Evidence under Bayes theorem
Encyclopedia
Bayes' theorem provides a way of updating the probability of an event in the light of new information. In the evidence law context, for example, it could be used as a way of updating the probability that a genetic sample found at the scene of the crime came from the defendant in light of a genetic test showing the frequency of the genetic sample (blood,hair, etc) in the general population.
scholars, the study of evidence in recent decades has become broadly interdisciplinary, incorporating insights from psychology
, feminism
, economics
, and probability theory
. One area of particular interest and controversy has been Bayes' theorem
.
Bayes' theorem is an elementary proposition of probability theory
. It provides a way of updating, in light of new information, one’s prior probability that a proposition is true. Evidence scholars have been interested in its application to their field, either to study the value of rules of evidence
, or to help determine facts at trial
.
Suppose, for example, that the proposition to be proven is that defendant was the source of a hair found at the crime scene. Before learning that the hair was a genetic match for the defendant’s hair, the factfinder believes that the odds are 2 to 1 that the defendant was the source of the hair. If she used Bayes’ theorem, she could multiply those prior odds by a “likelihood ratio” in order to update her odds after learning that the hair matched the defendant’s hair. The likelihood ratio is a statistic
derived by comparing the odds that the evidence (expert testimony of a match) would be found if the defendant was the source with the odds that it would be found if defendant was not the source. If it is ten times more likely that the testimony of a match would occur if the defendant was the source than if not, then the factfinder should multiply her prior odds by ten, giving posterior odds of 20 to one.
Bayesian skeptics have objected to this use of Bayes’ theorem in litigation on a variety of grounds. These run from jury confusion and computational complexity to the assertion that standard probability theory is not a normatively satisfactory basis for adjudication of rights.
Bayesian enthusiasts have replied on two fronts. First, they have said that whatever its value in litigation, Bayes' theorem is valuable in studying evidence rules. For example, it can be used to model relevance. It teaches that the relevance of evidence that a proposition is true depends on how much the evidence changes the prior odds, and that how much it changes the prior odds depends on how likely the evidence would be found (or not) if the proposition were true. These basic insights are also useful in studying individual evidence rules, such as the rule allowing witnesses to be impeached with prior convictions.
Second, they have said that it is practical to use Bayes' theorem
in a limited set of circumstances in litigation (such as integrating genetic match evidence with other evidence), and that assertions that probability theory
is inappropriate for judicial determinations are nonsensical or inconsistent.
Some observers believe that in recent years (i) the debate about probabilities has become stagnant, (ii) the protagonists in the probabilities debate have been talking past each other
, (iii) not much is happening at the high-theory level, and (iv) the most interesting work is in the empirical
study of the efficacy of instructions on Bayes’ theorem in improving jury accuracy. However, it is possible that this skepticism
about the probabilities debate in law rests on observations of the arguments made by familiar protagonists in the legal academy. In fields outside of law, work on formal theories relating to uncertainty continues unabated. One important development has been the work on "soft computing
" such as has been carried on, for example, at Berkeley under Lotfi Zadeh's BISC (Berkeley Initiative in Soft Computing). Another example is the increasing amount of work, by people both in and outside law, on "argumentation" theory. Also, work on Bayes nets continues. Some of this work is beginning to filter into legal circles. See, for example, the many papers on formal approaches to uncertainty (including Bayesian approaches) in the Oxford journal: Law, Probability and Risk http://lpr.oxfordjournals.org/. In retrospect it may appear that the discussion about formal argument about factual uncertainty in law was just beginning in the last quarter of the twentieth century.
can be applied.
Explanation
Among evidenceEvidence (law)
The law of evidence encompasses the rules and legal principles that govern the proof of facts in a legal proceeding. These rules determine what evidence can be considered by the trier of fact in reaching its decision and, sometimes, the weight that may be given to that evidence...
scholars, the study of evidence in recent decades has become broadly interdisciplinary, incorporating insights from psychology
Psychology
Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...
, feminism
Feminism
Feminism is a collection of movements aimed at defining, establishing, and defending equal political, economic, and social rights and equal opportunities for women. Its concepts overlap with those of women's rights...
, economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, and probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
. One area of particular interest and controversy has been Bayes' theorem
Bayes' theorem
In probability theory and applications, Bayes' theorem relates the conditional probabilities P and P. It is commonly used in science and engineering. The theorem is named for Thomas Bayes ....
.
Bayes' theorem is an elementary proposition of probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
. It provides a way of updating, in light of new information, one’s prior probability that a proposition is true. Evidence scholars have been interested in its application to their field, either to study the value of rules of evidence
Rules of evidence
Rules of evidence govern whether, when, how, and for what purpose, proof of a legal case may be placed before a trier of fact for consideration....
, or to help determine facts at trial
Trial (law)
In law, a trial is when parties to a dispute come together to present information in a tribunal, a formal setting with the authority to adjudicate claims or disputes. One form of tribunal is a court...
.
