# Bayesian “No Miracle Argument” and the Priors of Truth

Alai, Mario (2023) Bayesian “No Miracle Argument” and the Priors of Truth. [Preprint]

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## Abstract

According to a quick formulation of the “No Miracle Argument” (NMA), if a hypothesis H predicted a very risky (i.e., improbable) novel prediction np (like the existence and exact astronomic data of Neptune, the existence and exact properties of new chemical elements, the exact magnetic moment of the electron, the exact retard of clocks in motion, etc.), it cannot be false, because there was only a negligible chance to pick a false hypothesis entailing the true and risky prediction.
However, in order to avoid the base-rate fallacy, one must also consider the prior probability of picking a true hypothesis, as prescribed by Bayes’ theorem: if T is the truth of H, and S is the success in finding a hypothesis that entailed np, then the probability of T given S is:

(1) P(T│S)=(P(S│T) ∙P(T) )/(P(S│T)∙P(T)+P(S│¬T)∙P(¬T))

Yet, antirealists object that, because of the empirical underdetermination of theories, there are infinitely many false hypotheses and only a true one compatible with the empirical data. Therefore, P(T) =1/∞, hence in (1) the numerator is 1/∞ ≈ 0, hence, no matter what P(S│T) is, P(T│S) is also =1/∞: no prediction, no matter whether novel, or risky, or not, can ever raise the probability that a hypothesis is true.
This paradoxical result, however, follows only if P(T) =1/∞, for if P(T) is even slightly greater than 1/∞ and np is risky, P(T│S) increases dramatically, and by recursive updating it converges rapidly to 1. Besides, P(T) =1/∞ only if H is picked randomly, but obviously it is not so, because it was conceived on purpose and through the scientific method (SM). Moreover, we know two things: (1) SM is vastly more effective than chance in finding hypotheses entailing novel predictions, because it frequently produces novel scientific predictions so improbable that could be picked by chance only once in hundreds of thousands or millions of trials. (2) SM was actually successful in finding the hypothesis H which licensed this particular novel prediction np.
Now, why and how is SM so effective, and has actually succeeded in predicting np? I claim that this is so because (a) Only by finding true and fruitful hypotheses one can find hypotheses making risky novel predictions, and (b) SM is very effective in finding true and fruitful hypotheses; therefore, since scientists found a hypothesis (H) entailing np, H is very probably true.
Against (a) antirealists have maintained that, instead, novel predictions can be found just by looking for empirically adequate hypotheses (because an empirically adequate hypothesis makes all the predictions of the true one): SM does not aim for the truth, but for empirical adequacy, by postulating unobservable mechanisms which have the same empirical effects as the real ones.
On the contrary, I hold, scientists endeavor to propose only hypotheses which are consistent with the previously well-confirmed ones (see below for further details), and this shows that they aim for the truth. Otherwise, hypotheses incompatible but consistent with the same empirical data as the previously confirmed ones would be equally well accepted by them.
Moreover, there exists no method to pursue empirical adequacy (in particular, novel unexpected predictions) except via truth, because at any time t the unknown empirical data are largely heterogeneous to the known ones, and unrelated to them by whatever is known at t. Therefore, no empirical analogy to previous data or methodological analogy to past theorizing could help to discover such new heterogeneous data. In other words, you cannot find an unobservable mechanism with the same empirical effects as the real one, unless you discover which is the real one, i.e. unless you find the truth.
Against (b) antirealists object that we don’t know whether SM is actually effective in finding true hypotheses, because we cannot check the truth-value of scientific claims on unobservable systems by directly comparing them with those systems: all our beliefs about those systems are based only on our theories, or on the reports of instruments whose reliability is warranted just by those very theories.
On the contrary, we get information about unobservable systems through instruments whose reliability can be tested by unaided observation, and then extended to the unobservable realm by pure inductive extrapolation, like the optical microscope. Moreover, we can discover the properties of certain unobservable entities just by unaided observation and mathematics, as when Perrin measured the size of molecules, and Millikan the charge of the electron. Again, we can check certain hypotheses trough instruments whose reliability is attested by totally independent theories. Moreover, many more hypotheses are probably true, because they are not just consistent with the available empirical data, but also with the instrumentally confirmed hypotheses. Granted, some of these hypotheses have subsequently been found to be partly false, but partial truth is often enough to derive true novel predictions. Thus, on the basis of criteria acceptable to empiricists and scientific antirealists we may assume that at least a small portion of the hypotheses tried by scientists (say, 2%) are at least partly true. Thus, we assume 0,02 is the frequency at which SM finds (partly) true hypotheses, and this should provide the value for the prior probability P(T) in equation (1) above.
Therefore, if P(T│S) is the probability that H is true given that it predicted a novel prediction whose probability is around 0.0003 (like,e.g.,Neptune^' s prediction), it can be computed as follows:
(2) P(T│S)=(P(S│T)=1 ∙P(T)=0,02 )/([P(S│T)=1∙P(T)=0,02]+[P(S│¬T)=0.0003 ∙P(¬T)=0,98]) = 0,9855129595
Here P(S│T)=1 because H entails np, and true hypotheses entail true consequences; therefore, if we found a true hypothesis (H), ipso facto we found a hypothesis making the true novel prediction np. Notice, also, that often novel predictions are much less probable than Neptune’s prediction, and that P(T│S) increases for any further novel prediction licensed by H.

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Item Type: Preprint
Creators:
CreatorsEmailORCID
Alai, Mariomario.alai@libero.it
Keywords: No miracle argument. Bayes' theorem. Bare-rate fallacy. Novel predictions. Scientific realism. Scientific method. Recursive empirical foundation. Instrumental observation
Subjects: General Issues > Confirmation/Induction
General Issues > Evidence
General Issues > Experimentation
General Issues > Realism/Anti-realism
General Issues > Theory/Observation
Depositing User: Prof. Mario Alai
Date Deposited: 19 Feb 2024 04:36
Item ID: 23106
Subjects: General Issues > Confirmation/Induction
General Issues > Evidence
General Issues > Experimentation
General Issues > Realism/Anti-realism
General Issues > Theory/Observation
Date: June 2023
URI: https://philsci-archive.pitt.edu/id/eprint/23106

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