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Numb3rs 113: ManhuntIn this episode, Charlie uses Bayesian inference and Markov chains to help his brother figure out the cause of a bus crash which let two prisoners escape from a prison bus.Bayesian InferenceBayesian inference is the process of adjusting your belief in the probability of the truth of some statement based on new events. As a silly example, you may believe that it is extremely unlikely that there are any purple cows in the world. However, if you see a purple cow, then you would obviously change your belief to the belief that there is at least one purple cow in the world.Bayes' rule is attributed to the Reverend Thomas Bayes,
a Presbyterian minister who lived from 1701 to 1761 in a work entitled
"Essay Towards Solving a Problem in the Doctrine of Chances."
This was a response to a work entitled "The Doctrine of Chances"
by Abraham de Moivre, one of Bayes' mathematical collegues. A link to
Bayes original paper (which was published posthumously) can be found
here
To give a mathematical basis for this inference process, we need to state
Bayes' theorem. Suppose we have two different events, A and B. We'll write
P(A) for the probability that A occurs and P(B) for the probability that
B occurs. Also, we'll write P(A|B) for the probability that A happens
given that B happens. We can rewrite this as P(A|B) = P(A and B)/P(B), that
is, P(A|B) is the probability that A and B both happen divided by P(B).
Then Bayes rule is the following formula:![]() ![]()
The Monty Hall Problem: This problem is loosely
based on a situation that occurred in the tv show "Let's Make a
Deal." Let's say you're on a game show and the host presents you with
three doors. One has a shiney new car behind it, and the other two have goats.
The object (of course) is to get the car and not to get a goat. Now the game
goes like this. First, you pick a door. Then at least one of the two doors
you did not pick has a goat behind it. The game show host knows what the shut
doors have behind them and he deliberately opens one of the two that
you did not pick to show you a goat (if
both have a goat, he randomly picks the one to open). Then he lets you
either stay with your original choice of door or switch to the other
unopened door. Then you get what is behind that door. Now the goal of the
problem is to use Bayes' rule to figure out whether it is better to
switch or stay put. Let's say you picked door 1 and he opened door 2.
Let A be the event that door 3 has a car behind it and let B be the event
that he opened door 2.
Markov ChainsA Markov chain is a series of random events where the outcome of event n is only dependent on the outcome of the event n-1. For example, let's say there is a number line where each integer has a dot on it, and let's say that there is also an evil but useless robot standing on position 0. The robot has a quarter, and he flips it to determine whether he should go left or right. If he gets heads, he'll move to position 1, and if he gets tails he'll move to position -1. This is the first random event. Now he repeats this process over and over to generate a series of random events, where each time if he gets heads he moves right 1 step from his current position and if he gets tails he moves left 1 step from his current position (let's say right is the positive direction). This is a Markov chain because where the robot goes on his next step only depends on his current position and doesn't depend on where he's been in the past. The output of each event is the current position of the robot. Now let's study this Markov chain a little bit to see how it behaves. One question we might ask is what is the average position of the robot after n steps. (A more technical way of phrasing this question is to ask what the expected value![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |