You are an interplanetary search and rescue expert who has just received an urge

Question

You are an interplanetary search and rescue expert who has just received an urgent message: a rover on Mercury has fallen and become trapped in Death Kavine, a deep, narrow gorge on the borders of enemy territory. You zoom over to Mercury to investigate the situation. Death Ravine is a narrow gorge ? miles long, as shown below. There are volcanic vents al localius A ad D, idicale by e iangular sybls a ose locatioIS. The rover was heavily damaged in the fall, and as a result, most of its sensors are broken. The only ones still func tioning are its thermometers, which register only two levels: hul and culd. The rover sends back evidence E = hul when it is at a volcanic vent (A and D), and E cold othcrwisc. There is no chancc of a mistaken rcading. The rovcr fell into thc gorge at position A on day 1, so Xi-A·Let the rover's position on day t be X, c {A, B, C. D, E, F). The rover is still executing its original programming, trying to move 1 mile east (i.e. right, towards F) every day. However, because of the damage. it only moves east with probability 0.5, and it stays in place with probability 0.5. Your job is to figure out where the rover is, so that you can dispatch your rescue-bot. 1. Filtering: Three days have passed since the rover fell into the ravine. The observations were (E = hot. E2 = cold, E3-cold). What is P(X3 hoti,cold2, cold3), the probability distribution over the rover's position on day 3, given the observations? (This is a probability distribution over the six possible positions). 2. Smoothing: What is P(X hoti, coldi, colda). the probability distribution over the rover's position on day 2. given the observations? (This is a probability distribution over the six possible positions). vations (E1, E-vld, E3-cvld)? days 4,5,6 respectively, given the previous observations in days 1,2, and 3? (This is a single value, not a distribution). 3. Mosi Likely Explanation Wlial is 1K Illus1 likely seyu??? uf die ruver's positiulis ?ll de lliee days givell le ubser- 4. Prediction: What is P(hota,hots, colde hoti, cold2, cold3), the probability of observing hot and hots and colde in 5. Prediction: You decide to attempt to rescue the rover on day 4. However, the transmission of Ea seems to have been corrupted, and so it is not observed. What is the rover's position distribution for day 1 given the same evidence P(X hot,cold, cold)? The same thing happens again on day 5. What is the rover's position distrihution for day 5 given the same evidence P(X5 hoti,cold2, colds)?

Explanation / Answer

1.FILTERING:

FINDING THE PROBABILITIES OF EXISTENCE OF ROVER IN ANY "HOT" LOCATIONS

first, calculate the P(X2|E1=hot,E2=cold) ? P(E2=cold | X2) ?X1 P(X2|X1) P(X1|E1=hot) value for the locations X2 ? {A,B,…} in which the rover can move

Now calculate the x3 value(PROBABILITY OF MOVING AWAY FROM HOT REGION) for each location

X3 ? {A,B,…} and normalize it as follows: P(X3|E1=hot,E2=cold,E3=cold) ? P(E3=cold | X3) ?X2 P(X3|X2) P(X2|E1=hot,E2=cold)

on day 4,(if thr current rover's position considering at C)rescue of the rover can be attempted but it is not considered in the observations

2.SMOOTHING:

EVALUATING THE ROVER'S POSITION ON DAY 2

P (X2 | hot1, cold2, cold3)

P (X2 | hot1, cold2, cold3)

INITIAL CONFIGURATION

A B C D E F

X

X

X   

ESTIMATED POSITION AFTER SECOND DAY

A B C D E F

X   

X   

X

PROBABILITY OF RESCUE

P(NEWPOSITION| OLD POSITION) = 0.5 * 0.5 * 0.5 = 0.125

X1 P(X1 | E1=hot) A 1.0 B 0.0 C,D,E,F 0.0

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