An Outline of the Bayesian Decision Theory
aa r X i v : . [ s t a t . O T ] S e p AN OUTLINE OF THE BAYESIAN DECISION THEORY
H.R.N. VAN ERP, R.O. LINGER, AND P.H.A.J.M. VAN GELDER
Abstract.
In this fact sheet we give an outline on the Bayesian DecisionTheory. Introduction
The Bayesian decision theory is very simple in structure. Its algorithmic stepsare the following:(1) Use the product and sum rules of Bayesian probability theory to constructoutcome probability distributions.(2) If our outcomes are monetary in nature, then by way of the Bernoulli utilityfunction we may map utilities to the monetary outcomes of our outcomeprobability distributions.(3) Maximize the position of the resulting utility probability distributions.This is the whole of the Bayesian decision theory.2.
Constructing Outcome Probability Distributions
In the Bayesian decision theory each problem of choice is understood to consistof a set of decisions from which we must choose. Each possible decision has as-sociated with it its own set of possible outcomes, and each outcome has its ownplausibility of occurring relative to the other outcomes under that same decision.Stated differently, each decision in our problem of choice admits its own outcomeprobability distribution.In its most abstract form, we have that each problem of choice consists of a setof potential decisions D i = { D , . . . , D n } . Each decision D i we make may give rise to a set of possible events E j i = { E i , . . . , E m i } . These events E j i are associated with the decisions D i by way of the conditionalprobabilities P ( E j i | D i ). Furthermore, each event E j i allows for a set of potentialoutcomes O k ji = (cid:8) O ji , . . . , O l ji (cid:9) . These outcomes O k ji are associated with the events E j i by way of the conditionalprobabilities P (cid:0) O k ji (cid:12)(cid:12) E j i (cid:1) .By way of the product rule [5], we compute the bivariate probability distributionof an event and an outcome conditional on the decision taken: P (cid:0) E j i , O k ji (cid:12)(cid:12) D i (cid:1) = P ( E j i | D i ) P (cid:0) O k ji (cid:12)(cid:12) E j i (cid:1) . (2.1)The outcome probability distribution is then obtained by marginalizing, by way ofthe sum rule [5], over all the possible events: P (cid:0) O k ji (cid:12)(cid:12) D i (cid:1) = m i X j i =1 P (cid:0) E j i , O k ji (cid:12)(cid:12) D i (cid:1) . (2.2)The outcome probability distributions (2.2), for i = 1 , . . . , n , are the informationcarriers which represent our state of knowledge in regards to the objective conse-quences of our decisions.3. A Consistency Proof of the Bernoulli Utility Function
We will now derive the Bernoulli utility function, or, equivalently, the Weber-Fechner law, or, equivalently, in content, Steven’s Power law, using the desiderataof invariance and consistency. In this we follow a venerable Bayesian tradition,[2, 5, 7].Say, we have the positive quantities x , y , and z , of some stimulus or commodityof interest. Then these quantities, being numbers on the positive real, admit anordering. So, let quantities be ordered as x ≤ y ≤ z . We now want to find thefunction f that quantifies the perceived decrease associated with going from, say,the quantity z to the quantity x .The first functional equation is based on the desideratum that the unknownfunction f should be invariant for a change of scale in our quantities: f ( x, z ) = f ( cx, cz ) , (3.1)where c is positive constant.For example, if our quantities concern sums of money, then the perceived lossof going from ten dollars to one dollar should be the same perceived loss if wereformulate this scenario in dollar cents.The second functional equation is based on the desideratum of consistency, inwhich we state that the perceived decrease in going directly from z to x , ought tobe the same perceived decrease in going from z to x via y : f ( x, z ) = g [ f ( x, y ) , f ( y, z )] . (3.2) N OUTLINE OF THE BAYESIAN DECISION THEORY 3
