Variety of evidence and the elimination of hypotheses

Varied evidence for a hypothesis confirms it more strongly than less varied evidence, ceteris paribus. This epistemological Variety of Evidence Thesis enjoys long-standing widespread intuitive support. Recent literature has raised serious doubts that the correlational approach of explicating the thesis can vindicate it. By contrast, the eliminative approach due to Horwich vindicates the Variety of Evidence Thesis but only within a relatively narrow domain. I investigate the prospects of extending the eliminative approach to a larger domain by considering a larger class of sensible explications of evidential variety. For a large subclass class of such explications I show how to construct cases in which the less diverse body of evidence for a hypothesis confirms more strongly. I hence argue that these prospects are dire since the eliminative approach widely fails to vindicate the thesis.


Introduction
Varied evidence for a hypothesis confirms it more strongly than less varied evidence, ceteris paribus. This epistemological Variety of Evidence Thesis (VET) enjoys longstanding widespread intuitive support (Carnap 1962;Claveau 2013;Earman 1992;Hempel 1966;Horwich 1982;Hüffmeier et al. 2016;Meehl 1990;Keynes 1921;Fitelson 2001;Franklin and Howson 1984). The pretheoretical intuition is vividly captured by Meehl,(p. 111) "Any working scientist is more impressed with 2 replications in each of 6 highly dissimilar experimental contexts than he is with 12 replications of the same experiment." There is also empirical work investigating the descriptive adequacy of this thesis for ordinary reasoning (Heit et al. 2005).
There are two main Bayesian approaches in the general philosophy of science aiming to provide an account of this thesis, the correlation analysis originating with Earman (1992) and the analysis of Horwich in terms of the elimination of competing hypotheses. 1 Earman's idea is that one can explicate a degree of variety of items of evidence as their lack of correlation. Less correlated items of evidence for a hypothesis are thus thought to confirm more strongly. The eliminative approach builds on the idea that diverse evidence for a hypothesis is better at eliminating (disconfirming) the other hypotheses one entertains. Hence, diverse evidence confirms the hypothesis of interest more strongly.
Despite early successes reported in Franklin and Howson (1984) and Earman (1992), 2 the correlational approach faces serious problems raised in Hartmann (2002, 2003); Claveau (2013); Claveau and Grenier (2019); Landes and Osimani (2020); Landes (2021): they show that there are cases in the correlational approach in which less diversity of the evidence providing sources bestows, ceteris paribus, more confirmation upon the hypothesis. Landes (2019) attempts to rescue the VET in the correlational approach, but even he discovers cases in his model in which less varied evidence confirms more strongly than more varied evidence.
Given the widespread intuitive support for the VET it appears worthwhile to devise a widely applicable vindication of the VET. Maybe, such a vindication can instead be achieved by extending the less-investigated eliminative approach due to Horwich. This has recently been suggested in Schupbach (2015, §. 3), where it has been suggested that an expansion of Horwich's account may also provide parts of the foundations for robustness analysis. There may hence be an extra reward to be reaped, if such a vindication can be provided.
I thus investigate the prospects of extending Horwich's account to explicate and vindicate the VET: are there many more cases than Horwich found in which diverse evidence is better at ruling out competing hypotheses and hence better confirms the hypothesis of interest? I do so by investigating a large class of sensible explications of evidential variety. For a large subclass of such explications I show how to construct cases in which the less diverse body of evidence for a hypothesis confirms more strongly. I hence argue that these prospects are dire since the eliminative approach widely fails to vindicate the thesis.
The rest of the paper is organised as follows, I next introduce Horwich's approach and discuss its critiques, caveats and refinements (Section 2) and then revisit the notion of evidential variety (Section 3). Sections 4.1, 4.2 and 4.3 illustrate the ubiquity of cases in the eliminative approach in which, ceteris paribus, varied evidence is worse at ruling out competing hypotheses than narrow evidence. And so, less varied evidence confirms the hypothesis of interest more strongly than more varied evidence in these cases. Section 5 offers some conclusions.

