Accounting for Unknown Unknowns in Managing Multi-hazard Risks

Abstract

A significant challenge in managing multi-hazard risks is accounting for the possibility of events beyond our range of experience. Classical statistical approaches are of limited value because there are no data to analyze. Judgment or subjective assessments are also of limited value because they are derived from within our range of experience. This chapter proposes a new framework, Decision Entropy Theory, to assess probabilities and manage risks for possibilities in the face of limited information. The theory postulates a starting point for assessing probabilities that reflect having no information in making a risk management decision. From this non-informative starting point, all available information (if any) can be incorporated through Bayes’ theorem. From a practical perspective, this theory highlights the importance of considering how possibilities for natural hazards could impact the preferred alternatives for managing risks. It emphasizes the role for science and engineering to advance understanding about natural hazards and managing their risk. It ultimately underscores the importance of developing adaptable approaches to manage multi-hazard risks in the face of limited information.

Appendices

A.1 Appendix 1: Mathematical Formulation for Principles of Decision Entropy

The following appendix provides the mathematical formulation characterizing the entropy of a decision.

Principle Number 1

In a non-informative sample space, a selected alternative is equally probable to be or not to be preferred compared to another alternative.

Given that alternative A _i is selected and compared to A _j , the maximum lack of information for the decision corresponds to maximizing the relative entropy for the events that an alternative is and is not preferred:

$$ \begin{array}{l}\mathrm{Maximize}\;{H}_{\mathrm{rel}}\left(\mathrm{Preference}\;\mathrm{Outcome}\Big|{A}_i\;\mathrm{Selected}\;\mathrm{and}\;\mathrm{Compared}\;\mathrm{t}\mathrm{o}\;{A}_j\right)\\ {}=-P\left[{A}_i\;\mathrm{Preference}\;\mathrm{t}\mathrm{o}\;{A}_j\right] \ln \left\{P\left[{A}_i\;\mathrm{Preference}\;\mathrm{t}\mathrm{o}\;{A}_j\right]\right\}\\ {}-P\left[{A}_i\;\overline{\mathrm{Preference}}\;\mathrm{t}\mathrm{o}\;{A}_j\right] \ln \left\{P\left[{A}_i\;\overline{\mathrm{Preference}}\;\mathrm{t}\mathrm{o}\;{A}_j\right]\right\}- \ln (2)\end{array} $$

where H _rel(Preference Outcome| A _i Selected and Compared to A _j ) is the relative entropy of the decision preference, P[A _i Preferred to A _j ] is the probability that alternative A _i is preferred compared to alternative A _j , and $P\left[{A}_i\ \overline{\mathrm{Preferred}\ }\kern0.5em \mathrm{t}\mathrm{o}\ {A}_j\right]$ is the probability that alternative A _i is not preferred compared to alternative A _j or $1-P\left[{A}_i\ \mathrm{Preferred}\ \mathrm{t}\mathrm{o}\ {A}_j\right]$. If the preference for one alternative versus another is characterized by the difference in the utility values for each alternative, then the relative entropy of the decision preference can be expressed as follows:

$$ \begin{array}{l}\mathrm{Maximize}\;{H}_{\mathrm{rel}}\left(\mathrm{Preference}\;\mathrm{Outcome}\Big|{A}_i\;\mathrm{Selected}\;\mathrm{and}\;\mathrm{Compared}\;\mathrm{t}\mathrm{o}\;{A}_j\right)\\ {}=-P\left[u\left({A}_i\right)-u\left({A}_j\right)>0\Big|{A}_i\;\mathrm{Compared}\;\mathrm{t}\mathrm{o}\;{A}_j\right]\\ {}\times \ln \left\{P\left[u\left({A}_i\right)-u\left({A}_j\right)>0\Big|{A}_i\;\mathrm{Compared}\;\mathrm{t}\mathrm{o}\;{A}_j\right]\right\}\\ {}-\left\{1-P\left[u\left({A}_i\right)-u\left({A}_j\right)>0\Big|{A}_i\;\mathrm{Compared}\;\mathrm{t}\mathrm{o}\;{A}_j\right]\right\}\\ {}\times \ln \left\{1-P\left[u\left({A}_i\right)-u\left({A}_j\right)>0\Big|{A}_i\;\mathrm{Compared}\;\mathrm{t}\mathrm{o}\;{A}_j\right]\right\}- \ln (2)\end{array} $$

where u(A _i ) is the utility for alternative A _i .

