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In a set of notes, the following stated: "If we consider a measurement where the initial state of knowledge (or prior) $P(x)$ is mistaken and not equal to the true probability, then the measurement will cause the observer's state of knowledge to become increasingly peaked about the true value."

Assuming that the measurements are independent, can this effect be observed by using Baye's theorem for multiple measurements: $$P(x|y_1,...,y_N) = \frac{P(y_N|x)...P(y_1|x)P(x)}{\mathcal{N}}$$and noting that this "increased peak" is due to the correct knowledge of the likelihood functions $P(y_N|x)...P(y_1|x)$ which either correctly increases or decreases the incorrect prior. Is this on the right track?

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  • $\begingroup$ What does the curly $N$ on the denominator represent? A normalizing constant? $\endgroup$ – Glen_b Jul 25 '17 at 1:09
  • $\begingroup$ @Glen_b Yes exactly. $\endgroup$ – Alex Jul 25 '17 at 8:33
  • $\begingroup$ It would be useful to mention that I think; it's hardly a universal notation. $\endgroup$ – Glen_b Jul 25 '17 at 10:57
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That's right. The more observations you make, the more the mass of the posterior distribution clusters around the true probability of the event.

(Your notation is a bit odd in that $x$ seems to be standing for the observed data in some instances and the true probability to be estimated in other instances.)

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  • $\begingroup$ There would seem to be some important exceptions. One is where the prior assigns zero probability to the true value. Another is where there is no true value (because it is constantly changing). Another, which is quite general, is that initially the posterior can become more diffuse as it makes the transition from a prior mode to a true mode while more measurements are collected. Thus, we must understand your assertion to be an asymptotic one rather than in the universal, general sense in which it is couched. $\endgroup$ – whuber Jul 25 '17 at 18:16
  • $\begingroup$ @whuber By the mass clustering around the true value to a greater degree, I mean that its posterior mean gets closer to the true value, not that the posterior variance decreases. But it's all asymptotic, of course. Sampling error could easily lead the posterior mean in the wrong direction. $\endgroup$ – Kodiologist Jul 25 '17 at 18:24

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