Bayesianism is the claim that scientific reasoning is\ud
probabilistic, and that probabilities are adequately interpreted\ud
as an agent"s actual subjective degrees of belief\ud
measured by her betting behaviour.\ud
Confirmation is one important aspect of scientific reasoning.\ud
The thesis of this paper is the following: Given that\ud
scientific reasoning (and thus confirmation) is at all\ud
probabilistic, the subjective interpretation of probability has\ud
to be given up in order to get right confirmation, and thus\ud
scientific reasoning in general

Huber, Franz

English

Name not available

  143Degrees of Belief as Basis for Scientific Reasoning? Franz Huber, Konstanz 1. The Bayesian Approach to Scientific Reasoning Bayesianism is the claim that scientific reasoning is probabilistic, and that probabilities are adequately inter-preted as an agent’s actual subjective degrees of belief measured by her betting behaviour. Confirmation is one important aspect of scientific rea-soning. The thesis of this paper is the following: Given that scientific reasoning (and thus confirmation) is at all probabilistic, the subjective interpretation of probability has to be given up in order to get right confirmation, and thus scientific reasoning in general. This will be argued for as follows: First, an example will be considered which is an instance of a more general version of the problem of old evidence, POE. This suggests to look whether the existing solutions to POE provide a solution to the more general problem called C. The first result is that the existing solutions to POE are no genuine ones, because they do not provide a solution to C. More importantly, the attempts to solve C all have in common that they essentially depend on the agent’s absolutely first guess, her first degree of belief function p0. Therefore, C leads to the problem of prior probabilities, POPP. However, the standard solution to POPP – the “washing out of priors” relying on convergence to certainty and merger of opinion – is not applicable here, because the solutions to C never get rid of the agent’s first degree of belief function p0. By the subjective interpretation of probability, p0 is any arbitrary assignment of values in [0,1] to the atomic propositions of the underlying language. Thus, by choosing an appropriate p0 one can obtain more or less any degree of confirmation. In case evidence E is known and logically implied by hypothesis H and background knowledge B, the degree of confirmation is even uniquely determined by the agent’s first guesses in H and E. The only way out is some kind of objective or logical pro-bability function the agent could adopt as her first degree of belief function p0. However, the difficulty of determining such a logical probability function just was the reason for turning to the subjective interpretation of probability. 2. Bayesian Confirmation Theory According to Bayesian confirmation theory, the agent’s degree of confirmation of hypothesis H by evidence E relative to background knowledge B is measured by some function cp such that    > 0   ⇔  p(H | E∧B) > p(H | B) cp(H, E, B) = 0   ⇔ p(H | E∧B) = p(H | B)  < 0   ⇔  p(H | E∧B) < p(H | B), where p is the agent’s degree of belief function. Any such function cp is called a relevance measure (based on p). An example is the distance measure dp, dp(H, E, B) = p(H | E∧B) – p(H | B). 3. The Example An agent with degree of belief function p considers the hypothesis H = All Scots wear kilts. At time t1 she has the impression to see her friend Stephen wearing a kilt. As the agent is not wearing her glasses, her degree of belief in E = Stephen wears a kilt is not very high, say p1(E | B1) = .6, where p1 is her degree of belief function at t1. B1 is her background knowledge at that time containing the information that Stephen is Scot. Because of knowing that H and B1 logically imply E, the agent gets interested in whether Stephen is indeed wearing a kilt. So she takes on her glasses and has a careful second look at Stephen, who still seems to wear a kilt – this happening at time t2. In passing from t1 to t2 the only change in the agent’s degrees of belief is in E. Moreover, for some reason she cannot express her observation in terms of a proposition. So her degree of belief in E increases exogenously, say to p2(E | B2) = .9, where p2 is the agent’s degree of belief function at t2. Her background knowledge B2 at t2 is the same as at t1, because the only change is in E, and that change is exogenous, i.e. not due to any proposition on which the agent could condition. So B1 is logically equivalent to B2, B1 ≡ B2. 