For example, if our quantities concern sums of money, then the perceived loss ofgoing from ten dollars to one dollar should be the same perceived loss if we first gofrom ten dollars to five dollars, and then from five dollars to one dollar; seeing thatin both scenarios we start out with an initial wealth of ten dollars, only to end upwith a current wealth of one dollar.The general solution to (3.1) is [3]: f ( x, y ) = h (cid:18) xy (cid:19) , (3.3)were h is some arbitrary function.The general solution to (3.2) is [7]:Θ[ f ( x, z )] = Θ[ f ( x, y )] + Θ[ f ( y, z )] , (3.4)where Θ is some arbitrary monotonic function. Moreover, because of this arbitrari-ness, we may define Θ as [7]: Θ( x ) = log Ψ( x ) . (3.5)Using (3.5), we may rewrite (3.4), without any loss of generality, aslog Ψ[ f ( x, z )] = log Ψ[ f ( x, y )] + log Ψ[ f ( y, z )] , (3.6)or, equivalently, Ψ[ f ( x, z )] = Ψ[ f ( x, y )] Ψ[ f ( y, z )] . (3.7)Substituting (3.3) into (3.4) and (3.7) and letting, respectively, θ (cid:18) xy (cid:19) = Θ (cid:20) h (cid:18) xy (cid:19)(cid:21) , (3.8)and ψ (cid:18) xy (cid:19) = Ψ (cid:20) h (cid:18) xy (cid:19)(cid:21) , (3.9)we obtain the equivalent functional equations: θ (cid:16) xz (cid:17) = θ (cid:18) xy (cid:19) + θ (cid:16) yz (cid:17) (3.10)and ψ (cid:16) xz (cid:17) = ψ (cid:18) xy (cid:19) ψ (cid:16) yz (cid:17) . (3.11)If we assume differentiability, then (3.10), together with the two boundary con-ditions: f ( x, x ) = θ (cid:16) xx (cid:17) = 0 , (3.12)and f ( x, y ) = θ (cid:18) xy (cid:19) < , for x < y, (3.13)is sufficient to find the function f that quantifies the perceived decrease associatedwith going from the quantity y to the quantity x . H.R.N. VAN ERP, R.O. LINGER, AND P.H.A.J.M. VAN GELDER
This function θ turns out to be Bernoulli’s utility function, or, equivalently, theWeber-Fechner law of sense perception: f ( x, y ) = q log xy , q ≥ y is our initial asset position and x is the final asset position, and q is somearbitrary constant which has to be obtained by way psychological experimentation.So, Bernoulli’s utility function (3.14) is the only function that adheres to thedesiderata of unit invariance and consistency, respectively, (3.1) and (3.2), and theboundary conditions that a zero change should lead to a zero perceived loss and thata perceived loss should be assigned a negative value, respectively, (3.12) and (3.13).Any other utility function will be in violation with these fundamental desiderataand specific boundary conditions.Note that Fechner re-derived (3.14) in 1860 as the law that guides our sensoryperception. In the years that followed (3.14) proved to be so successful, as it,amongst other things, gave rise to our decibel scale, that it established psychol-ogy as a legitimate experimental science [4]. But as Fechner was very careful, formetaphysical reasons, or so we hazard to guess [3], to apply his Weber law, whichlater became the Fechner-Weber law, only to non-monetary stimuli, the implieduniversality of (3.14) was not recognized for the longest time.However, because of the here given consistency derivation of (3.14), it is nowshown that the Fechner-Weber, or, equivalently, Bernoulli’s utility function, is oneof the consistent functions that quantifies the distance between x and y ; thus,explaining the universal applicability of Bernoulli’s utility function.The other consistent distance function is Steven’s power law, which may bederived as follows. If we assume differentiability, then (3.11), together with the twoboundary conditions: f ( x, x ) = ψ (cid:16) xx (cid:17) = 1 , (3.15)and 0 < f ( x, y ) = ψ (cid:18) xy (cid:19) < , for x < y, (3.16)is sufficient to find the function f that quantifies the perceived decrease associatedwith going from the quantity y to the quantity x .This function f turns out to be Steven’s power law: f ( x, y ) = (cid:18) xy (cid:19) q , q ≥ . (3.17)where y is our initial asset position and x is the final asset position, and q is somearbitrary constant which has to be obtained by way psychological experimentation.So, Steven’s power law (3.17) is the only function that adheres to the desiderataof unit invariance and consistency, respectively, (3.1) and (3.2), and the boundary N OUTLINE OF THE BAYESIAN DECISION THEORY 5 conditions that a zero change should lead to a ratio of one between the initial andfinal ‘asset position’ and that a perceived loss should be assigned a value smallerthan 1, respectively, (3.15) and (3.16). Any other utility function will be in violationwith these fundamental desiderata and specific boundary conditions.We summarize, given the desiderata (3.1) and (3.2), the Fechner-Weber law(3.14) results from the boundary condition that negative increments result negativeutilities and a zero increment results in an utility of zero, (3.12) and (3.13); whereasSteven’s power law (3.17) results from the boundary condition that utilities mustbe greater than zero and that a zero increment results an utility of one, (3.15) and(3.16).Stated differently, the Fechner-Weber law and Steven’s power law are both equiv-alent in content, differing only in the proposed utility scale. A subtlety thatseems to have been overlooked by some, seeing that the Fechner-Weber law ver-sus the Steven’s power law has been a source of controversy in psycho-physicalcommunity[8].4.
