Using Bayesian principles, what prior odds are necessary to represent an ideal trier of fact (TOF); that is, a TOF who impartially and equitably weighs evidence and decides a case based only on the evidence adduced at trial?

 

Imagine the following situation in which a trial is held wherein a TOF is to decide a particular fact in issue. Imagine the TOF is to be presented with a single piece of evidence at trial that is relevant to that fact in issue. The piece of evidence has a particular weight, or probative value, that quantifies how helpful it is to the TOF to determine the fact in issue; that is, to make the fact in issue more or less probable. The evidence will either make the probability of the prosecution's hypothesis about the fact in issue, p(Hp), more probable, or it will make the probability of the defense's hypothesis about the fact in issue, p(Hd), more probable. The two hypotheses are mutually exclusive and exhaustive, so 1 = p(Hp) + p(Hd). The TOF must decide the case solely upon the evidence adduced at trial. Therefore, hearing the piece of evidence will influence them to change their prior probabilities about the fact in issue, specifically p(Hp)1 and p(Hd)1, into posterior probabilities about the fact in issue, specifically p(Hp)2 and p(Hd)2.

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If the TOF is exposed only to that single piece of evidence at trial and the TOF decides solely on the evidence adduced at trial, then hearing that evidence should cause them to change their probabilities in direct proportion to the probative value of the evidence. It should not matter if the evidence favors the prosecution's hypothesis or if the evidence favors the defense's hypothesis; it should have the same amount of effect on an impartial juror, albeit in different directions.

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Imagine a single, perfectly ideal, fair and unbiased TOF who has some prior beliefs, p(Hp)1 and p(Hd)1 about some fact in issue that is to be decided at court. The TOF could hear either one of two cases. In the first case, X, there is a single piece of evidence that favors the prosecution's hypothesis, and it has a weight equal to WX. Hearing this evidence, the perfectly ideal, fair and unbiased TOF would come to a posterior probability regarding the prosecution's hypothesis equal to p(Hp)2x, for which p(Hp)2x > p(Hp)1. In the second case, Y, there is a single piece of evidence that favors the defense's hypothesis, and it has a weight equal to WY. Hearing this evidence, the same perfectly ideal, fair and unbiased TOF would come to a posterior probability regarding the defense's hypothesis of p(Hd)2y, for which p(Hd)2y > p(Hd)1. Insomuch as the weights of the evidence are perfectly equal (albeit supporting opposite hypotheses), WX = WY, and insomuch as this single, perfectly ideal, fair and unbiased TOF must decide the fact in issue based solely on the evidence adduced at court, it is clear this requires that p(Hp)2x = p(Hd)2y. The question is, what values of p(Hp)1 and p(Hd)1 are required to satisfy the requirement that the TOF decides the case based only on the evidence adduced at trial according to its usefulness for determining the fact in issue, such that p(Hp)2x will equal p(Hd)2y in this scenario?

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Using Bayesian principles and the convention of placing the probability of the prosecution's hypothesis in the numerator and of placing the probability of the defense's hypothesis in the denominator, the process of a TOF incorporating evidence into their subjective probabilities of each hypothesis may be represented as follows:

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[1]                                                            {p(Hp)2/p(Hd)2} = { p(Hp)1/p(Hd)1} · {LR}

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In equation [1], the weight of the evidence is represented by the likelihood ratio, LR, which is the ratio of the probability of the evidence if the prosecution's hypothesis were true, p(E|Hp), to the probability of the evidence if the defense's hypothesis were true, p(E|Hd).