4. The Less Reliable the Source of Informa-tion, the Higher the Degree of Bayesian Confirmation Let us compare the agent’s degrees of confirmation at time t1 and at time t2. As the agent knows that H and B1 logically imply E (and does not forget this and that Stephen is Scot), pj(E | H∧Bj) = 1, for all points of time tj, j ≥ 0, even if it is not assumed that she is logically omniscient in the first sense that all logical truths are transparent to her (cf. Earman 1992, 122). Given Jeffrey conditionalisation (JC), i.e. assuming p1(H | ±E∧B1) = p2(H | ±E∧B2), it follows that H is more confirmed by E relative to B1 at t1 than (rela-tive to B2) at t2 if and only if the agent’s degree of belief in E at t1 is smaller than at t2, i.e. dp1(H, E, B1) > dp2(H, E, B2) ⇔ p2(E | B2) > p1(E | B1). Degrees of Belief as Basis for Scientific Reasoning? - Franz Huber    144 More generally, C dp1(H, E, B1) > dp2(H, E, B2) ⇔  p1(E | H∧B1) > p1(E | B1) and  p2(E | B2) > p1(E | B1) or p1(E | H∧B1) < p1(E | B1) and p2(E | B2) < p1(E | B1), where the only change in the agent’s degrees of belief in passing from t1 to t2 is exogenous and in E, whence B1 ≡ B2, and JC is used. Here and in the following the probabili-ties of all contingent propositions involved are assumed to be positive. C holds for the distance measure dp, the log-likelihood ratio lp, and the ratio measure rp, lp(H, E, B) = log[p(E | H∧B)/p(E | ¬H∧B)], rp(H, E, B) = log[p(H | E∧B)/p(H | B)]. The measure sp, sp(H, E, B) = p(H | E∧B) – p(H | ¬E∧B), is invariant w.r.t. exogenous belief changes in E (which yield B1 ≡ B2), i.e. sp1(H, E, B1) = sp2(H, E, B2). In case of cp, cp(H, E, B) = p(H∧E∧B)⋅p(B) – p(H∧B)⋅p(E∧B), something different (but not much better) holds: C’ cp1(H, E, B1) > cp2(H, E, B2) ⇔  p1(E | H∧B1) > p1(E | B1) and p1(E∧B1)/p2(E∧B2) > p2(¬E∧B2)/p1(¬E∧B1) or p1(E | H∧B1) < p1(E | B1) and p1(E∧B1)/p2(E∧B2) < p2(¬E∧B2)/p1(¬E∧B1). For the different measures and the problem of measure sensitivity cf. Fitelson 2001. 5. A More General Version of the Problem of Old Evidence C is a more general version of the problem of old evi-dence, POE. POE is that evidence E which is old in the sense of being assigned a degree of belief of 1 cannot provide any confirmation, since for any p, H, E and B: p(H | E∧B) = p(H | B), if p(E | B) = 1. POE is a problem, because there are historical cases where old evidence did provide confirmation (for an excellent discussion cf. chapter 5 of Earman 1992). And: If POE is a problem, then so is C.  This is important, because a Bayesian could simply refuse to consider C as counterintuitive. Is it not rational, she might say, that I take positively relevant E to provide the less confirmation for H, the more I already believe in E and have built this belief into my belief in H?1                                                      1 This point was made by Luc Bovens in personal correspondence. This reply is perfectly reasonable, but applies equally well to POE. However, a brief look at the literature shows that POE is taken to be a problem. Let us therefore look whether the existing solutions to POE give rise to a solution to C. Generally, there are two ways of approaching POE: 1) Conditioning on the entailment relation: Garber 1983 2) Counterfactual strategy: Howson and Urbach 1993 6. Conditioning on the Entailment Relation The idea here is to distinguish between a historical and an ahistorical POE, and to solve the former by noting that what increases [the agent]’s confidence in [H] is not E itself, but the discovery of some generally logical or mathematical relationship between [H] and E. (Garber 1983, 104) Then one shows that even if p(E | B) = 1,  the discovery that [H entails E] can raise [the agent]’s confidence in [H]. (Garber 1983, 123) Conditioning on the entailment relation does not provide a solution to C, for in the example the agent is interested in E just because of knowing that H and B1 logically imply E (and does not forget this and that Stephen is Scot), whence pj(H entails E | Bj) = 1, for every point of time tj, j ≥ 0. Moreover, by substituting ‘H entails E’ for E one gets another instance of C. 7. The Counterfactual Strategy Concerning POE, Howson and Urbach write: the support of [H] by E is gauged according to the effect which one believes a knowledge of E would now have on one’s degree of belief in [H], on the (counter-factual) supposition that one does not yet know E. (Howson and Urbach 1993, 404-405) Suppose B – E is the logically weakest proposition such that (B – E)∧E ≡ B, so that p(X | B – E) is the agent’s degree of belief in X “on the (counter-factual) supposition that [she] does not yet know E”. Then, if p(E | B) = 1, the agent’s degree of confirmation is given by d’p(H, E, B) = p(H | B) – p(H | B – E), “actual” – “counterfactual”. However, in case E is not known, it cannot be dropped from B. Therefore one has to generalize from the case of POE where p(E | B) = 1 to the case of C where p(E | B) need not be 1. The question is, of course, how the counterfactual strategy is adequately generalized. Apart from the above, there are the following (and uncountably many more) formulations of d’p(H, E, B): d’p(H, E, B) = p(H | (B – E)∧E)⋅p(E | B) + p(H | (B – E)∧¬E)⋅p(¬E | B) – p(H | B – E) Degrees of Belief as Basis for Scientific Reasoning? - Franz Huber    145 = p(H | (B – E)∧E)⋅p(E | B) – p(H | B – E)  = p(H | (B – E)∧E) – p(H | B – E)  = p(H | B∧E) – p(H | B – E) 8. Generalizing the