Bayesian Network Problems
(Chains, Trees, & Polytrees)

  1. You have installed an Alarm at your house.  This alarm is reasonably sensitive (Probability of the Alarm going off given that there is a Burlary, P(a | b) = 0.8).  The alarm is prone to false positives, however -- P(a | ~b) = 0.1.  The probability of getting Burgled on any particular day is 0.01 (Apparantly, burglaries are pretty common).  While you are at work, you cannon hear your alarm, but your neighbor John has promised to call you if he hears the alarm.  John might not hear the alarm (if he is listening to loud must, say), or he might call you if he does not hear the alam (a car alarm might sound like your house alarm).  Quantitatively P(j | a) = 0.8, P(j | ~a) = 0.01.

    a.  Find the probability of being burgled if John calls.  Show all lambda and pi messages.

    b.  A second neighbor, Mary, has also offered to call.  Mary will not call if the alarm does not go off (she's much better at telling the difference between house alarms and car alarms than John), but she is out of the house much of the time, and will only call you when the alarm does go off about half the time.  If John does not call, and Mary does call, what is the probability that the house was burgled?  Show all lambda and pi messages that you need to compute to solve the problem

    c.  What is the probability that Mary will call, if John calls?   Show all lambda and pi messages necessary to solve the problem

    d.  One of the reasons that the alarm will go off when there is no burglary is that a minor earthquake can confuse the sensors.  We can revise our conditional probability for A as follows:

    P(A|B,E)
    		 A = ~a
    		 A =a
    B = ~b, E = ~e
    		 0.99
    		 0.01
    B = ~b, E = e
    		 0.8
    		 0.2
    B = b, E = ~e
    		 0.2
    		 0.8
    B = b, E = e
    		 0.1
    		 0.9

    A minor earthquake will be reported by the news about 1/2 the time (slow news days only).  Newscasters don't report an earthquake when one did not occur.  Small earthquakes are reasonably common, and occur on about 3.65 days out of the year.

    John calls, Mary does not call, and there is no report on the news about earthquakes.  What is the probability of a Burglary, given all this information?  What is the probability of a minor Earthquake given all this information?

    Harder Question

  2. An enemy power has sent a message, consisting of 6 letters (S1, S2, S3, S4, and S5).  You have collected frequency information from past messages, and have discovered the probability of a particular letter following another in a message as follows:


    P(Sn | Sn-1)
    		 Sn=a
    		 Sn=b
    		 Sn=c
    Sn-1 = a
    		 0.1
    		 0.6
    		 0.3
    Sn-1 = b
    		 0.3
    		 0.1
    		 0.6
    Sn-1 = c
    		 0.6
    		 0.3
    		 0.1

    That is, P(S2=b | S1 = a) = 0.6,  P(S4 = b | S3 = c) = 0.1, and so on.

    Messages never start with c, but a and b are equally likely to start a message

    You have received an interception of this message, R1, R2, R3, R4, and R5.  Unfortuneatly, the message that you received may not have been the message that was sent, since letters could have been corrupted in transmission, as described by the following table:


    P(Rn | Sn)
    		 Sn=a
    		 Sn=b
    		 Sn=c
    Rn = a
    		 0.8
    		 0.1
    		 0.1
    Rn = b
    		 0.1
    		 0.8
    		 0.1
    Rn = c
    		 0.1
    		 0.1
    		 0.8

    You have received the message abcba  (R1 = a, R2 = b, R3 = c, R4 = b, R5 = a)

    a. Find the probability that S3 = c, and the probability that S5=c
    b. Find the probability that the message that was sent was abcba

    (HINT- P(S1=a,S2=b,S3=c,S4=b,S5=a | evidence) = P(S5 = a | e) * P(S4 = b | S5 = a,e) * P(S3 = c | S4 = b,S5 = a,e) * P(S2 = b | S3 = c,S4 = b,S5 = a,e) * P(S1 = a | S2 = b, S3 = c, S4 = b , S5 = a, e))