Horwich
Horwich investigates the Bayesian confirmation of a hypothesis of interest, H 1 , relative to a set of n − 1 competing less simple hypotheses, H 2 , . . . , H n . These n hypotheses are taken to be mutually exclusive. 3 Furthermore, "it is known that one of them is true" (Horwich 1982, p. 118). He considers two bodies of evidence, E D , E N , where E D is more diverse than the narrower E N , 'narrow' here means the opposite of 'diverse'. At Horwich (1982, p. 118) it is suggested that "evidence is significantly diverse to the extent that its likelihood is low, relative to many of the most plausible competing hypotheses." Horwich explicates this suggestion by the requirement that for most competing hypotheses H j . In particular, his explication of evidential variety is tied to the set of the hypotheses under consideration and the likelihood of the evidence under these hypotheses.
Making further assumptions, Horwich shows that the more diverse body of evidence E D is better at eliminating the competing hypotheses, hence E D confirms H 1 more strongly than E N : One of these assumptions is that the hypothesis of interest, H 1 , entails E D and E N , i.e., P (E D |H 1 ) = P (E N |H 1 ) = 1.

Horwich's critics
The first critique of Horwich's analysis was voiced by Wayne alleging to have found a counter-example to Horwich's approach. In his counter-example the narrow body of evidence, E N , is better at eliminating the competing hypotheses than the diverse body of evidence, P (H 1 |E D ) < P (H 1 |E N ). 4 This discovery is driven by replacing Horwich's assumption that 'H 1 entails the evidence' by a 'probabilistic constraint on P (E D |H 1 )/P (E N |H 1 )' (Wayne 1995, eq. 11).
It was quickly pointed out in Fitelson (1996, pp. 654-5) that in Wayne's "counterexample" the diverse body of evidence E D disconfirms H 1 ; whereas, the narrow body of evidence E N confirms H 1 . I agree with Fitelson who argues that in order 3 See Schupbach and Glass (2017) for more on competing hypotheses and see Fitelson (1996, pp. 657-8) for a discussion of the simplicity assumption. 4 Wayne also challenged the champions of the correlational approach to put forward an explication of evidential variety that incorporates more than just prior probabilities. Such explications are a topic of current discussion (Bovens and Hartmann 2003;Claveau 2013; Claveau and Grenier 2019; Landes 2019), see Section 3 for more details.
to be charitable to Horwich the ceteris paribus clause that E D , E N are equally likely given H 1 Ceteris Paribus Clause (3) needs to be imposed, if the eliminative approach is applied to bodies of evidence which do not follow from the hypothesis of interest. I now offer a second argument for imposing the Ceteris Paribus Clause which draws on (Steel 1996, p. 671). Suppose one does not impose the clause and consider a case with diverging likelihoods, P (E D |H 1 ) P (E N |H 1 ). By Bayes' Theorem P (H 1 |E D ) · P (E D ) P (H 1 |E N ) · P (E N ) holds. Unless P (E D ) P (E N ) holds, it is the narrow body of evidence which confirms H 1 more strongly than the diverse body of evidence. This has the unwelcome consequence that whether E N or E D confirm H 1 more strongly comes down only to the priors of E N , E D and the likelihoods given H 1 and does not depend on the diversity of evidence [or lack thereof].
Wayne's "counter-example" hence poses no major difficulty to Horwich's analysis because it fails to satisfy this Ceteris Paribus Clause.
However, Fitelson, pp. 656-9 also points out that Horwich's canonical example produces counter-intuitive results: in some cases, in which the Ceteris Paribus Clause (3) holds, the body of evidence which we intuitively judge to be more diverse is less diverse according to Horwich's explication. In these cases, the assumption that the hypothesis of interest entails the evidence is dropped and H 1 is sufficiently complex but still less complex 5 than the other hypotheses. In these cases, it is the intuitively less diverse body of evidence which bestows more confirmation upon the hypothesis of interest.
So, Fitelson pointed out that the eliminative approach applied to bodies of evidence which do not follow from the hypothesis of interest may possibly vindicate the VET, if two caveats apply: 1) the Ceteris Paribus Clause holds (3) and 2) the hypothesis of interest is really simple. If one of these two caveats is allowed to fail, then there are counter-examples to the eliminative approach in which the less diverse body of evidence confirms more strongly.
Schupbach criticises Horwich's "problematic assumptions" that the hypotheses are mutually inconsistent and exhaustive as, in many practical applications, unrealistic. 6 He goes on to raise the question of how far Horwich's analysis can be extended. Although, Schupbach does not discuss Fitelson's caveats, I take Schupbach's question with these caveats as the starting point of this paper. 5 Complexity and its opposite, simplicity, of a hypothesis, H , are here understood in terms of the degree of the smallest family of polynomials containing H . The greater this degree, the greater the complexity of the hypothesis. 6 I think that Horwich could easily defend himself against the criticism that the hypotheses are nonexhaustive in realistic cases. Horwich could point out that every set of non-exhaustive hypotheses can always be made exhaustive by adding a catch-all complement hypothesis, H : all other hypotheses are false, cf. Wenmackers and Romeijn (2016). Another way to deflect this criticism is to point out that "knowing that one of the hypotheses is true" (Horwich 1982, p. 118) does not entail that the disjunction of these hypotheses is a logical tautology.

