Bayes net CPTs that computes this distribution (or its unnormalized version). What is the maximum size of the resultant table if all marginalization is done at the end?

(c) We employ variable elimination to solve for the query above. Identify a variable ordering that i) yields the greatest number of operations possible, and ii) yields the fewest number of operations possible. Also give the max table sizes in each case.

(d) Following your second variable ordering above, numerically solve for Pr(Smoke | report) using the default parameters in the applet example. You may check your answer using the applet,but you should work it out yourself and show your work.

Problem 3: Bayes Net 2 (20 points)

The following Bayes net is the “Simple Diagnostic Example” from the Sample Problems of the Belief and Decision Networks tool on AIspace. All variables are binary.

(a) We can describe all guaranteed independences in the Bayes net by defining two or more subsets of nodes Si , such that all nodes in Si are independent of all nodes in Sj for i ̸=j. For example, we can define S1 = {Smokes} and S2 = {Influenza, Sore Throat, Fever} given no observations. Do the same for conditionally independent nodes i) given Influenza, ii) given Bronchitis, and iii) given both Influenza and Bronchitis. Make sure your answers capture all guaranteed independences.

(b) Consider using likelihood weighting to solve two queries, one in which Influenza and Smokes are observed, and one in which Coughing and Wheezing are observed. Explain how the two cases differ in the distribution of the resulting samples, as well as the weights that are applied to the samples.

(c) We perform Gibbs sampling and would like to resample the Influenza variable conditioned on the current sample (+s, +st, −f, −b, +c, −w). Give a minimal analytical expression for the sampling distribution Pr(Influenza | sample) (or its unnormalized form). What is the maximum size of the table that has to be constructed?

(d) Numerically solve for the sampling distribution Pr(Influenza | sample) using the default parameters in the applet example. You may check your answer using the applet, but you should work it out yourself and show your work.