8 - Back to Bayes: knowledge as a probability distribution

Summary

In this chapter the reader is requested to sit back and think. Think about what you are doing and why, and what your conclusions really mean. You have a theory, containing a number of unknown – or insufficiently known – parameters, and you have a set of experimental data. You wish to use the data to validate your theory and to determine or refine the parameters in your theory. Your data contain inaccuracies and whatever you infer from your data contains inaccuracies as well. While the probability distribution of the data, given the theory, is often known or derivable from counting events, the inverse, i.e., the inferred probability distribution of the estimated parameters given the experimental outcome, is of a different, more subjective kind. Scientists who reject any subjective measures must restrict themselves to hypothesis testing. If you want more, turn to Bayes.

Direct and inverse probabilities

Consider the reading of a sensitive digital voltmeter sensing a constant small voltage – say in the microvolt range – during a given time, say 1 millisecond. Repeat the experiment many times. Since the voltmeter itself adds a random noise due to the thermal fluctuations in its input circuit, your observations y_i will be samples from a probability distribution f(y_i − θ), where θ is the real voltage of the source. You can determine f by collecting many samples.