The entropy is a measure of the frequencies or probabilities of possible outcomes divided between the two states: A _i Preferred to A _j and ${A}_i\ \overline{\mathrm{Preferred}}\kern0.5em \mathrm{t}\mathrm{o}\ {A}_j$. The natural logarithm of the entropy is used for mathematical convenience by invoking Stirling’s approximation for factorials; note that maximizing the logarithm of the frequencies of possible outcomes is the same as maximizing the frequencies of possible outcomes since the logarithm is a one-to-one function of the argument. The relative entropy normalizes the entropy by the number of possible states (in this case two); the maximum value of the relative entropy is zero.

For two outcomes, A _i Preferred to A _j and ${A}_i\ \overline{\mathrm{Preferred}}\ \mathrm{t}\mathrm{o}\ {A}_j$, the relative entropy H _rel (Preference Outcome| A _i Selected and Compared to A _j ) is maximized in the ideal case where it is equally probable that one or the other alternative is preferred, or

$$ \begin{aligned}&P\left[\left.u\left({A}_i\right)-u\left({A}_j\right)>0\ \right|{A}_i\ \mathrm{Compared}\ \mathrm{t}\mathrm{o}\ {A}_j\right]\\ &\quad=1-P\left[\left.u\left({A}_i\right)-u\left({A}_j\right)>0\right|{A}_i\ \mathrm{Compared}\ \mathrm{t}\mathrm{o}\ {A}_j\right].\end{aligned} $$

When i = j (i.e., an alternative is compared to itself), the relative entropy becomes equal to its minimum possible value, $- \ln (2)$, because there is no uncertainty in the preference [$p \ln (p)\to 0\ \mathrm{a}\mathrm{s}\ p\to 0\ \mathrm{or}\ p\to 1$].

Principle Number 2

In a non-informative sample space, possible degrees of preference between a selected alternative and another alternative are equally probable.

To represent the maximum lack of information for the decision between two alternatives, maximize the relative entropy for the possible positive and negative differences in utility values between alternative A _i and A _j , $\Delta {u}_{i,j}=u\left({A}_i\right)-u\left({A}_j\right)$. The sample space where A _i is Selected and Compared to A _j is divided into two states, A _i Preferred to A _j and ${A}_i\ \overline{\mathrm{Preferred}\ }\ \mathrm{t}\mathrm{o}\ {A}_j$, which are respectively subdivided into ${n}_{\Delta {u}_{i,j}\Big|{A}_i\succ {A}_j}$ and ${n}_{\Delta {u}_{i,j}\Big|{A}_i\preccurlyeq {A}_j}$ substates for each interval of Δu _i,j , where the operator $\succ$ means “preferred to” and the operator $\preccurlyeq$ means “not preferred to” or the complement of $\succ$. Maximizing the relative entropy for degree of preference is then given by the following:

$$ \begin{array}{l}\mathrm{Maximize}\;{H}_{\mathrm{rel}}\left(\mathrm{Preference}\;\mathrm{Degrees}\Big|{A}_i\;\mathrm{Selected}\;\mathrm{and}\;\mathrm{Compared}\;\mathrm{t}\mathrm{o}\;{A}_j\right)\\ {}={H}_{\mathrm{rel}}\left(\mathrm{Preference}\;\mathrm{Degrees},\;{A}_i\;\mathrm{Preferred}\;\mathrm{t}\mathrm{o}\;{A}_j\right)\\ {}+{H}_{\mathrm{rel}}\left(\mathrm{Preference}\;\mathrm{Degrees},\;{A}_i\;\overline{\mathrm{Preferred}}\;\mathrm{t}\mathrm{o}\;{A}_j\right)\end{array}\vspace*{6pt} $$

where

$$ \begin{array}{l}{H}_{\mathrm{rel}}\left(\mathrm{Preference}\;\mathrm{Degrees},\;{A}_i\;\mathrm{Preferred}\;\mathrm{t}\mathrm{o}\;{A}_j\right)\\ {}={\displaystyle \sum_{n_{\Delta {u}_{i,j}\Big|{A}_i\succ {A}_j}}\begin{array}{l}-P\left(\Delta {u}_{i,j}\Big|{A}_i\;\mathrm{Preferred}\;\mathrm{t}\mathrm{o}\;{A}_j\right)P\left({A}_i\;\mathrm{Preferred}\;\mathrm{t}\mathrm{o}\;{A}_j\right)\\ {}\times \ln \left[P\left(\Delta {u}_{i,j}\Big|{A}_i\;\mathrm{Preferred}\;\mathrm{t}\mathrm{o}\;{A}_j\right)P\left({A}_i\;\mathrm{Preferred}\;\mathrm{t}\mathrm{o}\;{A}_j\right)\right]\end{array}}\\ {}-\left(\frac{1}{2}\right) \ln \left({n_{\Delta {u}_{i,j}}}_{\Big|{A}_i\succ {A}_j}\times 2\right)\end{array} $$