Counterfactual Strategy Instead of considering the (counter-factual) supposition that one does not yet know E (Howson and Urbach 1993, 405)  the quote suggests to consider the (counter-factual) supposition that one does not yet believe in E to degree p(E | B). However, the background knowledge at t1 and at t2 is the same, because the change in the agent’s degree of belief in E is exogenous. Therefore one cannot just drop something (say, all information bearing on E) from B2 to get a counterfactual supposition B2 \ E which could play a role analogous to that of B2 – E in the special case where p2(E | B2) = 1. Instead, one really has to adopt a new probability function pE! Suppose pE(X | B) is the agent’s degree of belief in X on the counterfactual supposition that she does not yet believe in E  to degree p(E | B). Then there are the following (and uncountably many more) ways of generalizing d’: g1p(H, E, B) = pE(H | B∧E)⋅p(E | B) + pE(H | B∧¬E)⋅p(¬E | B) – pE(H | B) g2p(H, E, B) = pE(H | B∧E)⋅p(E | B) – pE(H | B) g3p(H, E, B) = pE(H | B∧E) – pE(H | B) g4p(H, E, B) = p(H | B∧E) – pE(H | B) g5p(H, E, B) = p(H | B) – pE(H | B) 9. The Result to Follow – and a Necessary and Sufficient Condition for it According to Bayesian intuitions, the result to follow is that H is more confirmed by E relative to B2 at t2 than (rela-tive to B1) at t1 if and only if the agent’s degree of belief in E at t2 is greater than at t1, i.e. cp2(H, E, B2) > cp1(H, E, B1) ⇔ p2(E | B2) > p1(E | B1), provided E is positively relevant for H given B1 (≡ B2). More generally, this means either DC or DA, depending on how one construes “positively relevant”: DC cp2(H, E, B2) > cp1(H, E, B1) ⇔   p1E(E | H∧B1) > p1E(E | B1) and p2(E | B2) > p1(E | B1) or p1E(E | H∧B1) < p1E(E | B1) and p2(E | B2) < p1(E | B1) DA cp2(H, E, B2) > cp1(H, E, B1) ⇔  p1(E | H∧B1) > p1(E | B1) and p2(E | B2) > p1(E | B1) or p1(E | H∧B1) < p1(E | B1) and p2(E | B2) < p1(E | B1). Before continuing, note that it is plausible to assume that counterfactual degrees of belief are stable over time, i.e. E p1E(H | B1) = p2E(H | B2). The reason is that in going from t1 to t2 the only change is exogenous and in E, and piE(H | Bi) is the agent’s degree of belief in H on the counterfactual supposition that she does not yet believe in E to degree pi(E | Bi). Interestingly, E sheds positive light on g1 and g5 (here and in the following the index of the background knowledge is dropped, because B1 ≡ B2): 1) E is necessary and sufficient for g1 to satisfy DC, assuming “counterfactual Jeffrey conditionalisation”, i.e.  p1E(H | ±E∧B) = p2E(H | ±E∧B), and 2) E is necessary and sufficient for g5 to satisfy DA, assuming JC. Moreover, E sheds negative light on g2-4: Given counter-factual JC, 3) E is necessary and sufficient for g2 to satisfy F, and 4) E is necessary and sufficient for g3 to satisfy GC. Given JC,  5) E is necessary and sufficient for g4 to satisfy GA. Here F cp2(H, E, B) > cp1(H, E, B) ⇔ p2(E | B) > p1(E | B), GC cp2(H, E, B) = cp1(H, E, B) = piE(H | B∧E) – piE(H | B), GA cp2(H, E, B) = cp1(H, E, B) = pi(H | B∧E) – piE(H | B). F is odd, because it does not matter whether E is positively relevant for H given B. GC and GA are odd for a Bayesian, because confirmation is invariant w.r.t. exogenous belief changes in E. All things considered it seems fair to say that the proper generalisation of d’ is g1 or g5. In order to get confirmation right they both require counterfactual degrees of belief to be stable over time. So g1 and g5 reduce to g1pi(H, E, B) = p0E(H | B∧E)⋅pi(E | B) + p0E(H | B∧¬E)⋅pi(¬E | B) – p0E(H | B), g5pi(H, E, B) = pi(H | B) – p0E(H | B). 10. Actual Degrees of Belief Whether or not the preceding generalisations are appropriate, they are not satisfying, because it remains questionable how pE(X | B) is determined and related to the agent’s actual degree of belief function p(X | B). This question being unanswered, the counterfactual strategy is concluded to provide no genuine solution to C either. Let us therefore consider an account solely in terms of actual degrees of belief (and providing a possible answer to the mentioned question). Generally, the example in section 3 is one where E is positively relevant for H given B, and the agent’s degree of belief in E changes exogenously as time goes by. If there is an increase (decrease) in the agent’s degree of belief in Degrees of Belief as Basis for Scientific Reasoning? - Franz Huber    146 E, her degree of belief in H increases (decreases), too; and conversely, if E is negatively relevant for H given B. All Bayesian accounts of confirmation measure in some way the difference between p(H | E∧B) and p(H | B). Given Bayes or strict conditionalisation, this is just the difference between the agent’s prior and posterior degree of belief in H when she learns E and nothing else. The counterfactual strategy measures the difference between the agent’s actual or posterior degree of belief in H and her counterfactual one – the latter replacing her prior. The reason is that the prior and posterior degrees of belief coincide if E was already known. Solving C requires something more general, because there one does not learn or know E; there is only a change in the agent’s degree of belief in E. This suggests to consider the agent’s prior and posterior degree of belief in H when the only change is exogenous and in E. However, one cannot simply take the difference between pi(H | B) and pi-1(H | B). (B is the same, because all changes are exogenous.)  