 # 统计代写|贝叶斯统计代写Bayesian statistics代考|STA602

## 统计代写|贝叶斯统计代写Bayesian statistics代考|CONFIRMATION VS CORROBORATION

Perhaps no single issue has generated more controversy than the alleged distinctness of corroboration and confirmation. For a Bayesian, ‘confirmation’ means ‘raising the probability of’; for a Popperian, ‘corroboration’ (a term Popper coined to avoid confusion with probabilistic notions of confirmation) means ‘withstanding severe tests’. But what are ‘severe tests’? Their primary attribute seems to be this. ${ }^1$ Let $x$ be an outcome of the test or experiment that agrees with the hypothesis $H$. Then the test of $H$ is severe if $P(x)$ is small. The magnitude of the deflection of light by a large body predicted by relativity theory was double the Newtonian defection, and consequently, the Eddington expedition of 1919 to observe the solar eclipse of that year constituted a severe test of relativity. As this example rightly suggests, severity is as much a property of the theory under test as it is of the experiment designed to test it. The aim of experimentation, from this perspective, is to make it virtually impossible for the experimental outcome to fit the theory unless the theory is true (or, at any rate, an adequate representation of the experiment in hand). ${ }^2$ Now, as everyone knows, it is a consequence of Bayes’ rule that hypotheses are confirmed by their consequences (or, more generally, by outcomes which they afford a high probability), and the more so as the outcomes in question are otherwise surprising or improbable. In the extreme case where $x$ is a logical consequence of $H$, Bayes’ rule reduces to $P(H / x)=P(H): P(x)$, which expresses the posterior probability of $H$ as the ratio of its prior probability to the outcome probability $P(x)$. In short, Bayes’ rule is not only compatible with, but actually rationalizes the primary attribute of corroboration.

If that were the only attribute of corroboration, there would be no reason to distinguish it from confirmation. But, of course, there is more to it. Above all, there is Popper’s insistence that high corroboration must not be equated with high probability. But then, equally, high confirmation cannot be equated with high probability. For if the initial probability of an hypothesis is low, say $0.001$, then its probability may be greatly increased, say to $0.5$, without making its final probability high. Or, at the other extreme, tautologies are highly probable, but not at all confirmable.

Moreover, while Popper maintains that high corroboration only makes an hypothesis testworthy (never trustworthy), that heroic line is difficult to maintain in practice, or, more precisely, in spelling out the relation between theory and practice. Mendelian genetics is better corroborated than the blending theory, and of course that can be expected to have a bearing on eugenic proposals, genetic counseling, breeding and agricultural programs, nature vs nurture disputes, and so on. It is almost impossible to resist the conclusion that because the particulate theory is better corroborated, it is rational to premiss it in theoretical derivations and practical decisions. But to act as though that theory were true is to assign it a higher probability than any of its rivals (it is to bet on its being the best available approximation to the truth in this domain). ${ }^3$

## 统计代写|贝叶斯统计代写Bayesian statistics代考|DEM ARCA TION

The likelihood principle implies, as already mentioned, the irrelevance of predesignation, of whether an hypothesis was thought of beforehand or was introduced to explain known effects. Bayesians deny that any additional force attaches to agreeing outcomes predicted in advance (though they would not deny that the fertility of a theory shows itself in the novel experiments it suggests). But a belief in the peculiar virtue of prediction is a recurrent theme in Popper’s writings. ${ }^9$ It is a fundamental part of his proposed demarcation of science from metaphysics. As I understand it, that proposal consists of two strands, a logical and a methodological, as it were. The logical requirement is that the theory logically exclude some possible state of affairs (in order to be accounted ‘scientific’), or that it have ‘potential falsifiers’. The methodological requirement is that proponents of the theory be willing to countenance at least some of the potential falsifiers as sufficient to reject the theory. In a discussion of the scientific credentials of psychoanalysis Popper writes:
‘Clinical observations’, like all other observations, are interpretations in the light of theories…. and for this reason alone they are apt to seem to support those theories in the light of which they were interpreted. But real support can be obtained only from observations undertaken as tests (by ‘attempted refutations’); and for this purpose criteria of refutation have to be laid down beforehand: it must be agreed which observable situations, if actually observed, mean that the theory is refuted (Popper, 1963, p. 38, Note 3).

In other places Popper expresses this as a demand that one specify ‘crucial experiments’ by which to discriminate a new theory from its rivals in the field. Posed in this milder form, it reads like good counsel, but it is also good Bayesian counsel. In Bayesian terms, a decisive test of $H$ against $K$ is one for which the expected weight of evidence (i.e., the expected log likelihood ratio) for $H$ against $K$ is high conditional on $H$, and vice versa. But high likelihood ratios are enough to insure objectivity! Their evidential weight is in nowise augmented by use of a predesignated rejection rule; they speak for themselves. Which theories our practical decisions and theoretical derivations are predicated upon must be decided on the merits of the case at hand by using one’s evaluations of the probabilities and the consequences of error to guide one. No purpose is served by attempting to lay down conventions which state conditions under which the relevant scientific community should ‘accept’ or ‘reject’ a theory.

# 贝叶斯统计代考

## 统计代写|贝叶斯统计代写Bayesian statistics代考|DEM ARCA TION

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