The Criterion Of Choice as a Degree of Freedom
Let D and D be two decisions we have to choose from. Let o i , for i = 1 , . . . , n ,and o j , for j = 1 , . . . , m , be the monetary outcomes associated with, respectively,decisions D and D . Then in the Bayesian decision theory we first construct thetwo outcome distributions that correspond with these decisions: p ( o i | D ) , p ( o j | D ) , (4.1)where, if n = m , the outcomes o i and o j may or not may be equal for i = j .We then proceed, by way of the Bernoulli utility function (3.14), or, equivalently,the Weber-Fechner law, to map utilities to the monetary outcomes o i and o j in (4.1).This leaves us with the utility probability distributions: p ( u i | D ) , p ( u j | D ) . (4.2)Now, our most primitive intuition regarding the utility probability distributions(4.2) is that the decision which corresponds with the utility probability distributionwhich lies more to the right will also be the decision that promises to be the mostadvantageous. So, when making a decision we ought to compare the positions ofthe utility probability distributions on the utility axis and then choose that decisionwhich maximizes the position of these utility probability distributions.This all sounds intuitive enough. But how do we define the position of a prob-ability distribution? Ideally we would have some consistency derivation of whatconstitutes a position measure of a probability distribution, say, H n ( p , . . . , p n , x , . . . , x n ) (4.3) H.R.N. VAN ERP, R.O. LINGER, AND P.H.A.J.M. VAN GELDER where p i are the probabilities of the values x i , for i = 1 , . . . , n . But in the absenceof such a consistency derivation we have to take our recourse to ad hoc commonsense considerations. Stated differently, the criterion of choice in our decision theoryconstitutes a degree of freedom.4.1. The Expectation Value as a Position Measure.
From the introductionof expected outcome theory in the 17th century and expected utility theory in the18th century the implicit assumption has been that the measure of a position of aprobability distribution is given by its expectation value [5, 1]: E ( X ) = n X i =1 p i x i = H n ( p , . . . , p n , x , . . . , x n . ) (4.4)But this criterion of choice has proven to be so unsatisfactory that it has givenrise to the paradigm of behavioral economics which holds as its central tenet thathuman decision making does not adhere to the maximization of expectation values[6]. So, we set out to search for a more appropriate criterion of choice.4.2. The Confidence Bounds as a Position Measure.
Now we may imagine adecision problem in which we are only interested in the positions of the probabilisticworst or best case scenarios.The absolute worst case scenario is: a = min( x , . . . , x n ) . (4.5)The criterion of choice (4.5) is also known as the maximin criterion of choice.The k-sigma lower bound of a given probability distribution is a given as LB ( X ) = E ( X ) − k std( X ) , (4.6)where std( X ) = vuut n X i =1 p i [ x i − E ( X )] , (4.7)and where k is the sigma level of the lower bound. The probabilistic worst casescenario then is given as: LB ∗ ( X ) = E ( X ) − k std( X ) , LB ( X ) > a,a, LB ( X ) ≤ a. (4.8)So, we have that the probabilistic worst case scenario holds the maximin criterionof choice as a special case.The absolute best case scenario is: b = max( x , . . . , x n ) . (4.9)The criterion of choice (4.9) is also known as the maximax criterion of choice. N OUTLINE OF THE BAYESIAN DECISION THEORY 7
The k-sigma upper bound of a given probability distribution is a given as: UB ( X ) = E ( X ) + k std( X ) , (4.10)where k is the sigma level of the upper bound. The probabilistic best case scenariothen is given as: UB ∗ ( X ) = E ( X ) + k std( X ) , UB ( X ) < b,b, UB ( X ) ≥ b. (4.11)So, we have that the probabilistic best case scenario holds the maximax criterionof choice as a special case.If we take as our criterion of choice (4.8) then we only endeavor to minimizeour ‘losses’ and if we take as our criterion of choice (4.11) then we only endeavorto maximize our ‘gains’. A more rational, that is, balanced, criterion of choicewould be to make a trade-off between the losses/gains in the probabilistic worstcase scenarios (4.8) and the corresponding gains/losses in the probabilistic bestcase scenarios (4.11).4.3. The Sum of Confidence Bounds as a Position Measure.
