The Reverend and the Russian
The value of a good guess can hardly be overestimated. Many have tried to make a science of guessing. The fields of probability and statistics are the results of these attempts. Two men’s ideas in these areas have gained currency in recent years. The two are the Reverend Thomas Bayes and Andrey Andreyevich Markov. As with all ideas that gain popularity, there is a danger that these will become dogma, or at least second-nature, and replace critical thinking. Their approaches are similar in that they both take an axiomatic view of likelihood and its measurement. It’s important to know their assumptions to avoid mis-applications.
The British Reverend’s axiom is also known as “Bayes’ Theorem” or “Bayes’ Rule”. It can be considered a theorem because it can be derived from other “laws” or rules. It says that new observations should affect our probability estimates for guesses in a simple multiplicative way: the probability that a guess is true after a new observation is the simple product of the likelihood of the observation, with the guess taken for granted, and the ratio of the prior probabilities of the guess and the observation. This provides a very simple way to “update our state of knowledge” based on new observations. As you can guess, it is extremely difficult, maybe even impossible, to verify this general rule empirically in any reasonable way. Is the universe really such that uncertainties propagate in this simple way? There is evidence that it may be, but that’s a far cry from taking the rule for granted.
A Markov Model of Congestive Heart Failure Treatment (National Library of Medicine)
The Russian mathematician’s axiom is easier to state. It simply says that the probability of one set of observations immediately following another set is independent of history. At first blush this seems counterintuitive. After all, how can we ignore the past and hope to make a good guess? Markov, of course, never claimed that we could. His axiom just generalizes properties of simple chance events: the probability of winning the lottery does not depend on what the winning numbers have been in the past (in the absence of cheating). His generalization simply says that the right set of observations gives a snapshot that summarizes the history that led to it. In the case of the honest lotto numbers, observations of previous drawings are irrelevant. Markov’s theory then builds on this axiom to consider chains and networks of these history-independent observations (or “states”); systems like these are said to have the “Markov Property.” One can then find interesting conclusions about these Markov chains and networks in the aggregate. It is important to keep in mind that the Markov property is a hypothesis and should not be used indiscriminately because it simplifies the math, or, worse yet, just because it sounds sophisticated.
Again, what does this all have to do with you and me? Calculations based on Bayes’ and Markov’s theories are fairly common now. Some of these calculations are in very critical policy and engineering areas. Just intoning their names gives a claim or a theory a certain level of respectability. We should guard against accepting such derivative claims and theories uncritically and consider whether the underlying assumptions are really applicable, at least in our own work.
