Abstract
It can be demonstrated in a very straightforward way that confirmation and evidence as spelled out by us can vary from one case to the next, that is, a hypothesis may be weakly confirmed and yet the evidence for it can be strong, and conversely, the evidence may be weak and the confirmation strong. At first glance, this seems puzzling; the puzzlement disappears once it is understood that confirmation is of single hypotheses, in which there is an initial degree of belief which is adjusted up or down as data accumulate, whereas evidence always has to do with a comparison of one hypothesis against another with respect to the data and is belief-independent. Confusing them is, we suggest, a plausible source of the so-called “base-rate fallacy” identified by Kahneman and Tversky which leads most of us to make mistaken statistical inferences. It is also in the background, or so we argue in some detail, of the important policy controversies concerning human-induced global warming.
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Notes
- 1.
That is, we do not need to know the posterior probabilities of the mutually exclusive and jointly exhaustive H’s and not-H’s in order to calculate the posterior probability of H in the simple cases that we use to illustrate our point. It might be thought that there is a cryptic reference to two hypotheses in the determination of Pr(D) in the denominator of Bayes Theorem, since it involves averaging the data over H and ̴ H. But there is only one hypothesis, H, and the data are averaged over whether it is true or false. From this point of view, the expression “mutually exclusive and jointly exhaustive hypotheses” is misleading. Asserting “ ̴H” (is true) is just another way of saying that “H” is false. In more complex cases, we do need to know the priors of all of the hypotheses being considered in order to calculate the marginal probability of the data.
- 2.
Since it is the Bayesian measure most frequently encountered in the literature. As noted in Chap. 2, there are other measures of confirmation and evidence than those we are taking as paradigm.
- 3.
Subsequent research shows that this frequency-based prior probability still holds for the US population.
- 4.
Recall that a standard benchmark for “strong” evidence is an LR > 8.
- 5.
See IPCC (2007).
- 6.
- 7.
Although over the past 200 years, when a rise of temperatures and the vastly increased use of fossil fuels both occurred, the rise in CO2 levels invariably preceded the temperature rise.
- 8.
2014, however, has apparently been the hottest year since accurate temperature records began to be kept. This claim has been disputed by the well-known Harvard astrophysicist Willie Soon, but taking the observation at face value, the evidence for a lack of recent temperature increase is greatly weakened. Preliminary results for 2015 indicate that it was hotter still.
- 9.
There has been a great deal of controversy about the nature of the data and the ways in which correlations between CO2 levels and temperature changes are established. One among many criticisms of both data and correlations taken to support the human-induced warming hypothesis is in Douglass and Christy (2009). For a history of the data/correlation controversy by the person who was chiefly responsible for developing the initial data sources and correlation models, see Mann (2012). Mann’s methodology and his models have been revised, extended, but in the main confirmed by a number of subsequent studies, including (Marcott et al. 2013), which uses marine and terrestrial fossils, inter alia, from ocean and lakebed sediments, as well as ice cores and tree-ring data (which don’t extend over the entire period).
- 10.
See Lindzen (2010) who asserts that IPCC computer models don’t “accurately include any alternative sources of warming—most notably, the natural, unforced variability associated with phenomena like El Niňo, the Pacific Decadel Oscillation, etc”.
- 11.
Lindzen argues not so much that there are better alternative hypotheses but that the anthropogenic hypothesis incorporates assumptions about negative radiation feedback which have not been tested against their positive feedback rivals, i.e., the global thesis has indispensable components for which there is as yet no “evidence.” See Lindzen and Y-K Choi (2009). Lindzen’s argument has been much criticized. See, e.g., (Lin et al. 2010).
- 12.
- 13.
See, in particular, Dyson (2009).
- 14.
According to Christy (2014), these are the “two fundamental facts” that everyone must accept. Christy, a credible global warming skeptic, is, however, very dubious about the claim that there are strong correlations between a rise in surface temperatures and CO2 accumulation and critical of the way in which “surface temperatures” have been defined and measured.
- 15.
What follows draws from a very accessible overview by a Professor in the Department of Physics and Astronomy at the University of Las Vegas (Farley 2008).
- 16.
Mark Greenwood and his colleagues have, to give but one example, shown just how complex the application of statistical techniques to the detection of changes that provide support for the global warming hypothesis is. See Greenwood et al. (2011).
