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An introduction to Bayesian statistics in forensic problems

Thomas Bayes
Thomas Bayes died in 1761 at the age of 59. He was a Nonconformist minister and a mathematician, best known for his posthumously published "Essay Towards Solving a Problem in the Doctrine of Chances" (1763). biography

In most cases, forensic evidence doesn't make it possible to completely prove or falsify an hypothesis. Things are rarely black or white. How do you deal with such uncertainty in an optimal way? How can you apply logic when your knowledge is incomplete, using all the information you do have, and nothing more? To know how much information the evidence really contains you'll have to enter the realm of probabilities.
In this section we will give an introduction to Bayesian statistics. This introduction is aimed at a general audience, and will be at a very basic mathematical level. Even so, we will derive Bayes theorem, and we'll see how the Bayesian approach clarifies many things, such as:

  • the evidentiary value of an observation
  • the logic of interpreting and concluding (inference)
  • the role of the expert and other players in a case

The Bayesian approach

  • can be seen as an extension of common sense
  • deals with inverse probability, because it involves inferring backwards from effects to causes
  • uses probability to describe knowledge or lack thereof, rather than seeing probability as the outcome of repeated experiments
  • can use subjective probability or "degree of reasonable belief", or "plausibility"

But lets start with a medical example.

A medical example

Mr. X has been tested positive for some disease, and asks his doctor what the chances are that he really has the disease.
Here is what's known:

  • 1 out of 100 people have this disease.
  • 80 out of 100 people with the disease will get tested positive (the detection efficiency of the test is 80%).
  • 10 out of 100 people without Bayes' disease will still get tested positive (the false alarm rate with this test is 10%).

Before we continue to calculate the probability that Mr. X really has the disease, ask yourself whether your feeling tells you it's rather unlikely, likely, or very likely that he in fact has the disease. Actually, feel free to estimate the probability as a percentage.

So what is the chance that he really has the disease?

The question can be rephrased (without changing it): If 1000 people are tested, what fraction of people with positive test results will actually have the disease?
There are two groups with positive results:

  • The ones that have the disease and test positive (true positives): 10·80%=8
  • The ones that don't have it but still test positive (false positives): 990·10%=99

So the fraction of people with positive test results that actually has the disease is 8/(8+99)=7.5% which was lower than Mr. X had expected! If Mr. X were from a country where 500 of every 1000 people had the disease, the same result of the same test would have led to a probability of 400/(400+50)=89% of him actually having the disease.

Believe it or not, but you have just applied Bayes' theorem.