Bayes’ theorem and Bayesian reasoning

Bayes’ theorem (Bayes 1763) provides the theoretical foundation for a branch of

statistics based on conditional probability, in which probability is considered as a measure of belief in an event or series of events. It is conditional in that it is dependent

on available information (Redmayne 2001: 55). According to Aitken and Taroni (2004: 22), Bayes’ theorem is defined by two fundamental elements: (1) that belief can be

modified as new information emerges or existing information changes, and (2) that different individuals’ beliefs in the same event will vary due to differences in the

weights attached to each piece of information. Bayes’ theorem may be expressed as:

p(Hn|E) =p(Hn)p(E|Hn) (2.1)

from Lee (2004: 8)

wherepis probability, E is evidence, Hnis a sequence of events and |is given. Ac-

cording to Bayes’ theorem, the probability of a sequence of events given the evidence

p(Hn|E)is equivalent to the product of the prior probability of that sequence events p(Hn) and the probability of the evidence assuming that series of events p(E|Hn)

(Iversen 1984: 12).

The conceptual application of Bayes’ theorem is commonly referred to as Bayesian

inference or reasoning. Bayesian inference plays an important and natural role in daily life, since our beliefs and opinions relating to uncertainty change as we come

into contact with relevant information. Since the real world truths of the events of a crime are inherently uncertain, Bayesian inference provides the probabilistic model

for making judgements in criminal trials. On the basis of the combined weight of the evidence presented to the court, the trier-of-fact assesses the likelihood of the

defendant’s innocence or guilt (Good 1991: 89-90). Since the burden of proof lies with the prosecution, a guilty verdict may only be reached when the likelihood of guilt

assigned by the trier-of-fact is greater than thebeyond reasonable doubtthreshold (i.e. where likelihood of guilt approaches one).

The odds form of Bayes’ theorem as applied to criminal trials is given as: p(Hp) p(Hd) × p(E|Hp) p(E|Hd) = p(Hp|E) p(Hd|E) (2.2)

where Hp is the prosecution proposition (guilty), Hdis the defence proposition (inno-

cent) and E is evidence. The prior odds reflect the trier-of-fact’s assessment of the

probability of the competing propositions before the introduction of (new) evidence. The weight or strength of each piece of evidence is expressed as the ratio ofp(E|Hp)and p(E|Hd)(the LR or Bayes Factor) which modifies the prior odds to establish the poste-

rior odds. The posterior odds concern the “ultimate issue” (Lynch and McNally 2003:

96) of innocence or guilt; an assessment of the probability of the competing propositions given the combined weight of the evidence.

There are a number of advantages to using Bayes’ theorem in criminal trials. The theorem is flexible, allowing the trier-of-fact’s belief in the competing propositions to

be modified as new evidence is introduced. Further, in Bayes’ theorem conditional probability is subjective (Redmayne 2001: 54). Therefore, where a jury is entrusted

with interpreting the evidence, Bayes’ theorem allows each individual to assign different weights to that evidence and thus potentially generate different posterior probabilities.

For forensic evidence, where the trier-of-fact cannot reasonably be expected to interpret the evidence, the expert is responsible for assessing the weight of the evidence and,

following Bayes’ theorem, can do this using the LR. The trier-of-fact can then use this to generate posterior probability.

However, there are a number of issues with the practical application of Bayes’ theorem in criminal trials. The first is the appropriate definition of the prior odds, to reflect

the initial assumption that the suspect is innocent until proven guilty. Cohen (1982) highlights that the presumption of innocence requires the prior probability to be zero,

meaning posterior probability would also necessarily be zero, irrespective of the evi- dence. It is preferable, therefore, to think about the prior odds in terms of theisland

problem(Aitken and Taroni 2004: 117-118). A crime is committed on an island with a population N, of which the suspect is a member. Without any evidence, each member of

the population is assumed to be equally likely to have committed the crime. Robertson and Vignaux (1995b) argue that the prior odds can then be thought of as the ratio of the

the probability of choosing any other member of the population at random (p(Hd)): 1 N N−1 N = 1 N −1 (2.3)

Secondly, the assessment of posterior probability requires an arbitrary division between

innocence and guilt. p(H|E)is therefore necessarily susceptible to thecliff-edgeeffect since the trier-of-fact needs to make a categorical decision as to whether the defendant

is guilty beyond a reasonable doubt (Evett 1991: 12). It is not clear how the threshold for determining innocence and guilt is determined by juries. Finally, the formal quan-

tification of Bayes’ theorem in criminal trials (i.e. assigning numerical values to each piece of evidence to generate an overall probability of innocence and guilt) has largely

been rejected by the courts in England and Wales (see §2.1.4.5.