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

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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.

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

贝叶斯统计代考

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

也许没有一个问题比所谓的佐证和确认的独特性更能引起争议。对于贝叶斯,“确认”意味着“提高概率”;对于波普尔来说,“确证”(波普尔创造的一个术语是为了避免与确认的概率概念混淆)意味着“经受住严峻的考验”。但什么是“严峻考验”?他们的主要属性似乎是这个。1让X是符合假设的测试或实验的结果H. 然后测试H如果是严重的磷(X)是小。相对论预测的大天体对光的偏转幅度是牛顿偏离的两倍,因此,1919年爱丁顿远征观察当年的日食,对相对论构成了严峻的考验。正如这个例子正确地表明的那样,严重性既是被测试理论的属性,也是旨在测试它的实验的属性。从这个角度来看,实验的目的是使实验结果几乎不可能符合理论,除非理论是正确的(或者,无论如何,是手头实验的充分代表)。2现在,众所周知,这是贝叶斯规则的一个结果,即假设通过其结果(或更一般地说,通过它们提供高概率的结果)得到证实,并且当所讨论的结果令人惊讶或不可能的。在极端情况下X是一个合乎逻辑的结果H, 贝叶斯规则简化为磷(H/X)=磷(H):磷(X),表示后验概率H作为其先验概率与结果概率之比磷(X). 简而言之,贝叶斯规则不仅兼容,而且实际上合理化了佐证的主要属性。

如果那是确证的唯一属性,就没有理由将其与确认区分开来。但是,当然,还有更多。最重要的是,波普尔坚持认为,高确证绝不能等同于高概率。但是,同样,高确认不能等同于高概率。因为如果假设的初始概率很低,比如说0.001,那么它的概率可能会大大增加,比如说0.5,而不会使其最终概率变高。或者,在另一个极端,重言式很有可能,但根本无法证实。

此外,虽然波普尔坚持认为,高确证只会使假设值得检验(从不值得信赖),但这种英雄路线很难在实践中保持,或者更准确地说,在阐明理论与实践之间的关系时。孟德尔遗传学比混合理论得到了更好的证实,当然,这可能会对优生建议、遗传咨询、育种和农业计划、先天与后天的争议等产生影响。几乎不可能拒绝这样的结论,即由于微粒理论得到了更好的证实,因此在理论推导和实际决策中假设它是合理的。3

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

如前所述,似然原理暗示了预先指定的无关性,无论是事先考虑假设还是引入假设以解释已知效果。贝叶斯主义者否认任何额外的力量都与预先预测的一致结果有关(尽管他们不会否认一个理论的丰富性在它所建议的新颖实验中表现出来)。但是,相信预测的特殊美德是波普尔著作中反复出现的主题。9这是他提出的科学与形而上学划分的基本部分。据我了解,该提案由逻辑和方法两部分组成。逻辑要求是该理论在逻辑上排除了某些可能的事态(以便被认为是“科学的”),或者它具有“潜在的证伪者”。方法论要求是该理论的支持者愿意支持至少一些潜在的证伪者足以拒绝该理论。在讨论精神分析的科学凭证时,波普尔写道:
与所有其他观察一样,“临床观察”是根据理论进行的解释……。仅仅因为这个原因,他们似乎倾向于支持那些根据它们被解释的理论。但真正的支持只能从作为测试的观察中获得(通过“试图反驳”);为此目的,必须事先制定驳斥标准:必须商定哪些可观察的情况,如果实际观察到,则意味着该理论被驳斥(Popper,1963,第 38 页,注 3)。

在其他地方,波普尔将此表达为一种要求,即要求人们指定“关键实验”,以将一种新理论与该领域的竞争对手区分开来。以这种更温和的形式提出,它读起来像是好的忠告,但它也是好的贝叶斯忠告。用贝叶斯术语来说,一个决定性的检验H反对ķ是一个证据的预期权重(即预期的对数似然比)H反对ķ是高条件H,反之亦然。但高似然比足以确保客观性!通过使用预先指定的拒绝规则,它们的证据权重大大增加;他们为自己说话。我们的实际决策和理论推导基于哪些理论,必须根据手头案例的优点,通过对概率和错误后果的评估来指导一个人。试图制定规定相关科学界应该“接受”或“拒绝”理论的条件的约定是没有任何目的的。

统计代写|贝叶斯统计代写Bayesian statistics代考

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