Recent work
Some authors have criticised Horwich's explication of evidential variety. Wayne, p. 118 puts his criticism bluntly "As an explanation of the superior evidential value of a diverse data set this account is clearly circular. Diverse evidence is better, the explanation goes, because of its ability to eliminate more of the rival hypotheses, yet eliminating more of the rival hypotheses is exactly the definition of diverse evidence with which Horwich began. Such a circularity is tolerable if the resulting account yields a deeper understanding of the notion of diverse evidence. But in this case it does not". Fitelson  The notion of evidential variety or as Horwich already noted diversity of evidence has received considerable attention since Horwich's contribution, a number of different explications have been offered. Earman, (p. 78) understands evidential variety as the rate of the increase in the factor P (E n |E 1 & . . . &E n−1 &K) given background knowledge K, Hartmann(2002, 2003), Claveau (2013) and Claveau and Grenier (2019) explicated evidential variety in terms of the variety of the instruments used to generate the evidence, 7 Landes (2019) and Claveau and Grenier (2019) explicate evidential variety in terms of the variety of the consequences of the hypothesis of interest the available evidence pertains to, 8 medical methodologists have focussed on the variety of designs of medical studies often trumpeting the epistemological superiority of quality of study method over variety of study method Worrall (2007) (see Borm et al. (2009) for a rare exception), variety of data is one of the big V of Big Data Lukoianova and Rubin (2014) and Sagiroglu and Sinanc (2013); variety here refers to sources of data as well as the structure (or lack thereof) of data. 9 7 Intuitions supporting the "source variety" VET seem to be predicated on the assumption that more source variety must lead to more hypothesis confirmation. However, less source variety makes the evidence more informative about the source(s) of evidence. Learning that sources of confirmatory evidence are reliable also increases hypothesis confirmation. In some cases, hypothesis confirmation via learning about the source(s) of evidence trumps learning from independent sources. 8 Intuitions supporting the "consequence variety" VET seem to be predicated on the assumption that more consequence variety must lead to more hypothesis confirmation. However, lower consequence variety makes the evidence more informative about fewer consequences of the hypothesis. Learning that a probabilistic consequence of the hypothesis is likely true also increases hypothesis confirmation (in particular if the consequence is a priori relatively unlikely). In some cases, hypothesis confirmation via boosting the belief in fewer consequences trumps learning less about more consequences of the hypothesis. 9 Measuring diversity has recently become important to measure our progress towards a more diverse society. Measures of diversity for quantifying species diversity are of much interest in ecological settings (Tuomisto 2010). Quantitative measures of entropy rather than diversity are an area of active research in information studies, (formal) epistemology and psychology (Landes 2015;Crupi et al. 2018;Csiszár 2008).
We see that a variety of notions of diversity of evidence or data have recently been proposed. None of these recent suggestions followed Horwich by tying evidential variety to competing hypotheses or to likelihoods. Given the number of different approaches to evidential variety, it is (next to) impossible to offer a unified account of evidential variety; a point already made before Big Data was big (Howson and Urbach 2006, p. 125). Instead, I will put forward one intuitive judgement of comparative evidential variety which is compatible with the above mentioned notions of evidential variety and, I suspect, with many other plausible such notions one may consider. Rather than assessing the absolute variety of a body of evidence I consider which of two bodies of evidence is more varied.