and

$$ \begin{array}{l}{H}_{\mathrm{rel}}\left(\mathrm{Preference}\;\mathrm{Degrees},\;{A}_i\;\overline{\mathrm{Preferred}}\;\mathrm{t}\mathrm{o}\;{A}_j\right)\\ {}={\displaystyle \sum_{{\mathrm{n}}_{\Delta {u}_{i,j}\Big|{A}_i\preceq {A}_j}}\begin{array}{l}-P\left(\Delta {u}_{i,j}\Big|{A}_i\;\overline{\mathrm{Preferred}}\;\mathrm{t}\mathrm{o}\;{A}_j\right)P\left({A}_i\;\overline{\mathrm{Preferred}}\;\mathrm{t}\mathrm{o}\;{A}_j\right)\\ {}\times \ln \left[P\left(\Delta {u}_{i,j}\Big|{A}_i\;\overline{\mathrm{Preferred}}\;\mathrm{t}\mathrm{o}\;{A}_j\right)P\left({A}_i\;\overline{\mathrm{Preferred}}\;\mathrm{t}\mathrm{o}\;{A}_j\right)\right]\end{array}}\\ {}-\left(\frac{1}{2}\right) \ln \left({n}_{\Delta {u}_{i,j}\Big|{A}_i\preceq {A}_j}\times 2\right)\end{array} $$

The maximum value for H _rel(Preference Degrees| A _i Selected and Compared to A _j ) is equal to zero, and it is achieved when $P\left({A}_i\ \mathrm{Preferred}\ \mathrm{t}\mathrm{o}\ {A}_j\right)=P\left({A}_i\ \overline{\mathrm{Preferred}}\ \mathrm{t}\mathrm{o}\ {A}_j\right)=1/2$, all $P\left(\Delta {u}_{i,j}\Big|{A}_i\ \mathrm{Preferred}\ \mathrm{t}\mathrm{o}\ {A}_j\right)=1/{\mathrm{n}}_{\Delta {u}_{i,j}\Big|{A}_i\succ {A}_j}$ and all $P\left(\Delta {u}_{i,j}\Big|{A}_i\ \overline{\mathrm{Preferred}}\ \mathrm{t}\mathrm{o}\ {A}_j\right)=1/{\mathrm{n}}_{\Delta {u}_{i,j}\Big|{A}_i\preccurlyeq {A}_j}$. Note that the number of substates for intervals of Δu _i,j in A _i  Preferred to A _j or in ${A}_i\ \overline{\mathrm{Preferred}}\ \mathrm{t}\mathrm{o}\ {A}_j$ is not important in maximizing the relative entropy; the entropy is maximized when the possible intervals (however many there are) are as equally probable as possible.

Principle Number 3

In a non-informative sample space, possible expected degrees of information value for the preference between a selected alternative and another alternative are equally probable.

To represent the maximum lack of information for a value of information assessment for the decision between two alternatives, maximize the relative entropy for the possible positive and nonpositive expected changes with information in the differences in expected utility values between alternative A _i and A _j . These possible expected changes are termed the “information value” and denoted by${\Omega}_{E_{k,l}}$:

$$ \begin{aligned}{\left({\Omega}_{E_{k,l}}\right)}_{i,j}&=E\left(\Delta {u}_{i,j}\left|{E}_k\cap {S}_{i,j}\right.\right)-E\left(\Delta {u}_{i,j}\left|{E}_l\cap {S}_{i,j}\right.\right)\\ &=E\left[u\left({A}_i\right)-u\left(\overline{A_l}\right)\left|{\mathrm{E}}_k\cap {S}_{i,j}\right.\right]-E\left[u\left({A}_i\right)-u\left(\overline{A_i}\right)\left|{\mathrm{E}}_l\cap {S}_{i,j}\right.\right]\end{aligned} $$

