Bayes' Rule by James V Stone

Author:James V Stone
Language: eng
Format: epub
Publisher: Sebtel Press
Published: 2015-01-29T16:00:00+00:00

4.4. A Rational Basis For Bias

For a coin which lands heads up with probability θtrue, we use the observed proportion x of heads to estimate θtrue. But all measurement devices are imperfect (eg we can mis-count the number of heads), so the measured estimate x of the true proportion of heads xtrue is noisy. Additionally, the proportion xtrue of heads is usually probabilistically related to θtrue (eg the proportion xtrue of heads varies randomly around θtrue). So there are at least two sources of uncertainty: uncertainty in the measured value x, and uncertainty in the relation between x and the parameter Θ (see Figure 1.12, p27).

These sources of uncertainty translate to a corresponding uncertainty in the value of Θ, which defines a likelihood function p(x|Θ). However, if we know the underlying (ie prior) distribution p(Θ) of values of Θ in the world then we can use this as a guide to reduce the uncertainty in Θ. In essence, this is what Bayes’ rule does. So, if we seek the most probable value θMAP of Θ then, given a measurement x, Bayes’ rule tells us how to ‘adjust’ the estimated value θMLE of θtrue, so that (on average), the adjusted value θMAP is more accurate than θMLE.

Thus, the available information x is incomplete, either because our measurement x adds noise to xtrue, or because even a noise-free measurement x = xtrue is probabilistically related to the value θtrue of the parameter Θ. In both cases, Bayes’ rule provides a rational basis for imposing a particular choice of values (the prior, which may appear to be biased) on estimated parameter values, to arrive at a value that represents our estimate of the most probable state of the physical world.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.