4.16 Probabilities for propositions
We have been careful to refer to probabilities for evidence conditioned on propositions. The probative value of evidence is related to the probabilities for evidence assuming competing propositions and then comparing the probabilities with each other. We explicitly did not state probabilities for propositions, and particularly not probabilities for competing propositions conditioned on evidence.
This is because it is generally not within the role of the expert witness (in the UK) to comment on probabilities for propositions. The expert witness uses their scientific expertise to comment on the evidence in light of proposed versions of events. It is then within the fact finder’s remit to use this information to determine the truth of propositions. In other words, it is up to the fact finder to reason about their own uncertainty for the propositions in light of expert (and other) evidence that is put before the court. Whether the fact finder decides to reason about this uncertainty using probability theory or otherwise is a matter for them.
One well-known error of reasoning using conditional probabilities is the prosecutor’s fallacy. This occurs when the probability of evidence conditioned on a proposition is mistaken for the probability of a proposition conditioned on evidence. Since the conditioning information in these probabilities is incorrectly reversed, it is also known as illegitimately transposing the conditional. The term was coined by Thompson and Schumann (1987), and despite its name, it is not only prosecutors who are at risk of making this error.
A concrete example of the prosecutor’s fallacy can be shown using the disease testing example from Section 3.17. In this example, there was a diagnostic test for a disease with base rate of 0.01 in a population of 10,000 people. The test had a sensitivity and specificity of 0.95. There are multiple possible opportunities for the prosecutor’s fallacy to be made here but we choose the one that confuses the following two probabilities:
- the probability of a positive test assuming that the person tested truly does have the disease,
- the probability of an individual having the disease assuming that they test positive.
Probability 1 above is the definition of the sensitivity of the test, which is 0.95. In the example, we reasoned probability 2 to be roughly 0.28. In practice, probability 2 is the one that is useful for determining what healthcare actions should be taken, e.g. should the person be admitted to hospital, told to isolate, sent for further testing etc. depending on the disease. Mistaking these two probabilities means that the interpretation of a positive test changes from the correct interpretation of “even after testing positive, an individual is still unlikely to have the disease” to the incorrect interpretation of “almost everyone who tests positive has the disease.” This incorrect interpretation highly overestimates the meaning of a positive test and would result in many people who do not have the disease facing treatment as if they did have the disease.
When forensic evidence is the subject of the interpretation, then one consequence of the prosecutor’s fallacy is to misinterpret the probative value of the evidence. When the expert or legal counsel commits the prosecutor’s fallacy, then there are the additional consequences of encroaching on the role of the fact finder by stating a probability for a proposition, and also of misleading the court about the value of the evidence. Unambiguously stating propositions, background information, and the scientific evidence that is subject the probability helps to avoid this mistake.