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Therefore, for case X we have,

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[2]                                             {p(Hp)2x/p(Hd)2x} = {p(Hp)1/p(Hd)1} · {p(E|Hp)x/p(E|Hd)x}

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and for case Y we have

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[3]                                            {p(Hp)2y/p(Hd)2y} = {p(Hp)1/p(Hd)1} · {p(E|Hp)y/p(E|Hd)y}

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Solving for p(Hp)2x in Equation [2] we have,

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[4]                                            p(Hp)2x = {p(Hd)2x} · {p(Hp)1/p(Hd)1} · {p(E|Hp)x/p(E|Hd)x}

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and solving for p(Hd)2y in Equation [3] we have,

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[5]                                            p(Hd)2y = {p(Hp)2y} · {p(Hd)1/p(Hp)1} · {p(E|Hd)y/p(E|Hp)y}

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Therefore, satisfaction of the requirement that p(Hp)2x = p(Hd)2y yields

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[6]      {p(Hd)2x} · {p(Hp)1/p(Hd)1} · {p(E|Hp)x/ p(E|Hd)x} = {p(Hp)2y} · {p(Hd)1/p(Hp)1} · {p(E|Hd)y/p(E|Hp)y}

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Furthermore, in this scenario, two separate cases with different pieces of evidence are considered. In case X the evidence favors the prosecution's hypothesis, and in case Y the evidence favors the defense's hypothesis. The two cases present evidence that is precisely equal in weight, but opposite in direction. This entails the following equality:

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[7]                                                          {p(E|Hp)x/p(E|Hd)x}  = {p(E|Hd)y/p(E|Hp)y}

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Using the equality of Equation [7] in Equation [6] simplifies Equation [6] to

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[8]                                               {p(Hd)2x} · {p(Hp)1/p(Hd)1} = {p(Hp)2y} · {p(Hd)1/p(Hp)1}

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It is now useful to recognize that the requirement that p(Hp)2x=p(Hd)2y likewise entails the requirement that p(Hd)2x = p(Hp)2y, which simplifies Equation [8] to

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[9]                                                                   {p(Hp)1/p(Hd)1} = {p(Hd)1/p(Hp)1}

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Using simple mathematics we can solve for p(Hp)1 and p(Hd)1 as follows:

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[10]                                                                              {p(Hp)1/p(Hd)1}^2 =1

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[11]                                                                      sqrt[{p(Hp)1/p(Hd)1}^2] = sqrt[1]

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[12]                                                                                p(Hp)1/p(Hd)1 = 1

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[13]                                                                                  p(Hp)1 = p(Hd)1

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[14]                                                                               p(Hp)1 = 1 - p(Hp)1

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[15]                                                                                     2p(Hp)1 = 1

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[16]                                                                                     p(Hp)1 = 0.5

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And because 1 = p(Hp)1 + p(Hd)1, we have

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[17]                                                                                    p(Hd)1 = 0.5.

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Hence, prior odds of p(Hp)1/p(Hd)1 = 0.5/0.5 = 1 are required to represent an ideal TOF who impartially and equitably weighs all evidence and decides a case based solely on evidence adduced at trial.

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Some may ask, "But what about the presumption of innocence?". The presumption of innocence is not evidence, so it has no evidentiary weight and should not influence the TOF in the same manner as does evidence. Rather the presumption of innocence entails the combination of two factors: First, it requires that the TOF decide a case based only on the weight of evidence adduced at trial (requiring prior odds = 1, as shown above) and, second, that the TOF ultimately decide guilty or not guilty only after the prosecution has met the burden the burden of proof, meaning that the TOF's posterior beliefs of guilt become large enough that, in the opinion of the TOF, the guilt of the defendant has been proven beyond a reasonable doubt. Even if the evidence adduced at trial points towards guilt, if the evidence fails to surpass the prosecution's burden of proof (i.e., not beyond a reasonable doubt), the presumption of innocence operates in favor of the defendant, and the TOF decides "not guilty". The presumption of innocence does not influence the TOF's beliefs about the case in the same way that evidence does. The presumption of innocence only entails what evidence is weighed (that adduced at trial), how it is weighed (impartially and according to its probative value), and the criterion that the TOF's belief in guilt must surpass in order to deliver a guilty verdict (beyond a reasonable doubt).