by Bayesianism — confusing degree of confirmation with probability If we identify degree of corroboration or confirmation with probability, we should be forced to adopt a number of highly paradoxical views, among them the following clearly self-contradictory assertion: “There are cases in which x is strongly supported by z and y is strongly undermined by z while, at the same time, x is confirmed by z to a lesser degree than is y.” Consider the next throw with a homogeneous die. Let x be the statement ‘six will turn up’; let y be its negation, that is to say, let y = x; and let z be the information ‘an even number will turn up’. We have the following absolute probabilities: p(x)=l/6; p(y) = 5/6; p(z) = 1/2. Moreover, we have the following relative probabilities: p(x, z) = 1/3; p(y, z) = 2/3. We see that x is supported by the information z, for z raises the probability of x from 1/6 to 2/6 = 1/3. We also see that y is undermined by z, for z lowers the probability of y by the same amount from 5/6 to 4/6 = 2/3. Nevertheless, we have p(x, z) < p(y, z) … A report of the result of testing a theory can be summed up by an appraisal. This can take the form of assigning some degree of corroboration to the theory.
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Bayesianism — confusing degree of confirmation with probability
If we identify degree of corroboration or confirmation with probability, we should be forced to adopt a number of highly paradoxical views, among them the following clearly self-contradictory assertion:
“There are cases in which x is strongly supported by z and y is strongly undermined by z while, at the same time, x is confirmed by z to a lesser degree than is y.”
Consider the next throw with a homogeneous die. Let x be the statement ‘six will turn up’; let y be its negation, that is to say, let y = x; and let z be the information ‘an even number will turn up’.
We have the following absolute probabilities:
p(x)=l/6; p(y) = 5/6; p(z) = 1/2.
Moreover, we have the following relative probabilities:
p(x, z) = 1/3; p(y, z) = 2/3.
We see that x is supported by the information z, for z raises the probability of x from 1/6 to 2/6 = 1/3. We also see that y is undermined by z, for z lowers the probability of y by the same amount from 5/6 to 4/6 = 2/3. Nevertheless, we have p(x, z) < p(y, z) …
A report of the result of testing a theory can be summed up by an appraisal. This can take the form of assigning some degree of corroboration to the theory. But it can never take the form of assigning to it a degree of probability; for the probability of a statement (given some test statements) simply does not express an appraisal of the severity of the tests a theory has passed, or of the manner in which it has passed these tests. The main reason for this is that the content of a theory — which is the same as its improbability — determines its testability and its corroborability.
Although Bayesians think otherwise, to me there’s nothing magical about Bayes’ theorem. The important thing in science is for you to have strong evidence. If your evidence is strong, then applying Bayesian probability calculus is rather unproblematic. Otherwise — garbage in, garbage out. Applying Bayesian probability calculus to subjective beliefs founded on weak evidence is not a recipe for scientific akribi and progress.
Neoclassical economics nowadays usually assumes that agents that have to make choices under conditions of uncertainty behave according to Bayesian rules — that is, they maximize expected utility with respect to some subjective probability measure that is continually updated according to Bayes’ theorem.
Bayesianism reduces questions of rationality to questions of internal consistency (coherence) of beliefs, but – even granted this questionable reductionism – do rational agents really have to be Bayesian? As I have been arguing repeatedly over the years, there is no strong warrant for believing so.
In many of the situations that are relevant to economics one could argue that there is simply not enough of adequate and relevant information to ground beliefs of a probabilistic kind, and that in those situations it is not really possible, in any relevant way, to represent an individual’s beliefs in a single probability measure.
Bayesianism cannot distinguish between symmetry-based probabilities from information and symmetry-based probabilities from an absence of information. In these kinds of situations most of us would rather say that it is simply irrational to be a Bayesian and better instead to admit that we “simply do not know” or that we feel ambiguous and undecided. Arbitrary an ungrounded probability claims are more irrational than being undecided in face of genuine uncertainty, so if there is not sufficient information to ground a probability distribution it is better to acknowledge that simpliciter, rather than pretending to possess a certitude that we simply do not possess.
So, why then are so many scientists nowadays so fond of Bayesianism? I guess one strong reason is that Bayes’ theorem gives them a seemingly fast, simple and rigorous answer to their problems and hypotheses. But, as already Popper showed back in the 1950’s, the Bayesian probability (likelihood) version of confirmation theory is “absurd on both formal and intuitive grounds: it leads to self-contradiction.”