Question

When considering data obtained from flipping one coin four times and obtaining all tails, what will the maximum likelihood approach calculate?
(Consider that there are three models possible for this coin toss: 1. A fair coin model. 2. A coin with both sides heads. And 3. A coin with both sides tails. Priors are 1. 99.8%, 2. 0.1%, 3. 0.1%)

A. The probability of obtaining all tails, averaged over all possible models (i.e. ((.5)^4 * 0.998) + (0 * 0.001) + (1.0 * 0.001))

B. The probability of obtaining all tails, given the model that maximizes this probability (i.e. 100% and it will always choose the third model)

C. The probability of obtaining all tails when using a fair coin (i.e. (.5)^3 * 0.998))

D. The probability of obtaining all tails, without considering possible models. This is possible because a robot is used to explore probability space.

E. Maximum likelihood is not applicable to coin toss data, only nucleotide or amino acid sequence data can be used.

Answer 1

By law of total probability,

probability of obtaining all tails = P(Fair) * P(All tails | Fair) + P(Both heads) * P(All tails | Both heads) + P(Both tails) * P(All tails | Both tails)

= 0.998 * 0.5^4 + 0.001 * 0 + 0.001 * 1 = 0.063375

Given all tails, probability of obtaining  fair coin model = P(Fair | All tails) = P(All tails | Fair) * P(Fair) / P(All tails)

= 0.998 * 0.5^4 / 0.063375 = 0.9842209

Given all tails, probability of obtaining  coin with both sides tails= P(Both tails | All tails) = P(All tails | Both tails) * P(Both tails) / P(All tails)

= 0.001 * 1 / 0.063375 = 0.01577909

The maximum probability is for Model 1. Thus,

C. The probability of obtaining all tails when using a fair coin (i.e. (.5)^4 * 0.998))