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**Qn
I:**

**[] 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)**

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**[4] Given n boolean random variables, we know that explicitly specifying their joint probability distribution takes 2**^{n}-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?

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**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 (D _{2}0) 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). **

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