where E _k and E _l are two sets of possible information about the preference between A _i and A _j , $\Delta {u}_{\mathrm{i},\mathrm{j}}=u\left({A}_i\right)-u\left({A}_j\right)$. The sample space for E _k,l is divided into two subsets, an expected positive information value (i.e., ${\Omega}_{E_{k,l}}>0$) with ${n}_{\Omega_{E_{k,l}}>0}$ states and an expected nonpositive information value (i.e., ${\Omega}_{E_{k,l}}\le 0$) with ${n}_{\Omega_{E_{k,l}}\le 0}$ states. Maximizing the relative entropy for possible information values is then given by the following:

$$ \begin{aligned}&\mathrm{Maximize}\;{H}_{\mathrm{rel}}\left(\mathrm{Information}\;\mathrm{Value}\;\mathrm{Degrees}\Big|\mathrm{Information}\;\mathrm{about}\;{A}_i\;\mathrm{Selected}\;\mathrm{and}\;\right.\\ &\quad\left.\mathrm{Compared}\;\mathrm{t}\mathrm{o}\;{A}_j\right) ={H}_{\mathrm{rel}}\left(\mathrm{Information}\;\mathrm{Value}\;\mathrm{Degrees},\;\mathrm{Information}\;\mathrm{has}\;\mathrm{Positive}\;\right.\\ &\quad\left.\mathrm{Value}\;\mathrm{f}\mathrm{o}\mathrm{r}\;{A}_i\;\mathrm{Compared}\;\mathrm{t}\mathrm{o}\;{A}_j\right) +{H}_{\mathrm{rel}}\left(\mathrm{Information}\;\mathrm{Value}\;\mathrm{Degrees},\;\mathrm{Information}\;\right.\\ &\quad\left.\mathrm{has}\;\mathrm{N}\mathrm{o}\mathrm{n}\hbox{-} \mathrm{Positive}\;\mathrm{Value}\;\mathrm{f}\mathrm{o}\mathrm{r}\;{A}_i\;\mathrm{Compared}\;{A}_j\right)\end{aligned} $$

where

$$ \begin{aligned}&{H}_{\mathrm{rel}}\left(\mathrm{Information}\;\mathrm{Value}\;\mathrm{Degrees},\;\mathrm{Information}\;\mathrm{has}\;\mathrm{Positive}\;\mathrm{Value}\;\mathrm{f}\mathrm{o}\mathrm{r}\;{A}_i\;\mathrm{Compared}\;\right.\\ &\quad\left.\mathrm{t}\mathrm{o}\;{A}_j\right)\!=\!{\displaystyle \sum_{{n_{\Omega}}_{{}_{E_{k,l}}}\!>\!0}\begin{array}{l}-P\left({\Omega}_{E_{k,l}}\Big|{\Omega}_{E_{k,l}}\!>\!0\right)P\left({\Omega}_{E_{k,l}}\!>\!0\right)\\ {}\times \ln \left[P\left({\Omega}_{E_{k,l}}\Big|{\Omega}_{E_{k,l}}\!>\!0\right)P\left({\Omega}_{E_{k,l}}\!>\!0\right)\right]\end{array}}\!-\!\left(\frac{1}{2}\right) \ln \left({n}_{\Omega}{{}_{{}_{E_{k,l}}}}_{>0}\!\times\! 2\right)\end{aligned} $$

and

$$ \begin{aligned}&{H}_{\mathrm{rel}}\left(\mathrm{Information}\;\mathrm{Value}\;\mathrm{Degrees},\;\mathrm{Information}\;\mathrm{has}\;\mathrm{N}\mathrm{o}\mathrm{n}\hbox{-} \mathrm{Positive}\;\mathrm{Value}\;\mathrm{f}\mathrm{o}\mathrm{r}\;{A}_i\;\right.\\ &\quad\left.\mathrm{Compared}\;\mathrm{t}\mathrm{o}\;{A}_j\right)\!=\!{\displaystyle \sum_{n_{\Omega}{{}_{{}_{E_{k,l}}}}_{\!\le\! 0}}\begin{array}{l}-P\left({\Omega}_{E_{k,l}}\Big|{\Omega}_{E_{k,l}}\!\le\! 0\right)P\left({\Omega}_{E_{k,l}}\!\le\! 0\right)\\ {}\times \ln \left[P\left({\Omega}_{E_{k,l}}\Big|{\Omega}_{E_{k,l}}\!\le\! 0\right)P\left({\Omega}_{E_{k,l}}\!\le\! 0\right)\right]\end{array}}\\ &\qquad\qquad\quad\qquad\qquad-\left(\displaystyle\frac{1}{2}\right) \ln \left({n}_{\Omega}{{}_{{}_{E_{k,l}}}}_{\le 0}\!\times\! 2\right)\end{aligned} $$

