Bayes’ Theorem: A Troubled History and Contested Present
Bayes’ Theorem is one of the most powerful formulas used in statistics today but that does not mean it is concurrently agreed upon by everyone nor has it always been generally accepted. The article “The Mathematics of Changing Your Mind” describes the theorem’s beginning, growth, contention, and usage. The ideas behind the theorem originated in the 18th century from the mind of Thomas Bayes, a Presbyterian minister; however, the theorem itself was mainly developed by Bayes’ friend Richard Price. After its initial discovery, the theorem faded in and out of the mathematical world until somewhat recently in which it has achieved an incredible revival. As described in the article and in lecture, the theorem is used to calculate the conditional probability of an event usually in light of some newly discovered evidence. We have applied the formula to simple scenarios such as drawing balls from urns or choosing a good restaurant, and in both situations probabilities were given. In these cases, Bayes’ theorem is a definite truth and logical conclusions can be drawn easily from the results. However, the problems with Bayes’ theorem begin to manifest themselves when it comes to real life, complex situations, and this is also where the theorem runs into a fair amount of critics.
When applied to the real world, Bayes’ theorem is entirely based off an initial hypothesis. In order to establish this hypothesis though, one must assign some initial probability to it before they can continue adding new evidence to carry out the formula. As such, many argue that Bayes’ theorem is not as much a definite probability as it is a subjective assumption. The article points out one group in particular, frequentists, who are strong opponents to the theorem. Frequentists argue that the probabilities of past events must be given as near objective numbers based on the repeated occurrence of outcomes as defined by structured experiments and detailed observation. Although frequentists seem to be strict in their opposition to Bayes’ principle, they do provide a valid point that the theorem can produce results which may not always prove to be completely accurate.
While the theorem does have some practical flaws and opponents, it has nonetheless proven its validity in many scenarios throughout history. Some of Bayes’ successes include the search for nuclear weapons, the decryption of code during World War II, the improvement of computer imaging, and the determining of false positive rates of illnesses and other bodily disorders. In contest to its critics, this goes to show that given correct initial probabilities Bayes’ theorem can produce tremendous results.
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