A proposal for comparative variety
I put forward the idea that it should be compatible with a widely applicable notion of evidential variety that, ceteris paribus, a more varied body of evidence may constrain rational belief in the value of a parameter differently than a less varied body of evidence. 1011 More precisely, the evidential constraints arising from a more varied body of evidence constrain rational degrees of belief to a different interval than the less varied body of evidence, at least in some cases. That is, I claim that for many -if not most -reasonable construals of evidential variety, there is at least one case in which a more varied body of evidence constrains rational degrees of belief in the value of a parameter to a different interval than a less varied body of evidence. This is meant to ensure that there are a number of cases in which two bodies of evidence are different from the perspective of parameter estimation (constraining beliefs to different intervals) where these bodies of evidence have different degrees of evidential diversity. Call this class of explications C. The above discussion is meant to show that the class of explications C is not only non-empty but does in fact contain many -if not most -sensible explications of evidential variety.
For example, we are entertaining different hypotheses about the history of the universe which make different predictions about the numerical value of the redshift of stars. We thus investigated and obtained two bodies of evidence each pertaining to a randomly selected but different star in the same distant galaxy. As it turns out, the measurements disagree, after performing error calculations the stars' redshifts are seemingly located in disjoint intervals. Fortunately, both intervals are in good agreement with the hypothesis of interest and are thus evidence for this hypothesis. It seems plausible to me, that we should be able to say that there exists 10 I want the notion of (comparative) evidential variety to be widely applicable so that a (possible) vindication of the VET in terms of eliminating competing hypotheses employing this notion covers a large part of scientific inference. Parameter estimation is one key task of scientific and statistical inference. A widely applicable vindication of the VET should hence apply to a number of cases of parameter estimation. 11 I here take a Bayesian view of parameter estimation according to which the constraints on the parameter(s) of interest constrain rational beliefs. some scenario in which one of these bodies of evidence is more diverse than the other. 12