If we take asour criterion of choice LB ∗ ( X ) + UB ∗ ( X )2 = E ( X ) , LB ( X ) > a, UB ( X ) < b, a + E ( X )+ k std( X )2 , LB ( X ) ≤ a, UB ( X ) < b, E ( X ) − k std( X )+ b , LB ( X ) > a, UB ( X ) ≥ b, a + b , LB ( X ) ≤ a, UB ( X ) ≥ b, (4.12)then we have a position measure which makes a trade-off between the losses/gainsin the probabilistic worst case scenarios (4.8) and the corresponding gains/losses inthe probabilistic best case scenarios (4.11); see Appendix A.This alternative position measure, as an added benefit, also holds the traditionalcriterion of choice (4.4) as a special case, when no undershoot and overshoot of,respectively, the lower and upper sigma confidence bounds occur, as well as Hur-witz’s criterion of choice with a balanced pessimism factor of c = 1 /
2, when both anundershoot and an overshoot occur. Nonetheless, it may be found that the criterionof choice (4.12) is vulnerable to a simple counter-example.Imagine two utility probability distributions having equal lower and upper bounds,but one being right-skewed and the other being left-skewed. Then the criterion ofchoice (4.12) will leave us undecided between the two, whereas our intuition wouldgive preference to the decision corresponding with the left-skewed distribution, asthe bulk of the probability distribution of the left-skewed distribution will be moreto the right than that of the right-skewed distribution.
H.R.N. VAN ERP, R.O. LINGER, AND P.H.A.J.M. VAN GELDER
The Sum of Confidence Bounds Plus the Expectation Value as a Po-sition Measure.
What we seek to maximize in our decision theory is the positionof the utility probability distributions; as we have that the decision that puts ourutility probability distribution most to the right promises to be the most profitabledecision. In this there is little room for maneuvering. But in our choice of themeasure that captures the position of a given probability distribution there is allthe more.Taking a cue from the behavioral economists we have derived as an alternativeto (4.4) the criterion of choice (4.12) that also takes into account the standarddeviation of a given probability distributions, by way of the positions of the underand overshoot corrected lower and upper bounds. But only to find its universalitycompromised by the simple counter example of a right-skewed and a left-skeweddistribution which have the same lower and upper bounds.Now, also taking a cue from the intuitive results which flow forth from (4.12) [3],we may ‘repair’ our criterion of choice (4.12), albeit in an ad hoc fashion, by takingas our position measure for a probability distribution the weighted sum: LB ∗ ( u ) + E ( u ) + UB ∗ ( u )3 = E ( X ) , LB ( X ) > a, UB ( X ) < b, a +2 E ( X )+ k std( X )3 , LB ( X ) ≤ a, UB ( X ) < b, E ( X ) − k std( X )+ b , LB ( X ) > a, UB ( X ) ≥ b, a + E ( X )+ b , LB ( X ) ≤ a, UB ( X ) ≥ b, (4.13)For in this criterion of choice we not only take into account the trade-off betweenthe probabilistic worst and best case scenarios, but also the location of the bulkof the probability density in a uni-model probability distribution; thus, accommo-dating the intuitive preference for the left-skewed distribution of the above counterexample.The position measure (4.13) is the weighted sum of the positions of, respectively,the probabilistic worst, expected, and best case. The uncorrected lower and upperbounds, (4.6) and (4.10), have been traditionally used as simplifying proxies fortheir generating probability distributions, by way of confidence intervals:[ LB ( X ) , UB ( X )] . (4.14)We, alternatively, take as our simplifying proxy the corrected lower and upperbounds, (4.8) and (4.11), and the expectation value (4.4):[ LB ∗ ( X ) , E ( X ) , UB ∗ ( X )] . (4.15)Because of the corrections for lower bound undershoot and upper bound overshootin (4.15) we have that for skewed distributions the distance between E ( X ) and N OUTLINE OF THE BAYESIAN DECISION THEORY 9 LB ∗ ( X ) may differ from the distance between UB ∗ ( X ) and E ( X ); thus, reflectingthe asymmetry present in these distributions.The position of the generating probability distribution then is taken to be theweighted sum of the positions of the elements of our simple proxy distribution. Thisthen is the rationale behind the criterion of choice (4.13).5. Discussion
