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Appendix 3: Derivations of the Two Bayesian Estimation Rules,Rule BE-D andRule BE-C

We restate each rule,BE-D andBE-C, and provide proofsof their claims along the way.

Rule BE-D: Bayesian Estimation for Disjunctions of Alternative Hypotheses

Let \(H\) be a collection of \(z\) alternative hypotheses, \(z \ge2\), where the conjunction of any two of them is logicallyinconsistent. Let \(c\) be observational or experimental conditionsfor which \(e\) describes one of the possible outcomes. And suppose\(b\) is a conjunction of relevant auxiliary hypotheses andplausibility considerations. For each hypothesis \(h\) in \(H\), letits prior probability be non-zero: \(P[h \mid c \cdot b] \gt 0\).

Choose any \(k\) hypotheses from collection \(H\), where each one ofthem, \(h\), has a likelihood value \(P[e \mid h \cdot c \cdot b] >0\). Label these \(k\) hypotheses (in whatever order you wish) as\(\lsq h_1\rsq\), \(\lsq h_2\rsq\), …, \(\lsq h_k\rsq\). Thenlabel all the remaining hypotheses in \(H\) (in whatever order youwish) as \(\lsq h_{k+1}\rsq\), \(\lsq h_{k+2}\rsq\), …, \(\lsqh_z\rsq\).

Given these labelings of hypotheses in \(H\), let \((h_1 \vee \ldots\vee h_k)\) represent the disjunction of the first \(k\) hypotheseschosen from \(H\), and \((h_{k+1} \vee \ldots \vee h_z)\) representthe disjunction of the remaining hypotheses from \(H\). The expression\((h_1 \vee \ldots \vee h_z)\) represents the disjunction of allhypotheses in \(H\). Furthermore, let \(b\) logically entail thisdisjunction of all alternative hypotheses in \(H\): \(b \vDash (h_1\vee \ldots \vee h_z)\). So, both \(P[(h_1 \vee \ldots \vee h_z) \midc \cdot b] = 1\) and \(P[(h_1 \vee \ldots \vee h_z) \mid c \cdot e\cdot b] = 1\).

Then, the posterior probability of \((h_1 \vee \ldots \vee h_k)\)satisfies the following form of Bayes’ Theorem:

\[P[(h_1 \vee \ldots \vee h_k) \mid c \cdot e \cdot b] \; \; = \; \;\frac{\sum_{j = 1}^k P[e \mid h_j \cdot c \cdot b] \times P[h_j \mid c \cdot b]}{\sum_{i = 1}^z P[e \mid h_i \cdot c \cdot b] \times P[h_i \mid c \cdot b]}.\]

In cases where the values of all the prior probabilities, \(P[h_i \midc \cdot b]\), are known, or can be closely approximated, this equationsuffices to provide values for the argument strengths \(r\) of theposterior probabilities, \(P[(h_1 \vee \ldots \vee h_k) \mid c \cdot e\cdot b] = r\). But when no precise values of the priors areavailable, a useful estimate of bounds on the posterior probabilitiesmay be derived as follows.

Let \(K\) be (your best estimate of) an upper bound on the ratios ofprior probabilities, \(P[h_i \mid c \cdot b] / P[h_j \mid c \cdot b]\)for all \(h_j\) in \(\{h_1, h_2, \ldots, h_k\}\) and all \(h_i\) in\(\{h_{k+1}, h_{k+2}, \ldots, h_z\}\). That is, for whichever \(h_j\)in \(\{h_1, h_2, \ldots, h_k\}\) has the smallest value of \(P[h_j\mid c \cdot b]\), and for whichever \(h_i\) in \(\{h_{k+1}, h_{k+2},\ldots, h_z\}\) has the largest value of \(P[h_i \mid c \cdot b]\),let \(K\) be a real number that is large enough that \(K \ge P[h_i\mid c \cdot b] / P[h_j \mid c \cdot b]\).

Then, \[\Omega[\neg (h_1 \vee \ldots \vee h_k) \mid c \cdot e \cdot b] \; \; \le \; \;K \times \left[\frac{1}{\frac{\sum_{j = 1}^k P[e \;\mid\; h_j \cdot c \cdot b]}{\sum_{i = 1}^z P[e \;\mid\; h_i \cdot c \cdot b]}} - 1 \right].\]

Thus, a lower bound on the associated posterior probability of\((h_1 \vee \ldots \vee h_k)\) is given by the formula \[\begin{align}P[(h_1 \vee \ldots \vee h_k) \mid c \cdot e \cdot b] \; \; &= \; \;\frac{1}{1 + \Omega[\neg (h_1 \vee \ldots \vee h_k) \mid c \cdot e \cdot b]} \\&\ge \; \; \frac{1}{1 + K \times \left[\frac{1}{\frac{\sum_{j = 1}^k P[e \;\mid\; h_j \cdot c \cdot b]}{\sum_{i = 1}^z P[e \;\mid\; h_i \cdot c \cdot b]}} - 1 \right]}.\end{align}\]