The eliminative approach and comparative variety
Can Horwich's analysis in terms of eliminating competing hypotheses which vindicates the VET -albeit with the caveats discussed above -be extended to the class of explications C? To answer this question, I consider an arbitrary explication c ∈ C and two bodies of evidence E 1 , E 2 each constraining parameter ranges to different intervals, I 1 = I 2 , such that 1. E 1 and E 2 are evidence for some simple hypothesis H 1 , 2. the Ceteris Paribus Clause P (E 1 |H 1 ) = P (E 2 |H 1 ) (3) holds and 3. E 2 is more c-diverse than E 1 .
Can one show -as Horwich did -that H 1 is more likely given the more diverse body of evidence? One may initially hope so.
Instead, I show that the opposite is true. If I 2 is not a subset of I 1 (nor equal to I 1 ), then there exist a set of hypotheses {H 1 , H 2 , . . . , H n } and many plausible prior probability functions P such that P ( To do so, I shall consider three cases: α I 2 is larger than I 1 (Section 4.1), β I 2 is smaller than I 1 and I 2 is not a subset of I 1 , this means that there are some elements of I 2 which are not in I 1 (Section 4.2) and γ I 1 and I 2 have equal size and the intervals are different (Section 4.3).
In all of these cases I show that there exist a set of hypotheses {H 1 , H 2 , . . . , H n } and many plausible prior probability functions P such that P (H 1 |E 1 ) > P (H 1 |E 2 ).
Using that E 2 is more c-diverse than E 1 I obtain P (H 1 |E N ) > P (H 1 |E D ).
If I 2 is a proper subset of I 1 , then P (I 2 |H i ) ≤ P (I 1 |H i ) follows for all 1 ≤ i ≤ n from the axioms of (conditional) probability. We are hence in a case in which Horwich's explication of evidential variety applies, and given the correctness of the eliminative approach and given the caveats we conclude that P (H 1 |E N ) ≤ P (H 1 |E D ). 13 In these cases, it hence not possible to extend the eliminative approach since it is already known to cover them. 12 I'm grateful to an anonymous reviewer for pressing me to clarify the relation between evidential variety and comparative evidential variety. On the one hand, any notion of evidential variety which allowsat least in some cases -the comparison of degrees of evidential variety induces a canonical notion of comparative evidential variety. On the other hand, every notion of comparative evidential variety is -in some shape or form -based on a notion of evidential variety. To formulate a comparative notion however there is (a priori) no need to specify the underlying notion in full detail, a full-blown explication is not required. 13 To streamline my presentation I will say that the VET is vindicated in case P (H 1 |E N ) ≤ P (H 1 |E D ).
Noting that C is the disjoint union of C * and C H C * := {c ∈ C : there exist E 1 , E 2 such that E 2 is more c-diverse than E 1 and I 2 I 1 } C H := {c ∈ C : E 2 is more c-diverse than E 1 entails that I 2 is a proper subset of I 1 }, These counter-examples are particularly troubling, since they satisfy the caveats applying to Horwich's analysis (Section 2): i) the hypothesis of interest is simple (it is linear, the competing hypotheses are functions of power [of at least] two) 14 and ii) the Ceteris Paribus Clause (3) holds. Furthermore, they occur for mutually exclusive hypotheses, an assumption criticised by Schupbach for being too idealised. Finally, these cases are independent of the prior probability function as long as all probabilities are non-zero -with one exception we will encounter in Section 4.2.
The remainder of Section 4 is devoted to illustrating Proposition 1, the proofs are in Appendix A.

Varied evidence constraining less tightly
Let us thus consider an arbitrary explication c ∈ C * such that there exists a less constraining body of evidence which is more varied, |I D | > |I N |. Consider the above example again in which two stars in Galaxy G were randomly selected. For both stars 50 measurements of their redshifts were performed. After performing error calculations, we are in a position to locate a star's redshift within some compact interval, [a, b] ⊂ R. For the purposes of the following Bayesian calculations, we equate this situation with the evidence proposition that 'this star's redshift is within the interval [a, b]'.
Let us call the first body of evidence consisting of the first set of measurements E D and call the second body of evidence consisting of the second set of measurements E N . Suppose furthermore the interval associated with the first body of evidence, , is larger than the interval associated with the second body of evidence, 14 Mutatis mutandis, my formal analysis continues to apply for a hypothesis of interest which is more complicated than the competing hypotheses: replace f 1 [defined below] by a very complicated function f * 1 , which is a very good approximation of f 1 . 15 There is a natural sense of evidential diversity this explication c captures: The measurements in E D are more spread out, in other words, the measurements in E N are more clustered around their average. In this sense, the body of evidence E D is more diverse than the body of evidence E N . Consequences of the existence of a case in which b D − a D < b N − a N are considered in Section 4.2. Note however that the evidential variety of a body of evidence for an explication c ∈ C depends, in general, also on factors other than the spread of measurements.
Let us suppose that we currently only entertain hypotheses H 1 , H 2 , H 3 , H 4 with 4 i=1 P (H i ) = 1 such that: H i : a randomly selected star's redshift X in Galaxy G is given by the probability distribution Intuitively, the greater the value of f i (x), the greater the probability that the star's redshift is close to x. For example, an f i could be the distribution of redshifts of stars in some Galaxy G according to some model M i . Different models, the M i , make different predictions for the distribution of redshifts in Galaxy G. Let us also suppose that it is known that one of the H i is true. Figure 1 depicts such a scenario with a = 0 and b = 3.
The thrust of my argument is that, although, these bodies of evidence satisfy 1 -3, it is the narrow body of evidence E N which fits much better with the simple hypothesis H 1 . It is hence the narrow body of evidence which confirms the simple hypothesis H 1 more strongly than the diverse body of evidence, P (H 1 |E D ) < P (H 1 |E N ).
Note that Horwich's approach is meant to vindicate the VET and thus apply to arbitrary sets of hypotheses as long as H 1 is the simplest hypothesis, the H i (1 ≤ i ≤ n) form a partition and the Ceteris Paribus Clause (3) holds. For my purposes (constructing counter-examples to a vindication of the VET), it hence suffices to find some set of hypotheses (here represented by the f i ) for which the narrower body of evidence confirms H 1 more strongly.
The formal analysis of this example can be found in Appendix A.1. Appendix A.3 demonstrates how to prove the general case, i.e., how to show that for all bodies of evidence with b D − a D > b N − a N there exists a set of hypotheses in which H 1 is the simplest, Eq. 3 holds and which satisfies 1-3 such that P (H 1 |E D ) < P (H 1 |E N ).