It may be read in Jaynes’ [5], that to the best of his knowledge, there are as ofyet no formal principles at all for assigning numerical values to loss functions; noteven when the criterion is purely economic, because the utility of money remainsill-defined. In the absence of these formal principles, Jaynes final verdict was thatdecision theory can not be fundamental.The Bernoulli utility function, initially derived by Bernoulli, by way of commonsense first principles [1], has now been derived by way of a consistency argument.This consistency argument explains why it is that Bernoulli’s utility function, bothin its original Fechner-Weber law and in its alternative Steven’s power law form, hasproven to be so ubiquitous and successful the field of sensory perception research;simply because human sense perception, like the laws of Nature, adheres to thedesideratum of consistency.The history of Bayesian probability theory has taught us that the usefulness ofa theory, in terms of its practical and beautifully intuitive results, in the absence ofa compelling axiomatic basis, provides no safeguard against attacks by those whochoose to close their eyes to this usefulness. This is why we felt compelled to searchfor a consistency derivation of the Bernoulli utility function.Now, having presented a consistency proof for the Bernoulli utility function, thequestion now is: Is the Bayesian decision theory, just like the Bayesian probabilityand information theories, Bayesian in the strictest sense in the word, or, equiva-lently, an inescapable consequence of the desideratum of consistency? We will nowtry to answer this question.The first two algorithmic steps of the Bayesian decision theory, respectively, theconstruction of outcome probability distributions by way of the Bayesian probabil-ity theory and the construction of utility probability distributions by way of theBernoulli utility function, allow us no freedom.To construct our outcome and utility probability distributions otherwise, wouldbe to invite inconsistency. But there is one degree of freedom remaining in theBayesian decision theory as a whole. This remaining degree of freedom lies in thechoice of our position measure of a given probability distribution.In any problem of choice we will endeavor to choose that decision which has acorresponding utility probability distribution that is lying most the right on the utility axis; that is, we will choose to maximize our utility probability distributions.In this there is little freedom. But we are free, in principle, to choose the measures ofthe positions of our utility probability distributions any way we see fit. Nonetheless,we believe that it is always a good policy to take into account all the pertinentinformation we have.If we only maximize the expectation values of the utility probability distributions,then we will, by definition, neglect the information that the standard deviations ofthe utility probability distributions have to bear on our problem of choice, by wayof the symmetry breaking in the case of an overshoot of one of the bounds.Likewise, we are free to only maximize one of the confidence bounds of ourutility probability distributions, while neglecting the other. But in doing so, wewill be performing probabilistic maximin or maximax analyses, and, consequently,neglect the possibility of either the (catastrophic) losses in the lower bound or the(astronomical) gains in the upper bound.However, if we only maximize the sum of the lower and upper bound, or a scalarmultiple thereof, then we will make a trade-off between the probabilistic worst andbest case scenarios. But in the process, we will, for uni-modal distributions, beneglecting the location of the bulk of our probability distributions.This is why, in our minds, the scalar multiple the sum of the lower bound, ex-pectation value, and upper bound currently is the most all-round position measurefor a given probability distribution, as it reflects the position of the probabilisticworst and best case scenarios, as well as the position of the expected outcome.Having removed the degree of freedom of the utility function by way of a con-sistency derivation, we now should endeavor to find a like consistency derivationof the measure of the position of a given probability distribution (4.3); as such aconsistency derivation would make the Bayesian decision theory incontestable.But until that time, we will have to do with the kind of simplistic common sensereasoning that led us from the traditional position measure (4.4), to the positionmeasures (4.12) and (4.13), and make the disclaimer that our adopted criterion ofchoice, as a degree of freedom, is just a matter of choice.
