# Improved state of knowledge due to measurement

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?

• What does the curly $N$ on the denominator represent? A normalizing constant? – Glen_b Jul 25 '17 at 1:09
• @Glen_b Yes exactly. – Alex Jul 25 '17 at 8:33
• It would be useful to mention that I think; it's hardly a universal notation. – Glen_b Jul 25 '17 at 10:57

(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.)