Varied evidence constraining more tightly
Against the counter-examples outlined in the previous section it might be objected that evidential variety has been construed the wrong way round. It might be intuited that it is the more diverse evidence which constrains rational beliefs more tightly. The intuition may be that more diverse items of evidence, which are all evidence for a hypothesis, rule out more plausible parameter values and hence constrain beliefs more tightly than less diverse evidence. Hence, diverse evidence constrains beliefs to a smaller range than narrow evidence.
Consider now an arbitrary c ∈ C * for which there exists an E D which is more cdiverse than E N such that |I D | < |I N | and I D I N . However, even in this case the above argument applies -with small modifications. The picture is given in Fig. 2. The interval on the right remains unchanged compared to Fig. 1. Some points x < 0 were added to the interval on the left, so that it becomes larger than the interval on the right. The interval on the left again corresponds to the narrow body of evidence, E N , while the interval on the right represents the constraints on rational beliefs imposed by the more diverse body of evidence.
I introduce one new hypothesis, H 5 , which is compatible with negative values of x. The only purpose of introducing H 5 is to ensure that the set of hypotheses entertained does not allow us to rule out that X < 0. If we could infer that X ≥ 0, then I N would collapse to [0, 0.1]. The only relevant properties of H 5 are that i) it is given by a probability distribution which is positive on all of I N ∪ I D , ii) f 5 (x) is small everywhere and iii) H 5 has a tiny non-zero prior probability, P (H 5 ) is close to but different from zero. 16 Intuitively, since the points added to the interval on the left play no major role in the above example, it is still the interval on the left which confirms more strongly. Since this interval continues to represent the narrow evidence and the prior of H 5 is tiny, it is again the narrow body of evidence which confirms more strongly, P (H 1 |E D ) < P (H 1 |E N ).
So, here it is the case that |I N | > |I D | and P (H 1 |E D ) < P (H 1 |E N ); in the previous section it was the case that |I N | < |I D | and P (H 1 |E D ) < P (H 1 |E N ). The formal analysis of this example can be found in Appendix A.2, Appendix A.3 contains a proof sketch of the general case. 17

Equally tightly constraining bodies of evidence
Finally, I briefly consider an arbitrary c ∈ C * such that there exist two different bodies of evidence which constrain beliefs to different intervals of the same size, 16 The condition that P (H 5 ) is close to but different from zero is the only assumption I put on the prior probability function P . This condition is in agreement with the suggestion at Horwich (1982, pp. 70-71) that complex hypotheses have low prior probability. 17 I D ⊂ I N is the only exceptional case here since it is already covered by Horwich's explication of evidential variety, P (E D |H i ) ≤ P (E N |H i ) for all i follows from I D ⊂ I N and the axioms of (conditional) probability. |I N | = |I D | and I D = I N where E D is more c-diverse than E N . The degree of evidential variety for such a c ∈ C * must depend on other things than just the size of the interval I .
To construct the counter-examples one picks a flat function f 1 and functions f i which are smaller than f 1 on I D and even smaller on I N , f i (x) < f i (y) < f 1 (y) for all x ∈ I N , y ∈ I D , i ≥ 2. The intuition and the blue-print for the proof of the general case are obtained by inspecting Fig. 3. Clearly, it is the narrow body of evidence Fig. 3 Functions , , , f 4 and bodies of evidence associated with I D , I N which better confirms the simplest hypothesis H 1 . So, it holds that P (H 1 |E D ) < P (H 1 |E N ).