Acknowledgments:
We would like here to express our gratitude to Kevin H.Knuth, whose kind and patient feedback led us to our consistency proof of Bernoulli’sutility function and to Kevin M. Vanslette, whose simple but highly effective‘counter’ example led us to drop (4.12) and propose (4.13) instead. The researchleading to these results has received partial funding from the European Com-mission’s Seventh Framework Program [FP7/2007-2013] under grant agreementno.265138.
N OUTLINE OF THE BAYESIAN DECISION THEORY 11
References [1] Bernoulli D.:
Exposition of a New Theory on the Measurement of Risk . Translated fromLatin into English by Dr Louise Sommer from ‘Specimen Theoriae Novae de Mensura Sortis’,Commentarii Academiae Scientiarum Imperialis Petropolitanas, Tomus V, 175-192, (1738).[2] Cox R.T.:
Probability, Frequency and Reasonable Expectation , American Journal of Physics,14, 1-13, (1946).[3] Erp van H.R.N., Linger R.O., and Gelder van P.H.A.J.M.:
Fact Sheet on the Bayesian DecisionTheory , arXiv, (2015).[4] Fancher R.E.:
Pioneers of Psychology , W. W. Norton and Company, London, (1990).[5] Jaynes E.T.:
Probability Theory; the Logic of Science . Cambridge University Press, (2003).[6] Kahneman D.:
Thinking, Fast and Slow , Penguin Random House, UK, (2011).[7] Knuth K.H. and Skilling J.:
Foundations of Inference , arXiv: 1008.4831v1 [math.PR], (2010).[8] Stevens S.S.:
To Honor Fechner and Repeal His Law , Science, New Series, Vol. 133, No. 3446,pp 80-86, (1961).
Appendix A. Deriving the Sum of the Lower and Upper ConfidenceBound Measure
Now, the confidence bounds of (4.2), say:[ LB ( u | D ) , UB ( u | D )] , [ LB ( u | D ) , UB ( u | D )] , (A.1)may provide us with a numerical handle on the concept of more-to-the-right.For example, if we have that both LB ( u | D ) > LB ( u | D ) , UB ( u | D ) > UB ( u | D ) . (A.2)Then we will have an unambiguous preference for decision D over decision D ;seeing that under both the still probable worst and best case we will be better ifwe opt for D .Likewise, if we have that either LB ( u | D ) = LB ( u | D ) , UB ( u | D ) > UB ( u | D ) , (A.3)or LB ( u | D ) > LB ( u | D ) , UB ( u | D ) = UB ( u | D ) . (A.4)Then, again, we will have an unambiguous preference for decision D over decision D . In the constellation (A.3), we stand, all other things being equal, to be betterof under the still probable best case scenario; while in the constellation (A.4), westand, all other things being equal, to be less worse of under the still probable worstcase scenario.However, things become more ambiguous when, say, under decision D , we haveto make a trade-off between either a gain in the upper bound and a loss in the lowerbound LB ( u | D ) < LB ( u | D ) , UB ( u | D ) > UB ( u | D ) , (A.5) or a gain in the lower bound and a loss in the upper bound LB ( u | D ) > LB ( u | D ) , UB ( u | D ) < UB ( u | D ) . (A.6)We postulate here that a rational criterion of choice in the respective trade-offsituations (A.5) and (A.6), would be to pick that decision whose gain in either thelower or upper bound exceeds the loss in the corresponding upper or lower bound.So, if, say, under D we stand to gain more in the still probable best case scenariothan we stand to lose under the still probable worst case scenario, that is, (A.5): LB ( u | D ) − LB ( u | D ) < UB ( u | D ) − UB ( u | D ) , (A.7)then we will choose D over D . Likewise, if under D we stand to gain more inthe still probable worst case scenario than we stand to lose under the still probablebest case scenario, that is, (A.6): LB ( u | D ) − LB ( u | D ) > UB ( u | D ) − UB ( u | D ) , (A.8)then again we will choose D over D .Note that the gains and losses in this discussion pertain to gains and losseson the utility dimension, not on the monetary outcome dimension. On the utilitydimension the phenomenon of loss