For suppose the agent’s degree of belief in E increases enormously in going from ti-2 to ti-1, say from pi-2(E | B) = .01 to pi-1(E | B) = .9; and then it increases again in going to ti, but only slightly, say to pi(E | B) = .91. Then the difference between pi-2(H | B) and pi-1(H | B) is much greater than the difference between pi-1(H | B) and pi(H | B). Consequently, the difference between the prior and posterior degree of belief in H at ti-1 is much greater than that at ti, although the agent’s degree of belief in E at ti-1 is smaller than at t2, i.e. pi(H | B) – pi-1(H | B) < pi-1(H | B) – pi-2(H | B) and pi(E | B) > pi-1(E | B), where E is positively relevant for H given B, and all belief changes are exogenous. What one has to consider instead is the difference between the agent’s current degree of belief in H, pi(H | B), and her first degree of belief in H, p0(H | B), where the only change in going from t0 to ti is exogenous and in E. The proposal therefore is g6pi(H, E, B) = pi(H | B) – p0(H | B) = p0(H | E∧B)⋅pi(E | B) + p0(H | ¬E∧B)⋅pi(¬E | B) –     – p0(H | B) i times JC, which satisfies DA. g1, g5, and g6 coincide, if p0E(H | ±E∧B) = p0(H | ±E∧B) and p0E(H | B) = p0(H | B). 11. The Common Knock-Down Feature or Anything Goes All three measures g1, g5, and g6 have in common that their values essentially depend on the agent’s first degree of belief function p0. In case E is known and logically implied by H and B, the agent’s degree of confirmation of H by E relative to B at time ti (measured by g6) is even uniquely determined by her first guesses in E and H, p0(E | B) and p0(H | B)! Why the exclamation mark? First, because this shows that the idea behind any Bayesian theory of confirmation – namely to determine the degree of confirmation by the agent’s actual subjective degrees of belief – is shown to fail. Second, because – by the subjective interpretation – p0 is any arbitrary assignment of values in [0,1] to the atomic propositions of the underlying language, whence by choosing an appropriate p0 one can obtain more or less any degree of confirmation. 12. The Problem of Prior Probabilities Thus we are back at the problem of prior probabilities, the standard solution to which I take to be the “washing out of priors” relying on convergence to certainty and merger of opinion (cf. Earman 1992, esp. 57-59). However, the latter is not applicable here, because g6 and company never get rid of the agent’s first degree of belief function p0. The only way out is some kind of objective or logical probability function the agent could adopt as her p0. Yet the difficulty of determining such a logical probability function just was the reason for turning to the subjective interpretation. 2 3 References Earman, J. (ed.) 1983 Testing Scientific Theories, Minnesota Studies in the Philosophy of Science, vol. 10, Minnesota: University of Minnesota Press. Earman, J. 1992 Bayes or Bust? A Critical Examination of Bayesian Confirmation Theory, Cambridge, Mass.: The MIT Press. Fitelson, B. 2001 Studies in Bayesian Confirmation Theory, Univer-sity of Wisconsin-Madison: Dissertation. Garber, D. 1983 “Old Evidence and Logical Omniscience in Bayesian Confirmation Theory”, in J. Earman (ed.), Testing Scien-tific Theories, Minnesota Studies in the Philosophy of Science, vol. 10, Minnesota: University of Minnesota Press, 99-131. Howson, C. and Urbach, P. 1993 Scientific Reasoning: The Bayesian Approach, 2nd ed., Chicago/La Salle: Open Court.                                                      2 Research for this paper was supported by the Alexander von Humboldt Foundation, the Federal Ministry of Education and Research, and the Program for the Investment in the Future (ZIP) of the German Government. 3 I am grateful to Luc Bovens for comments on an earlier draft. 

Degrees of Belief as Basis for Scientific Reasoning?

Elektronisch archivierte Theorie - Sammelpunkt

  143
Degrees of Belief as Basis for Scientific Reasoning? 
Franz Huber, Konstanz 
1. The Bayesian Approach to Scientific 
Reasoning 
Bayesianism is the claim that scientific reasoning is 
probabilistic, and that probabilities are adequately inter-
preted as an agent’s actual subjective degrees of belief 
measured by her betting behaviour. 
Confirmation is one important aspect of scientific rea-
soning. The thesis of this paper is the following: Given that 
scientific reasoning (and thus confirmation) is at all 
probabilistic, the subjective interpretation of probability has 
to be given up in order to get right confirmation, and thus 
scientific reasoning in general. 
