contact@onbelief.orgCertainty, Uncertainty and Bayes Theorem
Insofar as we are rational in our beliefs, the intensity of belief will tend to correspond to the firmness of the available evidence. Insofar as we are rational, we will drop a belief when we have tried in vain to find evidence for it."
- Quine and Ullian, The Web of Belief Quoted from Evidence"I cannot give any scientist of any age better advice than this: the intensity of a conviction that a hypothesis is true has no bearing on whether it is true. The importance of the strength of our conviction is only to provide a proportionately strong incentive to find out if the hypothesis will stand up to critical examination".
- Peter B. Medwar, zoologist and immunologist Advice to a Young Scientist, 1979 Taken from www.editoreric.comProbably True
We do not have to be 100% convinced about the truth of an idea for it to be useful. Uncertainty is helpful for it opens us up to the possibility of considering new ideas. In short, doubt is healthy. There is nothing improper about thinking that an idea is probably true or probably false. If we are to really understand the concept of 'probably true' we should learn about the concept of 'probability' from mathematicians and statisticians who are professional thinkers in these matters. The fact that probability is best thought of in terms of numbers and ratios of numbers tend to put many people off. This need not be the case for we often think in percentages. Just think of certainty of the truth of ideas as 100% strength of belief and certainty of falseness as 0%. Using a different although equivalent scale the very famous 17th century German mathematician and philosopher Gottfried Leibniz realised that we could express the range of probability in the range between 1 and 0 as the extremes (on a different scale that would be 100% and 0%). A very strongly held belief might then be treated as having a value of 99% or 0.99. A belief that was almost certainly false might have a value of 1% or 0.01. In a technical sense we are then using subjective probability as a philosophical device. Rather than simply accept or reject an idea (or hypothesis) we can use a variety of complementary means to reach a probabilistic view of whether or not a belief is right or wrong. For the dogmatist and concrete thinker this can be an uncomfortable experience. However modern medicine and other intellectual disciplines teach us that in diagnostic thinking, in its widest sense, it is beneficial and even necessary to be open minded about the possibility of uncertainty.
Learning from the Professionals
If we then learn how scientists, engineers and medical researchers are advised by statisticians when they are testing matters of scientific, technical and medical belief we can benefit in our lives. For example, how convinced does a doctor have to be before deciding to routinely use a new medicine on patients? The answer might be 95 or 99% certain. How convinced do you, the patient, have to be that a medicine will be beneficial and not have terrible side effects before you as the patient take it. If you did not have a life-threatening cancer, the answer might be that you would take a medicine even if there was a substantial chance that it was not effective provided you were strongly convinced that it would not cause you terrible problems. In other words you might have different strengths of belief in the possible benefits and possible harm and act accordingly. The problem in that situation is that you might be better taking a medicine in which there was good reason to have a higher strength of belief that it will work in your case rather than just consider the minimisation of harm. The same is probably true for many people in matters of religion. Many possibly say to themselves why not believe the 'truth', with 100% strength of belief, if there is little chance of the idea doing me harm? I argue however that there is little to be gained by certainty. It is not just in matters of science or its medical applications where strength of belief is important. We should be able to take more general lessons from the practice of these subjects in our everyday lives.
It is unfortunate that here in the UK that rainfall predictions in weather forecasts for example are not expressed in terms of probabilities for that would condition us much more into thinking about the value of probability estimates. (To see how probability data can be used in weather forecasting click here>).
Probably True and Bayes Theorem
We will often say to one another that an idea is probably true if we are uncertain. In more formal language we are using epistemic or subjective probability as a belief condition that is qualitatively distinct from the 'facts' of observation. Facts by this definition are possible categories of observation, or descriptive beliefs, as referred to on a previous page. When this type of analysis is applied to matters of belief we are adopting a form of meta-probability to describe how strongly an individual believes an idea to be true or false. (for a more formal description see Bayesian Epistemology at Stanford Encyclopedia of Philosophy).
By a probabilistic approach, the strength of belief is not described in dichotomous terms, as true or false, but as an intermediate pseudo-stochastic probability of state. If I have a strong belief that might be restated as p=0.95 or 95% and a weakly held belief as 0.05 or 5% for example. Indeed such statistical confidence intervals are widely used in modern biological analysis in relation to the probability of random outcomes of tests in research.
