Estimating a coin's bias, both ways

Question

You flip a coin 10 times and see 8 heads. Estimate the coin's probability of heads as a frequentist and as a Bayesian, and explain why the two answers differ.

Accepted Answer

Frequentist. Treat p as a fixed unknown and maximize the likelihood. For a binomial, the MLE is the sample proportion: \hat p = \frac{8}{10} = 0.8. A quick (Wald) 95\% confidence interval is \hat p \pm 1.96\sqrt{\hat p(1-\hat p)/n} = 0.8 \pm 1.96\sqrt{0.016} \approx 0.8 \pm 0.25, i.e. roughly (0.55, 1.0), and note it is wide and behaves poorly near the boundary with only 10 flips. The point estimate is 0.8, with no formal room for "coins are usually fair." Bayesian. Put a prior on p. A natural weakly-informative choice is the uniform prior 	ext{Beta}(1, 1). Adding 8 successes and 2 failures …

Estimating a coin's bias, both ways

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