Inference with Imperfect Theories: The Evidence-Based Prior Probability Distribution

This section is concerned with a scenario where a Bayesian has gathered evidence $E_1$, and wishes to draw inferences on the basis thereof, about the truth or falsehood of theory $T$, having previously noted other evidence $E_0$. Ideally, the prior probability used in Bayes' theorem

$$P(T\vert E_1) = \frac{P(E_1\vert T)P(T)}{P(E_1)}$$ (51)

would (chapter 1) be

$$P(T) = P(T\vert E_0)\protect$$

$$= \frac{P(E_0\vert T)P_0(T)}{P_0(E_0)}\textrm{,}$$ (52)

where $P_0(T)$ is an initial, arbitrary (section 3.1) prior probability distribution, and $P_0(E_0)$ is the marginal likelihood (chapter 1) calculated on the basis of $P_0(T)$.

However, it may be found that some theories $T$ have not been developed in such a way as to generate values of $P(E_0\vert T)$, rendering this ideal method impossible¹⁸. In these cases, Bayesians often use [34] prior probability distributions $P(T)$, which attempt to estimate $P(T\vert E_0)$, taking account of $E_0$ in an approximate way, when they attempt to make inferences from $E_1$. The author has considered using this technique to take account of visual impressions from electron diffraction patterns, in determining the crystal structure of manganese-cobalt multi-layers.