Editing Raven paradox (section)

====Orthodox approach====
The orthodox [[Type I and type II errors|Neyman–Pearson]] theory of hypothesis testing considers how to decide whether to ''accept'' or ''reject'' a hypothesis, rather than what probability to assign to the hypothesis. From this point of view, the hypothesis that "All ravens are black" is not accepted ''gradually'', as its probability increases towards one when more and more observations are made, but is accepted in a single action as the result of evaluating the data that has already been collected. As Neyman and Pearson put it:

{{quote|Without hoping to know whether each separate hypothesis is true or false, we may search for rules to govern our behaviour with regard to them, in following which we insure that, in the long run of experience, we shall not be too often wrong.<ref>{{cite journal |last1=Neyman |first1=J. |last2=Pearson |first2=E. S. |date=1933 |title=On the Problem of the Most Efficient Tests of Statistical Hypotheses |journal=Philosophical Transactions of the Royal Society A| volume=231|issue=694–706 | page=289 |url=http://www.stats.org.uk/statistical-inference/NeymanPearson1933.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.stats.org.uk/statistical-inference/NeymanPearson1933.pdf |archive-date=2022-10-09 |url-status=live |doi=10.1098/rsta.1933.0009|bibcode=1933RSPTA.231..289N |jstor=91247|doi-access=free }}</ref>}}

According to this approach, it is not necessary to assign any value to the probability of a ''hypothesis'', although one must certainly take into account the probability of the ''data'' given the hypothesis, or given a competing hypothesis, when deciding whether to accept or to reject. The acceptance or rejection of a hypothesis carries with it the risk of [[Type I and type II errors|error]].

This contrasts with the Bayesian approach, which requires that the hypothesis be assigned a prior probability, which is revised in the light of the observed data to obtain the final probability of the hypothesis. Within the Bayesian framework there is no risk of error since hypotheses are not accepted or rejected; instead they are assigned probabilities.

An analysis of the paradox from the orthodox point of view has been performed, and leads to, among other insights, a rejection of the equivalence condition:

{{quote|It seems obvious that one cannot both ''accept'' the hypothesis that all P's are Q and also reject the contrapositive, i.e. that all non-Q's are non-P. Yet it is easy to see that on the Neyman-Pearson theory of testing, a test of "All P's are Q" is ''not'' necessarily a test of "All non-Q's are non-P" or vice versa. A test of "All P's are Q" requires reference to some alternative statistical hypothesis of the form <math>r</math> of all P's are Q, <math>0<r<1</math>, whereas a test of "All non-Q's are non-P" requires reference to some statistical alternative of the form <math>r</math> of all non-Q's are non-P, <math>0<r<1</math>. But these two sets of possible alternatives are different ... Thus one could have a test of <math>H</math> without having a test of its contrapositive.<ref>{{cite journal | last1 = Giere | first1 = R. N. | year = 1970 | title = An Orthodox Statistical Resolution of the Paradox of Confirmation | journal = Philosophy of Science | volume = 37 | issue = 3| pages = 354–362 | jstor=186464 | doi=10.1086/288313| s2cid = 119854130 }}</ref>}}