The maximum value for the relative entropy for possible information value is equal to zero and achieved when $P\left({\Omega}_{E_{k,l}}>0\right)=P\left({\Omega}_{E_{k,l}}\le 0\right)=1/2$, all $P\left({\Omega}_{E_{k,l}}\Big|{\Omega}_{E_{k,l}}>0\right)=1/{\mathrm{n}}_{\Omega_{E_{k,l}}>0}$ and all $P\left({\Omega}_{E_{k,l}}\Big|{\Omega}_{E_{k,l}}\le 0\right)=1/{\mathrm{n}}_{\Omega_{E_{k,l}}\le 0}$.

This principle is consistent with the first two principles where the alternative of obtaining information, ${E}_{E_{k,l}}$, is compared with the alternative of not obtaining information for a given preference comparison (i.e., ${E}_l={E}_0=\varnothing$): when the relative entropy of the information value is maximized, there is an equal probability that obtaining the information is preferred (i.e., has positive information value or ${\Omega}_{E_{k,l}}>0$) and is not preferred (i.e., has nonpositive information value or ${\Omega}_{E_{k,l}}\le 0$) and the possible positive and nonpositive degrees of information value are equally probable.

1.1 A.1.1 Multiple Pairs of Alternatives

The principles of decision entropy establish a sample space for the comparison of any two decision alternatives, A _i and A _j . The sample space for the set of all possibilities of comparison for a given decision problem is denoted the decision sample space. In this sample space, each possible combination of an alternative that is selected, A _i , and an alternative that could have been selected, A _j , is equally probable (Fig. 18.22). For n _A alternatives, there are n ² _A pairs of i, j and $P\left({A}_i\ \mathrm{Selected}\ \mathrm{and}\ \mathrm{Compared}\ \mathrm{t}\mathrm{o}\ {A}_j\right)=1/{n}_A^2$ and $P\left({A}_i\ \mathrm{Compared}\ \mathrm{t}\mathrm{o}\ {A}_j\left|{A}_i\ \mathrm{Selected}\right.\right)=1/{n}_A$. The preferred decision alternative has the maximum expected degree of preference compared to all other alternatives:

$$ E\left[u\left({A}_i\right)-u\left(\overline{A_l}\right)\left|{A}_i\ \mathrm{Selected}\right.\right]={\displaystyle \sum_{\mathrm{all}\ j}}\begin{array}{c}\hfill E\left(\Delta {u}_{i,j}\Big|{A}_i\ \mathrm{Selected}\ \mathrm{and}\ \mathrm{Compared}\ \mathrm{t}\mathrm{o}\ {A}_j\right)\hfill \\ {}\hfill \times P\left({A}_i\ \mathrm{Compared}\ \mathrm{t}\mathrm{o}\ {A}_j\left|{A}_i\ \mathrm{Selected}\right.\right)\hfill \end{array}\vspace*{-3pt} $$

where

$$ \begin{aligned}&E\left(\Delta {u}_{i,j}\Big|{A}_i\ \mathrm{Selected}\ \mathrm{and}\ \mathrm{Compared}\ \mathrm{t}\mathrm{o}\ {A}_j\right)\\ &\quad={\displaystyle \sum_{\mathrm{all}\ \Delta {u}_{i,j}}}\Delta {u}_{i,j}P\left(\Delta {u}_{i,j}\Big|{A}_i\ \mathrm{Selected}\ \mathrm{and}\ \mathrm{Compared}\ \mathrm{t}\mathrm{o}\ {A}_j\right)\end{aligned}\vspace*{-3pt} $$

Fig. 18.22

Sample space for decision with three alternatives

The use of the expected degree of preference (or utility difference) as a measure of preference is consistent with utility theory analysis (e.g., Von Neumann and Morgenstern 1944; Hurwicz 1951; Savage 1951; Hodges and Lehmann 1952; Luce and Raiffa 1957; Raiffa and Schlaifer 1961; Benjamin and Cornell 1970; Ang and Tang 1984, etc.). In a conventional decision analysis, the sample space for utility values is not conditioned on a particular alternative being selected, A _i , meaning that the expected utility for a selected alternative can be calculated without considering the alternatives to which it is being compared. However, the absolute magnitude of expected utility is irrelevant⁵; its relevance depends on comparing it with the expected utility values for other alternatives. Therefore, it is the differences between utility values that are of interest.