A mathematical formulation of this idea was proposed by the 18th century Scottish-educated mathematician and Presbyterian minister Thomas Bayes and has become known as Bayes' theorem. (see also a biography of Thomas Bayes.) His "Essay Towards Solving a Problem in the Doctrine of Chances" was published posthumously in 1763. The non-mathematical statements of Bayes theorem might seem extremely obscure for many people, however they have very profound implications for the way we should view the world. From the wikipedia article we read that the theorem is "The probability of an event A conditional on another event B is generally different from the probability of B conditional on A . However, there is a definite relationship between the two, and Bayes' theorem is the statement of that relationship." From the wikipedia article on conditional probability we learn that "The conditional probability fallacy is the assumption that P ( A | B ) is approximately equal to P ( B | A ), is a mistake often made even by doctors, lawyers, and other highly educated non-statisticians"
For the more technically minded
A very clearly argued statement of Bayes theorem is available at Bayesian Statistics for Dummies. In more concise terms the discussion of coin tossing above can be stated in more formal terms as follows:
" p(H|E,I) = p(H|I)*p(E|H,I)/p(E|I) [Bayes Rule]
The left-hand term, p(H|E,I) , is called the posterior probability, and it gives the probability of the hypothesis H after considering the effect of evidence E in context I. The p(H|I) term is just the prior probability of H given I alone; that is, the belief in H before the evidence E is considered. The term p(E|H,I) is called the likelihood, and it gives the probability of the evidence assuming the hypothesis H and background information I is true. The last term, 1/p(E|I) , is independent of H, and can be regarded as a normalizing or scaling constant. The information I is a conjunction of (at least) all of the other statements relevant to determining p(H|I) and p(E|I) . Note that all of these probabilities are conditional - they specify the degree of our belief in some proposition(s) under the assumption that some other propositions are true. "
[Taken from http://ic.arc.nasa.gov/ic/projects/bayes-group/html/bayes-theorem-long.html]
The term 'true' is reinterpreted here to mean instance or category of observational possibility. "Bayes Theorem can be derived from the Product Rule of probability. The Product Rule is P(A, B/I) = P(A/B, I) * P (B/I) = P(B/A,I) * P(A/I). "
Taken form http://www.mrs.umn.edu/~sungurea/introstat/history/w98/Bayes.html
Conclusion
The aim of this page has been to show that even logically precise thinking based on observations of the world should sometimes, or even often, result in fundamental uncertainty. Although we can learn from systematic academic study of the physical world, it is healthy to be suspicious of the veracity of claims of certain knowledge be they from current practice, scientific textbooks or revered ancient documents. Where possible we should design and apply tests of belief as research scientists and doctors do. If we decide to apply tests of belief we need to understand something of their usefulness. Where such tests of belief are not possible it is better to abandon the belief concerned and replace it with something that is testable. In situations where we cannot apply tests of belief and do not wish to abandon the ideas involved we can at least acknowledge our ignorance and so become less dogmatic and more open to the possibility of changing our ideas.
Online Resources
Probability. A discussion in BBC Radio 4's In Our Time Series between Melvyn Bragg and Marcus du Sautoy, Professor of Mathematics at the University of Oxford Colva Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews and Ian Stewart, Professor of Mathematics at the University of Warwick
Intuitive errors in the assessment of uncertainty a talk by Peter Donnelly, a mathematician, at TED
See parts of the introduction to the online manual for the Microsoft Bayesian Network Editor
Bayesian Statistics for Dummies (Highly recommended for the clarity of explanation)
Cause, Chance and Bayesian Statistics
When Did Bayesian Inference Become Bayesian"? by Stephen E. Fienberg
Bayesian Statistics , by José M. Bernardo
Bayes Euro Master, by Bernardo et al. (Provide a useful survey of applications)
Degrees of Belief: Subjective Probability and Engineering Judgment by Steven G. Vick
Some, especially non-engineers, will find this book incredibly 'dry' and unreadable. However it is really a little masterpiece of Natural Philosophy disguised as a geotechnical engineering text book since it requires us to look behind structured thinking in engineering to question the nature of its probabilistic assumptions. If, like me, you are not an engineer this paperback will also give you a chance to learn something of another area of human thinking. If you also think that philosophy should result in engagement with the world in a way that really counts then you might be sympathetic to the aims of the author. Unfortunately as it is not from a mass market publisher the price is excruciating. However it is full of material that helps to justify its cost even to the non-engineer. Click on the link above to sample it on Google. Ignore the preface.
On the Nature of Belief
www.onbelief.org
Scotland, 12th October 2007 and thereafter
Copyright 2007 onwards
