[15 marks A team of archaeologists from Cambridge university are searching for t

Question

[15 marks A team of archaeologists from Cambridge university are searching for the ruins of an ancient village. They aren't sure whether it even existed or not, but if it did, they know it must be in one of two regions. They consider three mutu- ally exclusive and exhaustive hypotheses, and assign the prior probabilities given below: Hypothesis Definition Prior probability 0.4 0.3 0.3 The village did not exist HiThe village was in region one The village was in region two Table 1: Hypotheses and prior probabilities. The archaeologists search each region for artifacts which would indicate the presence of the ancient village. Suppose that if they search region one, the probability of finding artifacts if the village eristed there is 0.8, and if they search region two, the probability of finding artifacts if the village eristed there is 0.6, perhaps because region two is bigger and they can't look everywhere. Also assume the probability of finding artifacts in a region if the village did not exist there is zero a) mark] Calculate the prior probability that the village existed. The archaeologists search the two regions, and the result is D: No artifacts were region found in region one, and no artifacts were found in Assuming conditional independence, the likelihood P(DIHi) is the probability of finding no artifacts in region one (assug H, multiplied by the probability of finding no artifacts in region two (assuming Hi) (1-0.8) 0.2. P(DIH) 1 b) 2 marks] Find the other two likelihoods, P(DIHo) and P(D|H2) c) 4 marks] Use a Bayes' Box to calculate the posterior probabilities for the three hypotheses, given D d) markl Find the posterior probability that the village existed. Aside: This result shows that the slogan "absence of evidence is not evidence of absence is oversimplified-it depends how hard you looked! e) 1 mark Write down Bayes' rule in a form that is equivalent to one row of your Bayes' Box f) 1 markj Write down the mathematical expression for the marginal likelihood in terms of the prior probabilities and likelihoods in this problem

Explanation / Answer

(a) The prior probability that the village existed is given by:

P(The village existed) =1- P(The village didnot exist)

P(The village existed) = 1- 0.4 = 0.6

P(The village existed) = 0.6

(b) The probaility P(D|H0) denotes the probability that no artifacts were found in region one and no arefacts were found in region two given that the village did not exist (Hypothesis H0).

It is certain that no artifacts will be found in region one and region two if the village did not exist.

So, P(D|H0) = 1

The probability P(D|H2) denotes the probability that no artifacts were found in region one and no artifacts were found in region two provided that the village existed in region two.

This probability is given by multiplying the probability of not finding an article in region two when the village existed in region two with the probabilty of not finding the article in region one when the village existed in region two

This is given by P(D|H2) = (1-0.6) X 1 = 0.4

Thus, P(D|H2) = 0.4

(c)We use the probabilities derived above to calculate the posterior probabilities.

Posterior probability for H0:

P(H0|D) = 1/(1+0.2+0.4) = 1/1.6 = 0.625

P(H1|D) = 0.2/(1+0.2+0.4) = 0.125

P(H2|D) = 0.4/(1+0.2+0.4) = 0.25

(d) Let us define the following events

A1: An artifact is found in region one

A2: An artifact is found in region two

E1: The village existed in region one.

E2: The village existed in region two.

The posterior probability that the village existed can be calculated as:

P(A1|E1)XP(E1) +P(A2|E2)XP(E2)

0.8X0.3 + 0.6X0.3 = 0.42

So, theposterior probability that the village existed is 0.42

(e) This is the rule of total probability

P(D) = P(D|H0)XP(H0) + P(D|H1)XP(H1) + P(D|H2)XP(H2)

P(D) = P(DH0) + P(DH1) + P(DH2)

(f) P(D|H0) = P(H0)XP(DH0)

Prior probability is given P(H0) and likelihood is given by P(DH0).

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