The Role of the Prior Probability in Experiment Selection

Let it be supposed that there exists some absolutely true theory¹⁵$T_0$, which has not yet been devised, and that the current probability distribution over theories is dominated by two theories, $T_1$ and $T_2$. Two experiments, $X$ and $Z$, are possible, and each is so costly that only one can be undertaken. The conceivable outcomes of $X$ are labelled $E_1$ and $E_2$, and the conceivable outcomes of $Z$ are labelled $F_1$ and $F_2$. The content of the theories is as follows:

$$P(E_1\vert T_0) = 1\textrm{,}$$ (29)

$$P(E_2\vert T_0) = 0\textrm{,}$$ (30)

$$P(F_1\vert T_0) = 1\textrm{,}$$ (31)

$$P(F_2\vert T_0) = 0\textrm{,}$$ (32)

$$P(E_1\vert T_1) = 1\textrm{,}$$ (33)

$$P(E_2\vert T_1) = 0\textrm{,}$$ (34)

$$P(F_1\vert T_1) = \frac{1}{3}\textrm{,}$$ (35)

$$P(F_2\vert T_1) = \frac{2}{3}\textrm{,}$$ (36)

$$P(E_1\vert T_2) = \frac{1}{3}\textrm{,}$$ (37)

$$P(E_2\vert T_2) = \frac{2}{3}\textrm{,}$$ (38)

$$P(F_1\vert T_2) = 1\textrm{,}$$ (39)

and

$$P(F_2\vert T_2) = 0\textrm{.}$$ (40)

When the likelihoods are marginalized over the set $S = \{T_1,T_2\}$ of available theories,

$$P(E_1) = P(T_1)+\frac{P(T_2)}{3}\textrm{,}$$ (41)

$$P(E_2) = \frac{2P(T_2)}{3}\textrm{,}$$ (42)

$$P(F_1) = \frac{P(T_1)}{3}+P(T_2)\textrm{,}$$ (43)

and

$$P(F_2) = \frac{2P(T_1)}{3}\textrm{.}$$ (44)

If the Shannon entropy is used as uncertainty function, the information of experiment $X$ is

$$I(X;S) = I(S;X)\protect$$

$$= H(X)-H(X\vert S)\protect$$

$$= -\left(P(T_1)+\frac{P(T_2)}{3}\right)\log\left(P(T_1)+\frac{P(T_2)}{3}\right)-\frac{2P(T_2)}{3}\log\left(\frac{2P(T_2)}{3}\right)\protect$$

$$+P(T_2)\left(\frac{2}{3}\log\left(\frac{2}{3}\right)+\frac{1}{3}\log\left(\frac{1}{3}\right)\right)\protect$$

$$= -\frac{1}{3}(1+2P(T_1))\log\left(\frac{1}{3}(1+2P(T_1))\right)-\frac{2}{3}(1-P(T_1))\log\left(\frac{2}{3}(1-P(T_1))\right)\protect$$

$$+(1-P(T_1))\left(\frac{2}{3}\log\left(\frac{2}{3}\right)+\frac{1}{3}\log\left(\frac{1}{3}\right)\right)\textrm{,}$$ (45)

and that of experiment $Z$ is

$$I(Z;S) = I(S;Z)\protect$$

$$= H(Z)-H(Z\vert S)\protect$$

$$= -\left(P(T_2)+\frac{P(T_1)}{3}\right)\log\left(P(T_2)+\frac{P(T_1)}{3}\right)-\frac{2P(T_1)}{3}\log\left(\frac{2P(T_1)}{3}\right)\protect$$

$$+P(T_1)\left(\frac{2}{3}\log\left(\frac{2}{3}\right)+\frac{1}{3}\log\left(\frac{1}{3}\right)\right)\protect$$

$$= -\frac{1}{3}(3-2P(T_1))\log\left(\frac{1}{3}(3-2P(T_1))\right)-\frac{2P(T_1)}{3}\log\left(\frac{2P(T_1)}{3}\right)\protect$$

$$+P(T_1)\left(\frac{2}{3}\log\left(\frac{2}{3}\right)+\frac{1}{3}\log\left(\frac{1}{3}\right)\right)\textrm{.}$$ (46)

Figure 1: Graph of the Information of the Experiments $X$ and $Z$ against the Prior Probability of Theory $T_1$.

If the prior probability $P(T_1) > \frac{1}{2}$, then (figure 1) $I(X;S) > I(Z;S)$, and experiment $X$ is chosen, with result $E_1$. This gives posterior probabilities, according to Bayes' theorem (equation 7,)

$$P(T_1\vert E_1) = \frac{P(T_1)}{P(T_1)+P(T_2)/3}\protect$$

$$= \frac{3P(T_1)}{1+2P(T_1)}\textrm{,}$$ (47)

and

$$P(T_2\vert E_1) = 1-\frac{3P(T_1)}{1+2P(T_1)}\protect$$

$$= \frac{1-P(T_1)}{1+2P(T-1)}\textrm{.}$$ (48)

Similarly, if the prior probability $P(T_1) < \frac{1}{2}$, then (figure 1) $I(X;S) < I(Z;S)$, and experiment $Z$ is chosen, with result $F_1$. This gives posterior probabilities, according to Bayes' theorem (equation 7,)

$$P(T_1\vert F_1) = \frac{P(T_1)}{3P(T_2)+P(T_1)}\protect$$

$$= \frac{P(T_1)}{3-2P(T_1)}\textrm{,}$$ (49)

and

$$P(T_2\vert F_1) = 1-\frac{P(T_1)}{3-2P(T_1)}\protect$$

$$= \frac{3-3P(T_1)}{3-2P(T_1)}\textrm{.}$$ (50)

The process that is taking place is starkly illustrated by a graph (figure 2) of the posterior probability of $T_1$, $P(T_1\vert E_1)$ or $P(T_1\vert F_1)$, against its prior probability, $P(T_1)$: the theory that has the higher prior probability is able to make one experiment, which tends to confirm it, appear more interesting than the other experiment, which tends to refute it, and thereby to enhance its probability further.

Figure 2: Graph of the Posterior Probability of $T_1$ against Its Prior Probability

While this statistical result does not relate specifically to any principle in standpoint epistemology, Kuhn's [40] recent epistemological observation, `Normal research... owes its success to the ability of scientists regularly to select problems that can be solved with conceptual... techniques close to those already in existence,' could be interpreted as a similar phenomenon, particularly since Kuhn's view of the normal research process is one in which strong belief in a theory also tends to be stable belief in that theory.

An investigation of the possibility that this phenomenon can arise with an absolutely true theory that is non-deterministic, or even without the need to use the content of an as yet undiscovered, absolutely true theory at all, will be of considerable interest. An investigation of the possibility that this phenomenon can arise with uncertainty functions derived from more realistic ideologies than that associated with the Shannon entropy will also be of considerable interest.