An estimator is called unbiased if its expected value is equal
to the true parameter value, regardless of the true parameter's
value. Is the Bayes estimate unbiased? - By part (f), E(θbayes)
= θ* if and only if 1-2θ* = 0, or θ* = 0.5. Therefore, the Bayes
estimate is NOT unbiased. In fact, it is biased towards central
values like 0.5 rather than extreme ones such as 0 or 1. This
reflects the uniform prior; even this choice makes an assumption
about the data. However, notice from part (f) that as N ->
infinity, the Bayes estimate does approach the true value. In
general, Bayes estimators are biased, but as long as the chosen
prior is "reasonable", they will eventually converge to the true
value when enough data is given. In statistics, there is a
division between "frequentist" and "Bayesian" philosophies and
techniques. Bayesian statistics tends to require stronger
assumptions, but can be more powerful as a result.