- 17.
IPCC (2004, Chap. 2, page 138).
- 18.
Another source of the misguided controversy has to do with the Bayesian account of confirmation that we have taken as paradigm. The prior probability of the hypothesis that global warming is not human-induced, is admittedly subjective, and for many people its rough determination depends not only on the plausibility of the “greenhouse gas” model, but on such otherwise extraneous considerations as that attempts to limit fossil fuel emissions will damage the economy and lead to the loss of jobs. Expected economic consequences often bleed over into the generation of priors concerning the existence and causes of global climate change, neither of which are in themselves economic hypotheses.
- 19.
See their classic paper, “Judgment Under Uncertainty: Heuristics and Biases,” re-published as Appendix A in Kahneman (2011).
- 20.
In the original question in the Harvard Medical School Test the subjects were told that nothing was known about the person’s symptoms or background, justifying the use of the base-rate prior, and they were asked to give the probability, on the assumption, that the person had the disease (see Howson (2000) for this discussion). We are thankful to Howson for some clarification here.
- 21.
We have argued that another heuristic short-cut, using the so-called “collapsibility principle” across the board, results in the celebrated Simpson Paradox (Bandyopadhyay et al. 2011) and have carried out our own experiments to confirm it, an exercise in “experimental philosophy”.
- 22.
Pr(D│H)/Pr(D│ ̴H) = 1/0.1 = 10.
- 23.
Referring to the Harvard Medical School Test, subjective Bayesians might contend that the answer the subjects gave were definitely wrong, and not just misunderstanding of what was required. They might add that in our version the person is already worried that they have TB, and so the prior is already much larger than the base rate. We will make two comments here. First, we agree with subjective Bayesians that subjects committed a probability error regarding the base-rate in our example. However, as contrasted with subjective Bayesians, we are able to provide an explanation for why subjects might have committed that sort of error. Second, if what we have stated is not the exact formulation of the base-rate fallacy typically understood then we need to know the entire probabilistic machinery at least conceptually required to address the way we discussed the fallacy because of the claim made by the subjective Bayesian. The onus is on the subjective Bayesian to offer that probabilistic mechanism regarding how the issue at stake can be handled within subjective Bayesianism.
- 24.
An anonymous referee of a paper containing the tuberculosis example claims that “the measures of confirmation literature…already provide the conceptual resources to acknowledge cases in which, by some measures, a hypothesis is greatly confirmed by a piece of evidence (e.g., when it ends up 100x as probable as it previously was), even though it does not end up highly probable nor does its probability change a great deal in absolute terms. This is what’s going on in the authors’ discussion of the TB example…” But it should be clear that “greatly confirming by a piece of evidence” (sic) is not at all tantamount to having strong evidence that the hypothesis is much better supported by the data than its rivals. In the TB case, the likelihood of a positive result on the hypothesis that the person tested has TB is much greater than on the hypothesis that she does not, independent of whether the first hypothesis is “greatly confirmed” by the data. As the referee’s criticism indicates, it is very difficult to shake the conflation of “evidence” with “data”.
- 25.
See Fitelson (2001).
- 26.
With multiple hypotheses it is easy to create scenarios where one model’s posterior probability is greater than its prior, in which case the model is confirmed, but have evidence against it. Consider three urns (A, B, and C) each containing 100 black or white balls. Urn A has one black ball and 99 white balls, Urn B has two black balls, and Urn C has no white balls. You are presented with an urn, but don’t know which it is. For whatever reason, you strongly believe that it is Urn C, but allow for the possibility that it could be A or B by assigning prior probabilities of 0.01, 0.1, and 0.89 to A, B, and C respectively. You draw a white ball randomly from the unidentified urn. Urn B is strongly confirmed because its posterior probability of 0.90 is much greater than its prior; however, there is weak evidence against B relative to A in that the LR B/A is 0.99.
- 27.
Bandyopadhyay and Brittan (2006).
- 28.
The following draws upon and is made more precise in Lele (2004).
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Bandyopadhyay, P.S., Brittan, G., Taper, M.L. (2016). Confirmation and Evidence Distinguished. In: Belief, Evidence, and Uncertainty. SpringerBriefs in Philosophy(). Springer, Cham. https://doi.org/10.1007/978-3-319-27772-1_3
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