This will be argued for as follows: First, an example will 
be considered which is an instance of a more general 
version of the problem of old evidence, POE. This 
suggests to look whether the existing solutions to POE 
provide a solution to the more general problem called C. 
The first result is that the existing solutions to POE are 
no genuine ones, because they do not provide a solution 
to C. 
More importantly, the attempts to solve C all have in 
common that they essentially depend on the agent’s 
absolutely first guess, her first degree of belief function p0. 
Therefore, C leads to the problem of prior probabilities, 
POPP. However, the standard solution to POPP – the 
“washing out of priors” relying on convergence to certainty 
and merger of opinion – is not applicable here, because 
the solutions to C never get rid of the agent’s first degree 
of belief function p0. 
By the subjective interpretation of probability, p0 is any 
arbitrary assignment of values in [0,1] to the atomic 
propositions of the underlying language. Thus, by 
choosing an appropriate p0 one can obtain more or less 
any degree of confirmation. In case evidence E is known 
and logically implied by hypothesis H and background 
knowledge B, the degree of confirmation is even uniquely 
determined by the agent’s first guesses in H and E. 
The only way out is some kind of objective or logical pro-
bability function the agent could adopt as her first degree 
of belief function p0. However, the difficulty of determining 
such a logical probability function just was the reason for 
turning to the subjective interpretation of probability. 
2. Bayesian Confirmation Theory 
According to Bayesian confirmation theory, the agent’s 
degree of confirmation of hypothesis H by evidence E 
relative to background knowledge B is measured by some 
function cp such that 
   > 0   ⇔  p(H | E∧B) > p(H | B) 
cp(H, E, B) = 0   ⇔ p(H | E∧B) = p(H | B) 
 < 0   ⇔  p(H | E∧B) < p(H | B), 
where p is the agent’s degree of belief function. Any such 
function cp is called a relevance measure (based on p). 
An example is the distance measure dp, 
dp(H, E, B) = p(H | E∧B) – p(H | B). 
3. The Example 
An agent with degree of belief function p considers the 
hypothesis 
H = All Scots wear kilts. 
At time t1 she has the impression to see her friend Stephen 
wearing a kilt. As the agent is not wearing her glasses, her 
degree of belief in 
E = Stephen wears a kilt 
is not very high, say 
p1(E | B1) = .6, 
where p1 is her degree of belief function at t1. B1 is her 
background knowledge at that time containing the 
information that Stephen is Scot. 
Because of knowing that H and B1 logically imply E, the 
agent gets interested in whether Stephen is indeed 
wearing a kilt. So she takes on her glasses and has a 
careful second look at Stephen, who still seems to wear a 
kilt – this happening at time t2. 
In passing from t1 to t2 the only change in the agent’s 
degrees of belief is in E. Moreover, for some reason she 
cannot express her observation in terms of a proposition. 
So her degree of belief in E increases exogenously, say to 
p2(E | B2) = .9, 
where p2 is the agent’s degree of belief function at t2. Her 
background knowledge B2 at t2 is the same as at t1, 
because the only change is in E, and that change is 
exogenous, i.e. not due to any proposition on which the 
agent could condition. So B1 is logically equivalent to B2, 
B1 ≡ B2. 
4. The Less Reliable the Source of Informa-
tion, the Higher the Degree of Bayesian 
Confirmation 
Let us compare the agent’s degrees of confirmation at time 
t1 and at time t2. 
As the agent knows that H and B1 logically imply E (and 
does not forget this and that Stephen is Scot), 
pj(E | H∧Bj) = 1, for all points of time tj, j ≥ 0, 
even if it is not assumed that she is logically omniscient in 
the first sense that all logical truths are transparent to her 
(cf. Earman 1992, 122). 
Given Jeffrey conditionalisation (JC), i.e. assuming 
p1(H | ±E∧B1) = p2(H | ±E∧B2), 
it follows that 
H is more confirmed by E relative to B1 at t1 than (rela-
tive to B2) at t2 if and only if the agent’s degree of belief 
in E at t1 is smaller than at t2, i.e. 
dp1(H, E, B1) > dp2(H, E, B2) ⇔ p2(E | B2) > p1(E | B1). 
Degrees of Belief as Basis for Scientific Reasoning? - Franz Huber 
 
 
 144 
More generally, 
C dp1(H, E, B1) > dp2(H, E, B2) ⇔  
p1(E | H∧B1) > p1(E | B1) and  
p2(E | B2) > p1(E | B1) 
or 
p1(E | H∧B1) < p1(E | B1) and p2(E | B2) < p1(E | B1), 
where the only change in the agent’s degrees of belief in 
passing from t1 to t2 is exogenous and in E, whence B1 ≡ 
B2, and JC is used. Here and in the following the probabili-
ties of all contingent propositions involved are assumed to 
be positive. 
C holds for the distance measure dp, the log-likelihood 
ratio lp, and the ratio measure rp, 
lp(H, E, B) = log[p(E | H∧B)/p(E | ¬H∧B)], 
rp(H, E, B) = log[p(H | E∧B)/p(H | B)]. 
