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- Subject: Education - Mathematics
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Bayes Probability

Can Bayes Confirmation Theory Give an Adequate Explanation for Confirmation of Scientific Theories?

Theorizing in science is a complex and time-consuming undertaking. The theorist uses collected evidence from some means of scientific inquiry to project a generalized case. However, there is a difficulty with this process. There is some amount of probability that the theory will be wrong. Even if this is not a harmful outcome, it is difficult for the theorist to overcome in their professional lives. So, researchers want to understand the probabilities involved in the success of their theories.

Bayes theorem discusses the probability that an event will occur, which in the use proscribed for this research is whether a theory is correct or not. Bayes looked at two different events one of which can be used to add to the probability that the other is correct. For example, say that a statement (any given statement) has a fifty percent chance of being true; that is not enough to build a theory on. Therefore, to increase the probability that the statement is true something else is added, a known quantity that relates to the previous statement, that gives a greater probability that the first statement (or thought) was true. As an illustration, one person (A) tells another (B) that they have just met with a friend and that they had lunch together. B hypothesizes that the friend was a woman, but there is only a fifty percent chance that this is so. More information is needed to ensure that this supposition is correct. In the course of the conversation person A says that the friend left to have a spa day. The chance that this the friend was a woman increases because B. knows that seventy-five percent of people who go to spas are women. Using Bayes theorem the second piece of evidence help to greater confirm the first supposition. This does not mean that B. is now completely convinced that the person was a woman because twenty-five percent of people who go to spas are men, but there is now a sixty seven percent chance that the individual was a female.

This logic can be used to investigate the probability that a theorem is correct also because the same pattern can be observed in both. Every theorem begins with a hypothesis. This hypothesis has some probability of being correct and some probability of being false. The theorist who is not satisfied with the randomness that is suggested by this probability wants to make ensure the correctness of their findings by using another piece of evidence to confirm or disprove the assumption. This piece of evidence has this power because it is able to increase the probability of the proof or the probability that the hypothesis will have to be abandoned. This paper examines Bayes theory in detail as regards its ability to help predict the correctness of a proposed theory.

Example of Bayes Theorem

Examples of the theory are easy to find in literature because it is one of the primary methods used to determine the probability of a statement being true or false. Simple probabilities such as will a coin turn up heads or tails are not affected by Bayes theorem because there is no data that can be added to predict whether the next throw can be better predicted as a head or a tail (of course given that the coin is not tampered with in some way). Since there are only two possibilities, in every circumstance, it is impossible to give one a higher value than the other. Both have a value of one in the Bayesian context. But, events are generally not as simple as that.

A good example is to try and determine what a number will be given that it is within a set of other numbers; such as the probability that a test subject will be male or female. This is a good example because "physical theories typically predict numerical values."[footnoteRef:1] Assuming that half of the population is male and the other half is female, the prior probability that a subject will be male is 0.50. However, newly obtained data says that said participant is also a cigar smoker. Given 9% of the male participants admittedly smoke cigars while only 2% of the female participants do, it is easy to determine that the probability that the chosen participant is male is approximately 82%.[footnoteRef:2] This shows how the theory works in practice, but it does not explain if it is possible to confirm a theory based on Bayes theorem. [1: PE Meehl, 'Theory-testing in psychology and physics: A methodological paradox', Philosophy of Science, vol. 34, 1967, pp. 103-115.] [2: MF Triola, Bayes theorem, 1997, Retrieved 24 March 2012 from http://faculty.washington.edu/tamre/BayesTheorem.pdf]

Bayes Theorem Challenges

In the lecture about Bayes Theorem several different evidences of its lack of fundamental utility were given. The values of the priors, the question of the zero prior needed for the equation, the problem of old evidence and the fact that scientists do not actually try to confirm theories in a probabilistic way. Bayes theorem is thought to be too simplistic because it needs some fundamental numbers to make it useful, but those numbers are not always available when a theorist is developing a hypothesis.[footnoteRef:3] The fact that so much evidence is needed prior to actually making Bayes theorem work both helps and hurts the predictive utility of the theorem. The problem of the priors is actually one in which the theory can be undone because there is no logical explanation for where the explanatory number came from. As for the zero priors, most probabilities have an initial number associated with them, but there is a chance that the value of H. Or H/E in the theorem could be zero. If this is the case, then no probability can be obtained. It is also difficult to know to which quantities a zero should be assigned. [3: D. Steel, Bayesian confirmation theory and the likelihood principle, 2007, Retrieved 25 March 2012 from https://www.msu.edu/user/steel/Bayes_and_LP.pdf]

The issue of old evidence seems to be the most puzzling of all to scientists. Rosenberg says, 'Scientists who construct hypotheses by intentional "curve fitting" are rightly criticized and their hypotheses are often denied explanatory power on the grounds that they are ad hoc.'[footnoteRef:4] [4: A Rosenberg, Philosophy of science: A contemporary introduction, 2nd edn, Routledge, New York, 2005, p. 137.]

This means that the effectiveness of the theory is brought into question if old (as the quote says ad hoc) evidence is used as a proof of said hypothesis. However, he goes on to say that scientists run into a problem with this criticism when trying to;

'distinguish cases like the confirmation of Newton's and Einstein's theories by old evidence from cases in which old evidence does not confirm a hypothesis because it was accommodated to the old evidence.'[footnoteRef:5] [5: A Rosenberg, Philosophy of science: A contemporary introduction, 2nd edn, Routledge, New York, 2005, p. 137.]

It appears that it is difficult for the critics to tell the difference between a theory that is actually confirmed by what is considered an 'old' piece of evidence, and one that was built around some sort of old evidence. This distinction is crucial when it comes to dismissing the theory outright.

The final challenge is that probabilistic theory confirmation is untenable with the method that is generally used to either prove or disprove a theorem. In empirical theory building, the researcher tries to gather as much evidence as possible with regard to the question, rather than trying to find confirmatory evidence which will strengthen the theory from the outside. The confirmatory evidence comes from the veracity of the initial hypothesis based on a great deal of empirical (observational) evidence.

Should Bayes Theory be Used

Neither the positive or negative aspects of the theory can be said to give a formal conclusion as to whether a Bayes theory should be used or not for confirmation of a theory. The only thing that can be conclusively proven from the fact that there are pros and cons to the theory is that some people believe that it is efficacious and some do not. Of course, in scientific inquiry if something is not useful part of the time it is better to shelve its use all of the time. This reduces the possibility that the user will be wrong in its use. However, there are definite instances when the theory has proven itself to be a useful tool because of the simplicity of its tenets. The theory helps theorists avoid the threat of underdetermination; that is, a problem with many hypothesis in fields such as theoretical physics.[footnoteRef:6] The fact that Bayes theory gives some concrete probability, in many but not all cases, is one of the factors in its favor. [6: R. Dawd, 'Scientific prediction and the undetermination of scientific theory building', PhilSci Archive, 2008, Retrieved 24 March 2012 from http://philsci-archive.pitt.edu/4008/]

In an article for the journal Philosophy of…[continue]

"Bayes_and_LP.pdf"

"Scientific-Prediction-and-the-Underdetermination-of-Scientific-Theory-Building---Philsci-Archive"

"Scientific-Prediction-and-the-Underdetermination-of-Scientific-Theory-Building---Philsci-Archive"

"The-Logical-Nature-of-Scientific-Confirmation-of-Theories"

"Bayes_and_LP.pdf"

"BayesTheorem.pdf"

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