aversion, that is, the phenomenon that monetarylosses may weigh heavier than equal monetary gains, has already been accountedfor. Stated differently, the utility scale is a linear loss-aversion corrected scale forthe moral value of monies.Now, if we look at the scenarios (A.5) and (A.6), and the corresponding postu-lated rational, because intuitive, criteria of choice (A.7) and (A.8), then we see thatwe will choose D over D whenever we have that LB ( u | D ) + UB ( u | D ) > LB ( u | D ) + UB ( u | D ) . (A.9)Moreover, this single criterion of choice is also consistent with the choosing of D over D in the scenarios (A.2), (A.3), and (A.4).Note that if the decision inequality (A.9) goes to an equality: LB ( u | D ) + UB ( u | D ) = LB ( u | D ) + UB ( u | D ) . (A.10)Then we have that we will be undecided when it comes to the decisions D and D .Now, the k -sigma bounds in (A.1) translate to[ E ( u | D i ) − k std( u | D i ) , E ( u | D i ) + k std( u | D i )] , (A.11)for i = 1 ,
2, which, if substituted in (A.9), give the inequality2 E ( u | D ) > E ( u | D ) , (A.12) N OUTLINE OF THE BAYESIAN DECISION THEORY 13 which brings us right back to Bernoulli’s expected utility theory, as proposed in1738, in which it is proposed that we choose that decision which maximizes theexpectation value of the utility probability distributions [1].Nonetheless, the criterion of choice, that the sum of the upper and lower boundshould be maximized, as proposed here, will deviate from Bernoulli’s initial 1738proposal when the k -sigma intervals overshoot either the minimal or the maximalvalue of the utility probability distributions, or both.Let a and b , respectively, be the minimal and maximal values of a given utilityprobability distribution. Then we may identify three additional cases, relative to(A.12): LB ∗ ( u ) + UB ∗ ( u ) = E ( u ) , LB ( u ) ≥ a, UB ( u ) ≤ ba + UB ( u ) , LB ( u ) < a, UB ( u ) ≤ bLB ( u ) + b, LB ( u ) ≥ a, UB ( u ) > ba + b, LB ( u ) < a, UB ( u ) > b (A.13)These additional cases correspond, respectively, to scenarios where the k -sigmaintervals (A.11) either undershoot, or overshoot, or both undershoot and overshootthe minimal and maximal values of a given utility probability distribution.In the case of a lower confidence bound undershoot (e.g cases two and four) atoo pessimistic worst case scenario is in play, and in the case of an upper confidencebound overshoot a too optimistic a best case scenario is in play, which is why wehave to readjust these confidence bounds by replacing them with more realisticworst and best case scenarios (e.g the minimal and maximal values, a and b , of agiven utility probability distribution).It may be readily seen that any scalar multiple of (A.13) will retain the transitiveordering of the criterion of choice (A.13). Now, if we take as our scalar multiple c = 1 /
2, then our criterion of choice may be interpreted as a location measure ofutility probability distribution: LB ∗ ( u ) + UB ∗ ( u )2 = E ( u ) , LB ( u ) ≥ a, UB ( u ) ≤ b a + UB ( u )2 , LB ( u ) < a, UB ( u ) ≤ b LB ( u )+ b , LB ( u ) ≥ a, UB ( u ) > b a + b , LB ( u ) < a, UB ( u ) > b (A.14)where we note that the first case of (A.14) corresponds with Bernoulli’s expectedutility theory criterion of choice, whereas the fourth case corresponds with Hurwitz’scriterion of choice with a balanced pessimism coefficient of α = 1 / the absence of lower bound undershoot and upper bound overshoot, we have as thelower and upper bounds whose sum is to be maximized: LB ∗ ( u ) = E ( u ) − k std( u ) , (A.15)and UB ∗ ( u ) = E ( u ) + k std( u ) . (A.16)Then (A.15) and (A.16) sum to: LB ∗ ( u ) + UB ∗ ( u ) = 2 E ( u ) + ( k − k ) std( u ) . (A.17)If in (A.17) we let k > k , then we put a premium caution; alternatively, if we set k > k , then we put a premium on opportunity; and if we let k = k2