Mathematically, comparing the expected degrees of preference, $E\left[u\left({A}_i\right)-u\left(\overline{A_i}\right)\right.\break\left.\left|{A}_i\ \mathrm{Selected}\right.\right]$, is the same as comparing expected utility values in a conventional decision analysis. In a conventional analysis where the probabilities for utility values given that an alternative has been selected do not depend on the alternative to which it is being compared, the expected degree of preference for an alternative can be expressed as follows:

$$ \begin{array}{l}E\left[u\left({A}_i\right)-u\left(\overline{A_l}\right)\Big|{A}_i\;\mathrm{Selected}\right]={\displaystyle \sum_{\mathrm{all}\;j}\begin{array}{l}E\left(\varDelta {u}_{i,j}\Big|{A}_i\;\mathrm{Selected}\;\mathrm{and}\;\mathrm{Compared}\;\mathrm{t}\mathrm{o}\;{A}_j\right)\\ {}\times P\left({A}_i\;\mathrm{Compared}\;\mathrm{t}\mathrm{o}\;{A}_j\Big|{A}_i\;\mathrm{Selected}\right)\end{array}}\\ {}={\displaystyle \sum_{\mathrm{all}\;j}\begin{array}{l}\left[E\left({u}_i\Big|{A}_i\;\mathrm{Selected}\right)-E\left({u}_j\Big|{A}_j\;\mathrm{Selected}\right)\right]\\ {}\times P\left({A}_i\;\mathrm{Compared}\;\mathrm{t}\mathrm{o}\;{A}_j\Big|{A}_i\;\mathrm{Selected}\right)\end{array}}\\ {}=E\left({u}_i\Big|{A}_i\;\mathrm{Selected}\right)-{\displaystyle \sum_{j=1\;\mathrm{t}\mathrm{o}\;{n}_A}E\left({u}_j\Big|{A}_j\;\mathrm{Selected}\right)\times \left(\frac{1}{n_A}\right)}\end{array}\vspace*{-3pt} $$

Therefore, the expected degree of preference for an alternative in a conventional decision analysis is equal to the expected utility for that alternative minus a constant (the average expected utility for all alternatives). Therefore, the order of comparisons is the same whether the expected utility values, $E\left({u}_i\Big|{A}_i\ \mathrm{Selected}\ \right)$, or the expected degree of preference values, $E\left[u\left({A}_i\right)-u\left(\overline{A_i}\right)\left|{A}_i\ \mathrm{Selected}\right.\right]$, are used in comparisons.

B.1 Appendix 2: Implementation of Bayes’ Theorem with Bernoulli Sequence for Two Possibly Related Sets of Data

Define F _A as the annual chance of occurrence in the period of available experience, Data Set A, and F _B as the annual chance of occurrence in the future for purposes of managing risk, Data Set B (Fig. 18.18). If the annual chance of occurrence is the same in the past and the future, then the likelihood of obtaining the available data from a Bernoulli sequence (i.e., x _A occurrences in n _A years) for a particular value of f _Bi is given by:

$$ P\left({x}_A\ \mathrm{in}\ {n}_A\Big|{f}_{Bi}={f}_{Ai}\right)=\left\{\left[\frac{n_A!}{x_A!\left({n}_A-{x}_A!\right)}\right]{f}_{Bi}^{x_A}{\left(1-{f}_{Bi}\right)}^{n_A-{x}_A}\right\} $$

Conversely, if the annual chance of occurrence is not the same in the past and the future, then the likelihood of obtaining the available data (i.e., x _A occurrences in n _A years) is given by:

$$ P\left({x}_A\ \mathrm{in}\ {n}_A\Big|{f}_{Bi}\ne {f}_{Ai}\right)={\displaystyle \sum_{\mathrm{all}\ {f}_{Ai}}}\left\{\left[\frac{n_A!}{x_A!\left({n}_A-{x}_A!\right)}\right]{f}_{Ai}^{x_A}{\left(1-{f}_{Ai}\right)}^{n_A-{x}_A}\right\}P\left(\left.{f}_{Ai}\right|{f}_{Bi}\ne {f}_{Ai}\right) $$