The measure sp, 
sp(H, E, B) = p(H | E∧B) – p(H | ¬E∧B), 
is invariant w.r.t. exogenous belief changes in E (which 
yield B1 ≡ B2), i.e. 
sp1(H, E, B1) = sp2(H, E, B2). 
In case of cp, 
cp(H, E, B) = p(H∧E∧B)⋅p(B) – p(H∧B)⋅p(E∧B), 
something different (but not much better) holds: 
C’ cp1(H, E, B1) > cp2(H, E, B2) ⇔  
p1(E | H∧B1) > p1(E | B1) and p1(E∧B1)/p2(E∧B2) > 
p2(¬E∧B2)/p1(¬E∧B1) 
or 
p1(E | H∧B1) < p1(E | B1) and p1(E∧B1)/p2(E∧B2) < 
p2(¬E∧B2)/p1(¬E∧B1). 
For the different measures and the problem of measure 
sensitivity cf. Fitelson 2001. 
5. A More General Version of the Problem 
of Old Evidence 
C is a more general version of the problem of old evi-
dence, POE. POE is that evidence E which is old in the 
sense of being assigned a degree of belief of 1 cannot 
provide any confirmation, since for any p, H, E and B: 
p(H | E∧B) = p(H | B), if p(E | B) = 1. 
POE is a problem, because there are historical cases 
where old evidence did provide confirmation (for an 
excellent discussion cf. chapter 5 of Earman 1992). 
And: If POE is a problem, then so is C.  
This is important, because a Bayesian could simply 
refuse to consider C as counterintuitive. Is it not rational, 
she might say, that I take positively relevant E to provide 
the less confirmation for H, the more I already believe in E 
and have built this belief into my belief in H?1 
                                                     
1 This point was made by Luc Bovens in personal correspondence. 
This reply is perfectly reasonable, but applies equally 
well to POE. However, a brief look at the literature shows 
that POE is taken to be a problem. 
Let us therefore look whether the existing solutions to 
POE give rise to a solution to C. Generally, there are two 
ways of approaching POE: 
1) Conditioning on the entailment relation: Garber 1983 
2) Counterfactual strategy: Howson and Urbach 1993 
6. Conditioning on the Entailment Relation 
The idea here is to distinguish between a historical and an 
ahistorical POE, and to solve the former by noting that 
what increases [the agent]’s confidence in [H] is not E 
itself, but the discovery of some generally logical or 
mathematical relationship between [H] and E. (Garber 
1983, 104) 
Then one shows that even if p(E | B) = 1,  
the discovery that [H entails E] can raise [the agent]’s 
confidence in [H]. (Garber 1983, 123) 
Conditioning on the entailment relation does not provide a 
solution to C, for in the example the agent is interested in 
E just because of knowing that H and B1 logically imply E 
(and does not forget this and that Stephen is Scot), 
whence 
pj(H entails E | Bj) = 1, for every point of time tj, j ≥ 0. 
Moreover, by substituting ‘H entails E’ for E one gets 
another instance of C. 
7. The Counterfactual Strategy 
Concerning POE, Howson and Urbach write: 
the support of [H] by E is gauged according to the effect 
which one believes a knowledge of E would now have 
on one’s degree of belief in [H], on the (counter-factual) 
supposition that one does not yet know E. (Howson and 
Urbach 1993, 404-405) 
Suppose B – E is the logically weakest proposition such 
that 
(B – E)∧E ≡ B, 
so that p(X | B – E) is the agent’s degree of belief in X “on 
the (counter-factual) supposition that [she] does not yet 
know E”. 
Then, if p(E | B) = 1, the agent’s degree of confirmation is 
given by 
d’p(H, E, B) = p(H | B) – p(H | B – E), 
“actual” – “counterfactual”. 
However, in case E is not known, it cannot be dropped 
from B. Therefore one has to generalize from the case of 
POE where p(E | B) = 1 to the case of C where p(E | B) 
need not be 1. 
The question is, of course, how the counterfactual 
strategy is adequately generalized. Apart from the above, 
there are the following (and uncountably many more) 
formulations of d’p(H, E, B): 
d’p(H, E, B) = p(H | (B – E)∧E)⋅p(E | B) + p(H | (B – 
E)∧¬E)⋅p(¬E | B) – p(H | B – E) 
Degrees of Belief as Basis for Scientific Reasoning? - Franz Huber 
 
 
 145
 = p(H | (B – E)∧E)⋅p(E | B) – p(H | B – E) 
 = p(H | (B – E)∧E) – p(H | B – E) 
 = p(H | B∧E) – p(H | B – E) 
8. Generalizing the Counterfactual Strategy 
Instead of considering 
the (counter-factual) supposition that one does not yet 
know E (Howson and Urbach 1993, 405)  
the quote suggests to consider 
the (counter-factual) supposition that one does not yet 
believe in E to degree p(E | B). 
