Question

1. Coin flipping: MLE and MAP. (10 points) Suppose you observed coin flips consisting of NH heads and NT tails. You model the coin flips as independent Bernoulli random variables with a common parameter ? that represents the probability of heads. In this problem, you'll estimate ? in two different ways. Maximum Likelihood 1. Write the likelihood, ?(?) = p(X|?) = ? (NH + NT, i=1) p(xi|?). The result should be a formula involving ?, NH, and NT. 2. Find the parameter ?MLE that maximizes the likelihood, ?MLE = arg max ? ?(?). (Hint: take the derivative and set it to zero). 3. What is ?MLE when NH = 3 and NT = 2? When NH = 30 and NT = 20? (Hint: use the expression you found in (2).)

          1. Coin flipping: MLE and MAP. (10 points) Suppose you observed coin flips consisting of NH heads and NT tails. You model the coin flips as independent Bernoulli random variables with a common parameter ? that represents the probability of heads. In this problem, you'll estimate ? in two different ways.
Maximum Likelihood
1. Write the likelihood,
?(?) = p(X|?) = ? (NH + NT, i=1) p(xi|?).
The result should be a formula involving ?, NH, and NT.
2. Find the parameter ?MLE that maximizes the likelihood,
?MLE = arg max ? ?(?).
(Hint: take the derivative and set it to zero).
3. What is ?MLE when NH = 3 and NT = 2? When NH = 30 and NT = 20? (Hint: use the expression you found in (2).)

1. Coin flipping: MLE and MAP. (10 points) Suppose you observed coin flips consisting of NH heads and NT tails. You model the coin flips as independent Bernoulli random variables with a common parameter ? that represents the probability of heads. In this problem, you'll estimate ? in two different ways.
Maximum Likelihood
1. Write the likelihood,
?(?) = p(X|?) = ? (NH + NT, i=1) p(xi|?).
The result should be a formula involving ?, NH, and NT.
2. Find the parameter ?MLE that maximizes the likelihood,
?MLE = arg max ? ?(?).
(Hint: take the derivative and set it to zero).
3. What is ?MLE when NH = 3 and NT = 2? When NH = 30 and NT = 20? (Hint: use the expression you found in (2).)

Added by Dylan M.

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