where P(f _Ai ) is the prior probability for the annual chance of occurrence in the experience. If the experience is relevant, then the prior probability for F _A is the same as that for F _B : $P\left(\left.{f}_{Ai}\right|{f}_{Bi}={f}_{Ai}\right)=P\left({f}_{Bi}\right)$. Furthermore, $P\left(\left.{f}_{Ai}\right|{f}_{Bi}\ne {f}_{Ai}\right)=P\left(\left.{f}_{Ai}\right|{f}_{Bi}={f}_{Ai}\right)=P\left({f}_{Ai}\right)=P\left({f}_{Bi}\right)$ since the probability for F _A does not depend on whether or not the experience is relevant (i.e., the prior probability of obtaining a particular set of data from Set A is the same whether or not Sets A and B are the same, and the probability of obtaining a particular set of data from Set A does not depend on the chance of occurrence in Set B if the two sets are different). Therefore,

$$ P\left({x}_A\ \mathrm{in}\ {n}_A\Big|{f}_{Bi}\ne {f}_{Ai}\right)={\displaystyle \sum_{\mathrm{all}\ {f}_{Bi}}}\left\{\left[\frac{n_A!}{x_A!\left({n}_A-{x}_A!\right)}\right]{f}_{Bi}^{x_A}{\left(1-{f}_{Bi}\right)}^{n_A-{x}_A}\right\}P\left({f}_{Bi}\right) $$

meaning that the likelihood of the information from the experience is a constant with respect to f _Bi (i.e., the updated probability distribution will be same as the prior probability distribution for F _B if the data are not relevant). The composite likelihood function considering the possibility that the data may or may not be relevant is given by:

$$ \begin{array}{l}P\left({x}_A\;\!\mathrm{in}\!\;{n}_A\Big|{f}_{Bi}\right)\!=\!P\left({x}_A\;\!\mathrm{in}\!\;{n}_A\Big|{f}_{Bi}\!=\!{f}_{Ai}\right)P\left({f}_{Bi}\!=\!{f}_{Ai}\right)\!+\!P\left({x}_A\;\!\mathrm{in}\!\;{n}_A\Big|{f}_{Bi}\!\ne\! {f}_{Ai}\right)\!P\left({f}_{Bi}\!\ne\! {f}_{Ai}\right)\\ {}=P\left({x}_A\;\mathrm{in}\;{n}_A\Big|{f}_{Bi}={f}_{Ai}\right)P\left({f}_{Bi}={f}_{Ai}\right)+P\left({x}_A\;\mathrm{in}\;{n}_A\Big|{f}_{Bi}\ne {f}_{Ai}\right)\left[1-P\left({f}_{Bi}={f}_{Ai}\right)\right]\end{array} $$

where $P\left({f}_{Bi}={f}_{Ai}\right)$ is the prior probability that the experience is relevant.

If the experience is relevant, then the greatest possible is learned from the experience. If the experience is not relevant, then the least possible (nothing) is learned from the experience. Therefore, the third principle of Decision Entropy Theory establishes that the probability that the experience is relevant is 0.5, or $P\left({f}_{Bi}={f}_{Ai}\right)=0.5$.

If information could be obtained about the annual chance of hazard occurrence in the period of the decision (F _B ) before making the decision (Fig. 18.18), then the likelihood of obtaining a particular set of data (i.e., x _A occurrences in n _A years and x _B occurrences in n _B years) is given by the following:

$$ \begin{array}{l}P\left(\left.{x}_A\ \mathrm{in}\ {n}_A\ \mathrm{and}\ {x}_B\ \mathrm{in}\ {n}_B\right|{f}_{Bi}\right)\\ {}=P\left({x}_A\ \mathrm{in}\ {n}_A\;\mathrm{and}\ {x}_B\ \mathrm{in}\ {n}_B\Big|{f}_{Bi}={f}_{Ai}\right)P\left({f}_{Bi}={f}_{Ai}\right)\\ {} + P\left({x}_A\ \mathrm{in}\ {n}_A\kern0.5em \mathrm{and}\ {x}_B\ \mathrm{in}\ {n}_B\Big|{f}_{Bi}\ne {f}_{Ai}\right)\left[1-P\left({f}_{Bi}={f}_{Ai}\right)\right]\end{array} $$

where

$$ \begin{array}{l}P\left({x}_A\ \mathrm{in}\ {n}_A\;\mathrm{and}\ {x}_B\ \mathrm{in}\ {n}_B\Big|{f}_{Bi}={f}_{Ai}\right)\\ {}=\left\{\left[\frac{n_A!}{x_A!\left({n}_A-{x}_A!\right)}\right]{f}_{Bi}^{x_A}{\left(1-{f}_{Bi}\right)}^{n_A-{x}_A}\right\}\times \left\{\left[\frac{n_B!}{x_B!\left({n}_B-{x}_B!\right)}\right]{f}_{Bi}^{x_B}{\left(1-{f}_{Bi}\right)}^{n_B-{x}_B}\right\}\end{array} $$