However, the background knowledge at t1 and at t2 is the 
same, because the change in the agent’s degree of belief 
in E is exogenous. Therefore one cannot just drop 
something (say, all information bearing on E) from B2 to 
get a counterfactual supposition B2 \ E which could play a 
role analogous to that of B2 – E in the special case where 
p2(E | B2) = 1. 
Instead, one really has to adopt a new probability 
function pE! Suppose pE(X | B) is the agent’s degree of 
belief in X on the counterfactual supposition that she does 
not yet believe in E  to degree p(E | B). 
Then there are the following (and uncountably many 
more) ways of generalizing d’: 
g1p(H, E, B) = pE(H | B∧E)⋅p(E | B) + pE(H | B∧¬E)⋅p(¬E | 
B) – pE(H | B) 
g2p(H, E, B) = pE(H | B∧E)⋅p(E | B) – pE(H | B) 
g3p(H, E, B) = pE(H | B∧E) – pE(H | B) 
g4p(H, E, B) = p(H | B∧E) – pE(H | B) 
g5p(H, E, B) = p(H | B) – pE(H | B) 
9. The Result to Follow – and a Necessary 
and Sufficient Condition for it 
According to Bayesian intuitions, the result to follow is that 
H is more confirmed by E relative to B2 at t2 than (rela-
tive to B1) at t1 if and only if the agent’s degree of belief 
in E at t2 is greater than at t1, i.e. 
cp2(H, E, B2) > cp1(H, E, B1) ⇔ p2(E | B2) > p1(E | B1), 
provided E is positively relevant for H given B1 (≡ B2). 
More generally, this means either DC or DA, depending on 
how one construes “positively relevant”: 
DC cp2(H, E, B2) > cp1(H, E, B1) ⇔   
p1E(E | H∧B1) > p1E(E | B1) and p2(E | B2) > p1(E | B1) 
or 
p1E(E | H∧B1) < p1E(E | B1) and p2(E | B2) < p1(E | B1) 
DA cp2(H, E, B2) > cp1(H, E, B1) ⇔  
p1(E | H∧B1) > p1(E | B1) and p2(E | B2) > p1(E | B1) 
or 
p1(E | H∧B1) < p1(E | B1) and p2(E | B2) < p1(E | B1). 
Before continuing, note that it is plausible to assume that 
counterfactual degrees of belief are stable over time, i.e. 
E p1E(H | B1) = p2E(H | B2). 
The reason is that in going from t1 to t2 the only change is 
exogenous and in E, and piE(H | Bi) is the agent’s degree 
of belief in H on the counterfactual supposition that she 
does not yet believe in E to degree pi(E | Bi). 
Interestingly, E sheds positive light on g1 and g5 (here and 
in the following the index of the background knowledge is 
dropped, because B1 ≡ B2): 
1) E is necessary and sufficient for g1 to satisfy DC, 
assuming “counterfactual Jeffrey conditionalisation”, i.e.  
p1E(H | ±E∧B) = p2E(H | ±E∧B), and 
2) E is necessary and sufficient for g5 to satisfy DA, 
assuming JC. 
Moreover, E sheds negative light on g2-4: Given counter-
factual JC, 
3) E is necessary and sufficient for g2 to satisfy F, and 
4) E is necessary and sufficient for g3 to satisfy GC. 
Given JC,  
5) E is necessary and sufficient for g4 to satisfy GA. 
Here 
F cp2(H, E, B) > cp1(H, E, B) ⇔ p2(E | B) > p1(E | B), 
GC cp2(H, E, B) = cp1(H, E, B) = piE(H | B∧E) – piE(H | 
B), 
GA cp2(H, E, B) = cp1(H, E, B) = pi(H | B∧E) – piE(H | 
B). 
F is odd, because it does not matter whether E is positively 
relevant for H given B. GC and GA are odd for a Bayesian, 
because confirmation is invariant w.r.t. exogenous belief 
changes in E. 
All things considered it seems fair to say that the proper 
generalisation of d’ is g1 or g5. In order to get confirmation 
right they both require counterfactual degrees of belief to 
be stable over time. 
So g1 and g5 reduce to 
g1pi(H, E, B) = p0E(H | B∧E)⋅pi(E | B) + p0E(H | 
B∧¬E)⋅pi(¬E | B) – p0E(H | B), 
g5pi(H, E, B) = pi(H | B) – p0E(H | B). 
10. Actual Degrees of Belief 
Whether or not the preceding generalisations are 
appropriate, they are not satisfying, because it remains 
questionable how pE(X | B) is determined and related to 
the agent’s actual degree of belief function p(X | B). This 
question being unanswered, the counterfactual strategy is 
concluded to provide no genuine solution to C either. 
Let us therefore consider an account solely in terms of 
actual degrees of belief (and providing a possible answer 
to the mentioned question). 
Generally, the example in section 3 is one where E is 
positively relevant for H given B, and the agent’s degree of 
belief in E changes exogenously as time goes by. If there 
is an increase (decrease) in the agent’s degree of belief in 
Degrees of Belief as Basis for Scientific Reasoning? - Franz Huber 
 
 
 146 
E, her degree of belief in H increases (decreases), too; 
and conversely, if E is negatively relevant for H given B. 
