Probability and Frequency- what to use?

As I write in my book How the Brain Thinks, ‘We float through life on a sea of uncertainty where emotion rules the waves; fear and anxiety are rife, and happiness is elusive and ephemeral. The God of Small Things constantly gives and constantly takes away.’

One way to reduce our anxiety and get closer the truth is to use the numbers of probability and frequency.

Probability is the likelihood of an event happening, and the percentage chance of an event happening within a sample space, expressed as a number between zero and one. We use phrases such as ‘there’s a probability of 0.7 that event A will happen, given certain conditions’.
We can also use the term ‘risk’ when we talk about probabilities and we use percentages when describing it — we say ‘there is a 70% risk of event A happening given certain conditions’.

Frequency is the rate at which something occurs over a particular period of time or in a given sample.

Conditional probability can be defined as ‘the likelihood that something is true, given a piece of information’ — if event B happens, what is the likelihood that event A is present? It is all about interpreting what we can see, trying to find the truth or predict the future of that we can’t see. What can I infer from this information? What is the likelihood that what I see before me really does mean that this or that is going to happen? What is this person really like, based on what I can see? We try and evaluate the probability of uncertain events happening so we can make decisions about what to do.

Note that this is not the prior probability (the prevalence of the disease in the practice population) but the conditional probability (the probability of the disease, given a patient’s symptoms). A synonym for conditional probability is predictive value: the predictive value of a symptom for a disease. Predictive value varies with prevalence.

The original theorem describing conditional probability is Bayes’ theorem, named after Thomas Bayes (c.1702–17 April 1761) , a Presbyterian minister who was also a mathematician. It captures uncertainty in terms of probability: Bayes’s theorem, or rule, is a device for rationally updating prior beliefs and uncertainties based on observed evidence. Reverend Bayes set out his ideas in “An Essay Toward Solving a Problem in the Doctrine of Chances,” published posthumously in 1763; it was refined by the preacher and mathematician Richard Price and included Bayes’s theorem. A couple of centuries later, Bayesian frameworks and methods, powered by computation, are at the heart of various models in epidemiology and other scientific fields

It is an equation about conditional probability where the probability of an event occurring is conditional on the probability of another related event.

In medicine it is used in situations where we want to know what the probability of diagnosis is when, say, we receive a test result. If we label
the test result as event B, we estimate the probability that an event A, such as the diagnosis, has occurred, based on the result of the test. We can then assess what we think the patient has, and can decide to start treatment or seek more evidence. It is about extrapolating the true meaning out of a piece of information within a particluar context.

One of the ways of stating the formula is:

P(A\B)=P(B\A) x P(A)/ P(B)

It expresses the probability P of some event A happening given the occurrence of another event B.

Bayes’ theorem can be used to calculate post-test probabilities (or odds) based on pretest probabilities and the sensitivity and specificity of a test. Sensitivity is a measure of how many people with the problem will test positive and specificity is a measure of how many people who test negative
won’t have the disorder. All are measures of how effective a symptom or sign or test is in identifying a disease and in discriminating between it and other diseases or an otherwise normal state of health.

Sensitivities and specificities are not constants for a particular event. They are context specific. Consider the example of urine testing for pregnancy. It is not until two to three weeks after conception, or four to five weeks after the last period that most urine tests become positive in a normal pregnancy. The test has different sensitivity and specificity depending on the gestation and health of the pregnancy.

The important thing to realise is that probabilities give you guesstimates only about what a piece of information means. Few examples come near a probability of one or zero. Probability can be actual (objective) or perceived (subjective). Objective probability is produced by external measurement.
Subjective probability is our own innate best guess.

In How to think like an Epidemiologist in the New York Times, science writer Siobhan Roberts discusses using Bayes Theorem.

And in Discussion points for Bayesian inference in Nature Human Behaviour, by Aczel, B., Hoekstra, R., Gelman, A. et al., the various questions that Bayesian analysis can answer, and the various ways in which it can be used are discussed .

Frequency is the rate at which something occurs over a particular period of time or in a given sample. The data can be displayed using a frequency box, and there is an excellent article in Wikipedia about Bayes theorem and comparing probability estimations and frequency estimations using the same data set. It demonstrates graphically that displaying the same data set using a frequency box is much easier to understand at a glance than the complicated ruminations of probabilists. We think in pictures. ‘Zhu and Gigerenzer found in 2006 that whereas 0% of 4th, 5th, and 6th-graders could solve word problems after being taught with formulas, 19%, 39%, and 53% could after being taught with frequency boxes, and that the learning was either thorough or zero.’

And in the complicated world of human decision making and judgment there are simpler models as I mention in How the Brain Thinks – see Reasoning the fast and frugal way: Models of bounded rationality by the frequentists Gerd Gigerenzer and Daniel G Goldstein. They suggest that ‘this result is an existence proof that cognitive mechanisms capable of successful performance in the real world do not need to satisfy the classical norms of rational inference’, i.e probability. We are not ‘naive intuitive statisticians’.

So, are you a frequentist or a probabilist? An ability to understand both and an understanding of how to present the same information in different formats to different audiences is probably all that frequently matters.

Take care.