Rule BE-C: Bayesian Estimation for a Continuous Range of Alternative Hypotheses

Let \(H\) be a continuous region of alternative hypotheses \(h_q\),where \(q\) is a real number, and where the conjunction of any two ofthese hypotheses is logically inconsistent. Let \(c\) be observationalor experimental conditions for which \(e\) describes one of thepossible outcomes. And suppose \(b\) is a conjunction of relevantauxiliary hypotheses and plausibility considerations. For each pointhypothesis \(h_q\) in \(H\), we take \(p[e \mid h_q \cdot c \cdot b]\)to be an appropriate likelihood.

Let \(p[h_q \mid c \cdot b]\) and \(p[h_q \mid c \cdot e \cdot b]\) beprobability density functions on \(H\), where these two densityfunctions are related as follows: \[p[h_q \mid c \cdot e \cdot b] \times P[e \mid c \cdot b] \;=\; p[e \mid h_q \cdot c \cdot b] \times p[h_q \mid c \cdot b].\]

We suppose throughout that prior probability density \(p[h_q \mid c\cdot b] > 0\) for all values of \(q\).

The prior probability that the true point hypothesis \(h_r\) lieswithin measurable region \(R\) is given by

\[\begin{align}P[h_R \mid c \cdot b] \; &= \; \int_R p[h_r \mid c \cdot b] \; dr,\;\; \text{where} \\P[h_H \mid c \cdot b] \; &= \; \int_H p[h_q \mid c \cdot b] \; dq \: =\: 1.\end{align}\]

The posterior probability that the true point hypothesis \(h_r\) lieswithin measurable region \(R\) is given by

\[\begin{align}P[h_R \mid c \cdot e \cdot b] \; &= \; \int_R p[h_r \mid c \cdot e \cdot b] \; dr, \;\; \text{where} \\P[h_H \mid c \cdot e \cdot b] \; &= \; \int_H p[h_q \mid c \cdot e \cdot b] \; dq \: =\: 1.\end{align}\]

Then, the posterior probability satisfies the following equation foreach measurable region \(R\): \[P[h_R \mid c \cdot e \cdot b] = \frac{\int_R p[e \mid h_r \cdot c \cdot b] \times p[h_r \mid c \cdot b] \; \; dr}{\int_H p[e \mid h_q \cdot c \cdot b] \times p[h_q \mid c \cdot b] \; \; dq}.\]

In cases where a precise model of the prior probability density,\(p[h_q \mid c \cdot b]\), is available, this equation suffices toprovide values for the posterior probabilities, \(P[h_R \mid c \cdot e\cdot b]\). However, when no precise model of the priors is available,bounds on the values of posterior probabilities may be evaluated inthe following way.

Let \(K\) be (your best estimate of) an upper bound on the ratios ofthe probability density values, \(p[h_q \mid c \cdot b] / p[h_r \mid c\cdot b]\), for each \(h_r\) in region \(R\) and \(h_q\) in \((H-R)\).That is, for whichever \(h_r\) in \(R\) has the smallest value of\(p[h_r \mid c \cdot b]\), and for whichever \(h_q\) in \((H-R)\) hasthe largest value of \(p[h_q \mid c \cdot b]\), let \(K\) be a realnumber such that \(K \ge p[h_q \mid c \cdot b] / p[h_r \mid c \cdotb]\).

Then, \[\Omega[\neg h_R \mid c \cdot e \cdot b] \; \; \le \; \;K \times \left[\frac{1}{\frac{\int_{R} p[e \;\mid\; h_r \cdot c \cdot b] \; \; dr}{\int_{H} p[e \;\mid\; h_q \cdot c \cdot b] \; \; dq}} - 1 \right].\]

Thus, a lower bound on the associated posterior probability of \(h_R\)is given by the formula

\[\begin{align}P[h_R \mid c \cdot e \cdot b] \;\; &= \;\; \frac{1}{1 + \Omega[\neg h_R \mid c \cdot e \cdot b]} \\&\ge \; \; \frac{1}{1 + K \times \left[\frac{1}{\frac{\int_{R} \; p[e \;\mid\; h_r \;\cdot\; c \cdot b] \; \; dr}{\int_{H} \; p[e \;\mid\; h_q \cdot c \cdot b] \; \; dq}} - 1 \right]}.\end{align}\]