and

$$ \begin{array}{l}P\left({x}_A\;\mathrm{in}\;{n}_A\;\mathrm{and}\;{x}_B\;\mathrm{in}\;{n}_B\Big|{f}_{Bi}\ne {f}_{Ai}\right)\\ {}\!\!=\!\left\{{\displaystyle \sum_{\mathrm{all}\;{f}_{Bi}}\left\{\left[\!\frac{n_A!}{x_A!\left({n}_A\!-\!{x}_A!\right)}\!\right]{f}_{Bi}^{x_A}{\left(1\!-\!{f}_{Bi}\right)}^{n_A-{x}_A}\right\}\!P\left({f}_{Bi}\right)\!}\right\}\!\times\! \left\{\left[\!\frac{n_B!}{x_B!\left({n}_B-{x}_B!\right)}\!\right]{f}_{Bi}^{x_B}{\left(1\!-\!{f}_{Bi}\right)}^{n_B-{x}_B}\right\}\end{array} $$

Hence, the updated probability distribution for the annual chance of occurrence for the hazard in the risk management decision (F _B ) is obtained from Bayes’ theorem as follows:

$$ \begin{array}{l}P\left({F}_B={f}_{Bi}\Big|{x}_A\;\mathrm{in}\;{n}_A\;\mathrm{and}\;{x}_B\;\mathrm{in}\;{n}_B\right)\\ {}=\frac{\left\{\begin{array}{l}P\left({x}_A\;\mathrm{in}\;{n}_A\;\mathrm{and}\;{x}_B\;\mathrm{in}\;{n}_B\Big|{f}_{Bi}={f}_{Ai}\right)P\left({f}_{Bi}={f}_{Ai}\right)\\ {}+P\left({x}_A\;\mathrm{in}\;{n}_A\;\mathrm{and}\;{x}_B\;\mathrm{in}\;{n}_B\Big|{f}_{Bi}\ne {f}_{Ai}\right)\left[1-P\left({f}_{Bi}={f}_{Ai}\right)\right]\end{array}\right\}}{{\displaystyle {\sum}_{\mathrm{all}\;{f}_{Bj}}\left\{\begin{array}{l}P\left({x}_A\;\mathrm{in}\;{n}_A\;\mathrm{and}\;{x}_B\;\mathrm{in}\;{n}_B\Big|{f}_{Bj}={f}_{Aj}\right)P\left({f}_{Bj}={f}_{Aj}\right)\\ {}+P\left({x}_A\;\mathrm{in}\;{n}_A\;\mathrm{and}\;{x}_B\;\mathrm{in}\;{n}_B\Big|{f}_{Bj}\ne {f}_{Aj}\right)\left[1-P\left({f}_{Bj}={f}_{Ai}\right)\right]\end{array}\right\}}}\end{array} $$

Likewise, the probability that the data from the experience (Data Set A) is relevant is updated with the data obtained from the period of the decision (Data Set B):

$$ \begin{array}{l}P\left({f}_B={f}_{Ai}\Big|{x}_A\;\mathrm{in}\;{n}_A\;\mathrm{and}\;{x}_B\; in\;{n}_B\right)\\ {}=\frac{P\left({x}_A\;\mathrm{in}\;{n}_A\;\mathrm{and}\;{x}_B\;\mathrm{in}\;{n}_B\Big|{f}_{Bi}={f}_{Ai}\right)P\left({f}_{Bi}={f}_{Ai}\right)}{{\displaystyle {\sum}_{\mathrm{all}\;{f}_{Bj}}\left\{\begin{array}{l}P\left({x}_A\;\mathrm{in}\;{n}_A\;\mathrm{and}\;{x}_B\;\mathrm{in}\;{n}_B\Big|{f}_{Bj}={f}_{Aj}\right)P\left({f}_{Bj}={f}_{Aj}\right)\\ {}+P\left({x}_A\;\mathrm{in}\;{n}_A\;\mathrm{and}\;{x}_B\;\mathrm{in}\;{n}_B\Big|{f}_{Bj}\ne {f}_{Aj}\right)\left[1-P\left({f}_{Bj}={f}_{Ai}\right)\right]\end{array}\right\}}}\end{array} $$