Suppose, for example, that the proposition to be proven is that defendant was the source of a hair found at the crime scene. Before learning that the hair was a genetic match for the defendant’s hair, the factfinder believes that the odds are 2 to 1 that the defendant was the source of the hair. If she used Bayes’ theorem, she could multiply those prior odds by a “likelihood ratio” in order to update her odds after learning that the hair matched the defendant’s hair. The likelihood ratio is a statistic
Statistic
A statistic is a single measure of some attribute of a sample . It is calculated by applying a function to the values of the items comprising the sample which are known together as a set of data.More formally, statistical theory defines a statistic as a function of a sample where the function...
derived by comparing the odds that the evidence (expert testimony of a match) would be found if the defendant was the source with the odds that it would be found if defendant was not the source. If it is ten times more likely that the testimony of a match would occur if the defendant was the source than if not, then the factfinder should multiply her prior odds by ten, giving posterior odds of 20 to one.
Bayesian skeptics have objected to this use of Bayes’ theorem in litigation on a variety of grounds. These run from jury confusion and computational complexity to the assertion that standard probability theory is not a normatively satisfactory basis for adjudication of rights.
Bayesian enthusiasts have replied on two fronts. First, they have said that whatever its value in litigation, Bayes' theorem is valuable in studying evidence rules. For example, it can be used to model relevance. It teaches that the relevance of evidence that a proposition is true depends on how much the evidence changes the prior odds, and that how much it changes the prior odds depends on how likely the evidence would be found (or not) if the proposition were true. These basic insights are also useful in studying individual evidence rules, such as the rule allowing witnesses to be impeached with prior convictions.
Second, they have said that it is practical to use Bayes' theorem
Bayes' theorem
In probability theory and applications, Bayes' theorem relates the conditional probabilities P and P. It is commonly used in science and engineering. The theorem is named for Thomas Bayes ....
in a limited set of circumstances in litigation (such as integrating genetic match evidence with other evidence), and that assertions that probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
is inappropriate for judicial determinations are nonsensical or inconsistent.
Some observers believe that in recent years (i) the debate about probabilities has become stagnant, (ii) the protagonists in the probabilities debate have been talking past each other
Talking past each other
Talking past each other is an idiomatic expression describing a situation in which people in a discussion proceed at cross purposes. There is a mismatch....
, (iii) not much is happening at the high-theory level, and (iv) the most interesting work is in the empirical
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
study of the efficacy of instructions on Bayes’ theorem in improving jury accuracy. However, it is possible that this skepticism
Skepticism
Skepticism has many definitions, but generally refers to any questioning attitude towards knowledge, facts, or opinions/beliefs stated as facts, or doubt regarding claims that are taken for granted elsewhere...
about the probabilities debate in law rests on observations of the arguments made by familiar protagonists in the legal academy. In fields outside of law, work on formal theories relating to uncertainty continues unabated. One important development has been the work on "soft computing
Soft computing
Soft computing is a term applied to a field within computer science which is characterized by the use of inexact solutions to computationally-hard tasks such as the solution of NP-complete problems, for which an exact solution cannot be derived in polynomial time.-Introduction:Soft Computing became...
" such as has been carried on, for example, at Berkeley under Lotfi Zadeh's BISC (Berkeley Initiative in Soft Computing). Another example is the increasing amount of work, by people both in and outside law, on "argumentation" theory. Also, work on Bayes nets continues. Some of this work is beginning to filter into legal circles. See, for example, the many papers on formal approaches to uncertainty (including Bayesian approaches) in the Oxford journal: Law, Probability and Risk http://lpr.oxfordjournals.org/. In retrospect it may appear that the discussion about formal argument about factual uncertainty in law was just beginning in the last quarter of the twentieth century.
Examples
There are some famous cases were Bayes' theoremBayes' theorem
In probability theory and applications, Bayes' theorem relates the conditional probabilities P and P. It is commonly used in science and engineering. The theorem is named for Thomas Bayes ....
can be applied.
- In the medical examples, a comparison is made between the evidence of cancerCancerCancer , known medically as a malignant neoplasm, is a large group of different diseases, all involving unregulated cell growth. In cancer, cells divide and grow uncontrollably, forming malignant tumors, and invade nearby parts of the body. The cancer may also spread to more distant parts of the...
suggested by mammograms (5% show positive) versus the general risk of having cancer (1% in general): the ratio is 1:5, or 20% risk, of having breast cancer when a mammogram shows a positive result. - A court case which argued the probabilities, with DNA evidence, is R v AdamsR v AdamsR v Adams [1996] 2 Cr App R 467, [1996] Crim LR 898, CA and R v Adams [1998] 1 Cr App R 377, The Times, 3 November 1997, CA, are rulings that ousted explicit Bayesian statistics from the reasoning admissible before a jury in DNA cases.-Facts:...
. - In cases where background noise and transfer occur (fibers, glass, gunshot residueGunshot residueGunshot residue is principally composed of burnt and unburnt particles from the explosive primer, the propellant, as well as components from the bullet, the cartridge case and the firearm used...
)