All Bayesian accounts of confirmation measure in some 
way the difference between 
p(H | E∧B) and p(H | B). 
Given Bayes or strict conditionalisation, this is just the 
difference between the agent’s prior and posterior degree 
of belief in H when she learns E and nothing else. 
The counterfactual strategy measures the difference 
between the agent’s actual or posterior degree of belief in 
H and her counterfactual one – the latter replacing her 
prior. The reason is that the prior and posterior degrees of 
belief coincide if E was already known. 
Solving C requires something more general, because 
there one does not learn or know E; there is only a change 
in the agent’s degree of belief in E. 
This suggests to consider the agent’s prior and posterior 
degree of belief in H when the only change is exogenous 
and in E. 
However, one cannot simply take the difference between 
pi(H | B) and pi-1(H | B). 
(B is the same, because all changes are exogenous.)  
For suppose the agent’s degree of belief in E increases 
enormously in going from ti-2 to ti-1, say from 
pi-2(E | B) = .01 to pi-1(E | B) = .9; 
and then it increases again in going to ti, but only slightly, 
say to 
pi(E | B) = .91. 
Then the difference between 
pi-2(H | B) and pi-1(H | B) 
is much greater than the difference between 
pi-1(H | B) and pi(H | B). 
Consequently, the difference between the prior and 
posterior degree of belief in H at ti-1 is much greater than 
that at ti, although the agent’s degree of belief in E at ti-1 is 
smaller than at t2, i.e. 
pi(H | B) – pi-1(H | B) < pi-1(H | B) – pi-2(H | B) 
and 
pi(E | B) > pi-1(E | B), 
where E is positively relevant for H given B, and all belief 
changes are exogenous. 
What one has to consider instead is the difference 
between the agent’s current degree of belief in H, pi(H | B), 
and her first degree of belief in H, p0(H | B), where the only 
change in going from t0 to ti is exogenous and in E. 
The proposal therefore is 
g6pi(H, E, B) = pi(H | B) – p0(H | B) 
= p0(H | E∧B)⋅pi(E | B) + p0(H | ¬E∧B)⋅pi(¬E | B) –  
   – p0(H | B) i times JC, 
which satisfies DA. 
g1, g5, and g6 coincide, if 
p0E(H | ±E∧B) = p0(H | ±E∧B) and p0E(H | B) = p0(H | B). 
11. The Common Knock-Down Feature or 
Anything Goes 
All three measures g1, g5, and g6 have in common that 
their values essentially depend on the agent’s first degree 
of belief function p0. 
In case E is known and logically implied by H and B, the 
agent’s degree of confirmation of H by E relative to B at 
time ti (measured by g6) is even uniquely determined by 
her first guesses in E and H, p0(E | B) and p0(H | B)! 
Why the exclamation mark? 
First, because this shows that the idea behind any 
Bayesian theory of confirmation – namely to determine the 
degree of confirmation by the agent’s actual subjective 
degrees of belief – is shown to fail. 
Second, because – by the subjective interpretation – p0 
is any arbitrary assignment of values in [0,1] to the atomic 
propositions of the underlying language, whence by 
choosing an appropriate p0 one can obtain more or less 
any degree of confirmation. 
12. The Problem of Prior Probabilities 
Thus we are back at the problem of prior probabilities, the 
standard solution to which I take to be the “washing out of 
priors” relying on convergence to certainty and merger of 
opinion (cf. Earman 1992, esp. 57-59). 
However, the latter is not applicable here, because g6 
and company never get rid of the agent’s first degree of 
belief function p0. 
The only way out is some kind of objective or logical 
probability function the agent could adopt as her p0. 
Yet the difficulty of determining such a logical probability 
function just was the reason for turning to the subjective 
interpretation. 2 3 
References 
Earman, J. (ed.) 1983 Testing Scientific Theories, Minnesota 
Studies in the Philosophy of Science, vol. 10, Minnesota: University 
of Minnesota Press. 
Earman, J. 1992 Bayes or Bust? A Critical Examination of 
Bayesian Confirmation Theory, Cambridge, Mass.: The MIT Press. 
Fitelson, B. 2001 Studies in Bayesian Confirmation Theory, Univer-
sity of Wisconsin-Madison: Dissertation. 
Garber, D. 1983 “Old Evidence and Logical Omniscience in 
Bayesian Confirmation Theory”, in J. Earman (ed.), Testing Scien-
tific Theories, Minnesota Studies in the Philosophy of Science, vol. 
10, Minnesota: University of Minnesota Press, 99-131. 
Howson, C. and Urbach, P. 1993 Scientific Reasoning: The 
Bayesian Approach, 2nd ed., Chicago/La Salle: Open Court. 
                                                     
2 Research for this paper was supported by the Alexander von Humboldt 
Foundation, the Federal Ministry of Education and Research, and the Program 
for the Investment in the Future (ZIP) of the German Government. 
3 I am grateful to Luc Bovens for comments on an earlier draft. 


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