Consider the following:
An IQ-test has been carried out, and the test result is T. However, since the test result is known to be normal distributed around the true IQ Q, the following reasoning is done:
The IQ for people taking this test is known to be normal distributed around 100. Actually, the IQ is distributed with the following probability function:
where A and are constants. The test result is then known to be normal distributed around the IQ with the following probability function:
where B and are constants. Taking all of this into account, the probability function, for that a person taking the test has the IQ Q and gets the test result T, is
where is a variable depending only on T. What this tells us is basically that if a person has got the test result T, the most likely value for his IQ, and it's expected value, is actually:
(which is closer to 100 than what T is) for a fixed value of T. However, this method of estimating the IQ from a test result is not consistent, since a sequence of estimators would converge in probability to and not to Q (note that for real IQ tests, B is most often many times bigger than A).
Now, what I'm wondering is, do you say that this inconsistency is due to a bias? Or what do you say it is caused by? And when tests like these are carried out, what is most often used to estimate the measured parameter, the actual test value or the "adjusted" test value?
The final estimator you have is just the Bayes estimator of Q with a normal prior on it, but your formula is only valid for a single test observation. You have right, but you need to calculate if you want to talk about what happens asymptotically (note that is complete sufficient, so it suffices to only condition on that). If you take the test a bunch of times and use all the data, you'll end up with a instead of just T the term will grow as n does.
One way to think about this is to imagine that there is actually very little variability in the population but very high variability in the test. Say the variance of the population is 1 and the variance of the test score is 40. Then, if someone gets a score of 140, I'm still going to be pretty sure that his true value is near 100. My propensity to pull him towards the prior mean is what creates the bias - after all, the test is assumed unbiased; it just isn't very precise in this example. But, as he takes the test infinitely often, if his true IQ is 140, I will become more and more convinced that his real IQ is 140 i.e. asymptotically I have unbiasedness.