Question

Discuss an example of applying probability to investigating burglaries . What are some ways you could measure or express that probability using the Basic Law of Probability.

Answer 1

We are now beginning to glimpse the power and variety of the potential applications of statistical inference in the administration of criminal justice. It must be stressed, however, that statistical inferences are ultimately only as good as their underlying data, which in turn depends upon

(1) the appropriateness of the research design (including sampling methodology)

(2) the integrity of the processes and procedures employed in data collection. Conversely, if data-collection was sloppy and incomplete or samples were poorly chosen, the validity of the inferences drawn from statistical data may be seriously compromised.

When statistics are being presented and interpreted in forensic contexts (or for that matter, in any other context), there are always two principal dimensions of analysis to be borne in mind

  (i) Research methodology and data collection: Do statistical data faithfully represent and reliably summarise the underlying phenomena of interest? Do they accurately describe relevant features of the empirical world?

(ii) The (epistemic) logic of statistical inference: Do statistical data robustly support the inference(s) which they are said to warrant? Is it appropriate to rely on particular inferential conclusions derived from the data?

2. Basic Concepts of Probabilistic Inference and Evidence

The starting point for thinking about information which is statistical or presented in the form of a probability is exactly the same as the starting point for interpreting evidence of any kind. The essential issue is: what does the evidence mean? The meaning of evidence is a function of the purpose(s) for which it was adduced in the proceedings, which in turn are defined by the issues in the case

These key concepts include:

(a) (absolute and relative) frequencies;

(b) likelihood of the evidence;

(c) the likelihood ratio;

(d) base rates for general issues (prior probabilities);

(e) posterior probabilities;

(f) Bayes’ Theorem; and

(g) independence.

The likelihood ratio

As previously stated (and as its name transparently implies), the “likelihood ratio” is an expression of the ratio of two relevant likelihoods (or probabilities).

Here is one example (with emphasis) that might be encountered in criminal proceedings

: “The blood-staining on the jacket of the defendant is approximately ten times more likely to be seen if the wearer of the jacket had, rather than had not, hit the victim.”

Posterior probabilities

All probabilities are predicated (or “conditioned”) on specified assumptions. This is merely another way of expressing the inherent conditionality of probability as a species of reasoning under uncertainty. Thus, for example, one might calculate the probability that the accused is guilty, given the evidence that has been presented in the trial – in mathematical notation, p(G|E). Whereas base rates for general variables inform prior probabilities, conditional probabilities conditioned on case-specific events or evidence can be described as posterior probabilities – such as the probability that the accused is guilty after (posterior to) having heard all the evidence. The ultimate posterior probability, of guilt or innocence and their corresponding legal verdicts, is always a question for the factfinder in English and Scottish criminal proceedings.

Bayes’ Theorem

Bayes’ Theorem is a mathematical formula that can be applied to update probabilities of issues in the light of new evidence. One begins with a prior probability of an issue and some pertinent item of evidence. Bayes’ Theorem calculates a posterior probability for the issue, conditioned on the combined value of the prior probability and the likelihood ratio for the evidence. This posterior probability can then be treated as a new prior probability to which a further additional piece of evidence can be added, and a new posterior probability calculated (now taking account of the original prior probability and the likelihood ratios for both pieces of evidence

posterior odds = likelihood ratio × prior odds