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.
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