Uncertainty

Qn I:

1. [] In class, we said that bayesian inference can be seen as a general case of the propositional inference. This means that propositional inference should be subsumed by bayesian inference. Suppose we know that A => B, and that B is not true. We know from the modus tollens inference that A is not true. Show that this also follows from bayesian inference. (Hint: We can model the propositional implication A =>B as P(B/A) = 1)

2. [4] Given n boolean random variables, we know that explicitly specifying their joint probability distribution takes 2ⁿ-1 probabilities. By exhibiting their inter-relationships in the form of a bayesian network, we can often get by with specification of fewer (conditional probability) numbers. What is the MAXIMUM number of probabilities that might be needed in an n-variable bayes net? How will the net look like in that case? What is the MINIMUM number of probabilities that might be needed in an n-variable bayes net? What will the structure of network look like in that case?

Qn II .[] We have decided to use the bayes-network technology to model the Springfield nuclear power plant (where Homer Simpson is gainfully employed). Here is our understanding of the domain (this is all official based on my conversations with Bart). The presence of inferior plutonium or low quality heavy water (D₂0) in the plant reactor leads to a core melt-down in the reactor. When core-meltdown occurs, it tends to irradiate the employees (such as Homer) causing them to glow in the dark. Core-meltdown also tends to shutoff the power grid which in turn causes the squishees in Apu's convenience store (called Squishees as Kwikee Mart) to melt and get watery.

Here are a bunch of probabilities I got from the Springfield Statistics Beureau: If inferior plutonium alone is present, core-meltdown occurs with probability 0.3; If low quality heavy water alone is present, the core-meltdown occurs with probability 0.3. If both inferior plutonium and low-quality heavy water are present, then the core-meltdown occurs with probability 0.7. Even under completely normal conditions, core-meltdown can occur with probability 0.01. Core-meltdown leads to glowing-in-the-dark employees with probability 0.5. What with lax quality control over at Springfield plant, even under normal circumstances, Homer and his buddies tend to glow in the dark with probability 0.05. Core-meltdown causes squishee liquification with 0.9, and over at Apu's squishees tend to get watery even without a core-meltdown with probability 0.1. Finally, the probability that Springfield plant gets inferior-quality plutonium is 0.3 and that it gets low-quality heavy water is 0.4 (you know that wacky Burns--he is always trying to buy cheap stuff and make more bucks).

Part A:[] We want to represent all his knowledge conveniently as a nice little bayes network. Show the configuration of the network along with the requisite conditional probabilities.

Part B.[] Given your network above, which of the following (conditional) independences hold? Briefly justify your answer in each case:

                            i.                         Independence of inferior-quality plutonium and low-quality heavy water in Springfield nuclear plant

                        ii.                         Independence of inferior-quality plutonium and low-quality heavy water in Springfield nuclear plant, given watery squishees over at Apu's.

                    iii.                         Independence of low-quality heavy-water and wateryness of squishees at Apu's, given core-meltdown

Part C. [] Consider the following probabilities: p1: Probability that low-quality heavy water is present, p2: probability that low-quality heavy water is present, given that core-meltdown has occurred, p3: probability that low-quality heavy water is present given that core-meltdown occurred and we know that inferior-quality plutonium is present. Given the network above, what would be the relative ordering of these probailities (Note: I don't need the exact numbers. I am just looking for a statement such as "pi less than or equal to pj and pj is less than or equal to pk"). Briefly explain your reasoning.

Part D.[] It is afterall the holiday season and we really would like to make sure that Springfield will have a merry season devoid of any untoward incidents. What is the probability that this is going to be the case? (That is, what is the probability that there is no inferior-grade plutonium in the plant, and there is no low-grade heavy water in the plant, and there is no core-meltdown, and Homer ain't glowing in the dark, and Apu's squishees are not watery).

We just found that there the squishees at Apus’s place have liquefied. We also found out, from reliable sources, that the batch of heavy water they are using at the reactor is INDEED low quality one. Compute the probability distribution on Homer glowing in the dark. Use the enumeration method to compute this probability (notice that this involves marginalization and normalization). [If you want to be sure, You can check your answer by plugging this network in the bayesnet tool and doing the query).