Conclusions
I set out to investigate how far Horwich's analysis of the VET in terms of the elimination of competing hypotheses can be extended to vindicate the VET. Achieving such a vindication would not only shore up support for the embattled VET but may also provide support for robustness analysis (Schupbach 2015). To do so, I considered possible extensions of Horwich's eliminative approach to a large class of explications of evidential variety.
I showed that there are ubiquitous counter-examples to the VET in the eliminative approach for a large sub-class of sensible explications of evidential variety (Proposition 1). The ubiquitous counter-examples preventing a vindication of the VET are all the more troubling since they rear their heads in situations in which the VET should hold par excellence (strong idealisations are made, the Ceteris Paribus Clause (3) holds, H 1 is really simple and the failures are independent of the Bayesian prior probability function). Therefore, the prospects for extending Horwich's eliminative approach to vindicate the VET are dire.
Those sharing my dissatisfaction with Horwich's notion of evidential variety, possibly because they seek a more informative, foundational account of what diversity per se is (see Section 3.1), may find the prevalent counter-examples particularly unfortunate, since they seem to efficiently block a vindication of the VET in the eliminative approach for a large class of sensible explications of evidential variety.
Summing up, I find that explications of the VET that were once deemed to be "textbook Bayesian success stories" (Bovens and Hartmann 2002, p. 44) are today in dire straits. Paraphrasing (Paris 2014, p. 6193) I conclude that: Most of us would surely prefer modes of reasoning which we could follow blindly without being required to make much effort, ideally no effort at all. Variety of Evidence reasoning was long thought to be a mode of reasoning one could follow blindly. Unfortunately, it is not such a paradigm.
It only remains show that the more diverse body of evidence confirms less strongly, P (H 1 |E D ) < P (H 1 |E N ):

A.2 Varied Evidence constraining more tightly
The formal analysis for this case resembles the previous one and his hence given in a more condensed form. x ∈ [0, 5] . 3. Let P (H 5 ) be tiny but non-zero, P (H 5 ) = 1 10000 say. Mutatis mutandis, the proof above (Appendix A.1) applies here, too. After going through the calculations again we obtain P (H 1 |E D ) < P (H 1 |E N ), since the prior of H 5 is tiny and all other hypotheses assign probability to zero to the points added to the left interval. 19 Clearly, 5 −5 f 5 (x)dx = 1 holds and f 5 is the most complicated hypothesis and thus H 1 remains the simplest hypothesis entertained.

A.3 The general case
The two proofs above apply to a special case with fixed E D , E N . I now show how to prove the general case for varied evidence constraining less tightly, i.e., arbitrary E D , E N satisfying |I N | < |I D |. This proof shows how to modify the functions f i for the general case. The case for varied evidence constraining more tightly is all but analogous and is only sketched here.
In the general case of varied evidence constraining less tightly we have that |I N | < |I D |. The ceteris paribus condition P 1 (E D ) = P 1 (E N ) is also assumed throughout.
Let us first assume that I N and I D are disjoint. By flipping the x-axis (f i (x) → f i (−x)) if necessary, we may assume that I D is to the right of I N .
If γ > 1, then put g i (x) := γf i (γ x). Geometrically, this shrinks the x-axis by the factor γ while the y-axis is dilated by the factor γ . Hence, all ordinal comparisons are preserved for all x ∈ R: f i (γ x) < f j (γ x), if and only if g i (x) < g j (x). It only remains to check that the g i define probability functions; we hence calculate for all i ∈ {1, 2, 3, 4}   (3) it holds that P (I D \I N |H 1 ) = 0 (almost everywhere). Now simply devise functions f i for i ∈ {2, 3, 4} which take large values on I D \I N . Then, these H i are more compatible with E D than with E N . Hence, E N confirms H 1 stronger than E D , P (H 1 |E N ) > P (H 1 |E D ). Now proceed as above.