# Bayes's Theorem and the Problem of the "Old evidence"

## (December 1995)

### contents

What is Bayesianism?
The "Old Evidence" Problem
Bibliography

Recently so called "Bayesianism" is getting popular as a formalization of scientific reasoning. However there have been many objections to Bayesianism, and Clark Glymour added a new one, namely the "old evidence" problem (1980). The purpose of this paper is to examine Glymour's and his opponents' arguments and suggest a new solution to the problem.

Bayesianism claims that Bayes's theorem gives a formal structure to inductive logic. Bayes's theorem ( BT) governs the relationship between a theory and its evidence;

(BT) P(T|E)=P(T)P(E|T)/P(E)

P(T) is called the prior probability, the probability of the theory before we get the evidence; P(E) is the expectedness of the evidence, the degree the evidence is likely to happen; P(E|T) is the likelihood of E, given that the theory T is true; P(T|E) is called the posterior probability of T, the probability of T given that E has happened. If the theory predicts that something unexpected (P(E) is low) will occur ( P(E|T) is high), and if this really happens, the theory will increase its authenticity (so P(T|E)>P(T)); and the more the prediction is surprising ( P(E) gets closer to 0), the stronger support this will give to the theory (P(T|E) gets closer to 1) These results seem to agree with our intuition of the relation between a theory and its evidence. This theorem can be easily proved from axioms of probability and the definition of conditional probability.
Another important claim of Bayesianism is that these probabilities should be regarded as subjective, though rational. with this claim Bayesianism can avoid the problem of assigning prior probability to a theory. We can assign this value according to our intuitive, personal degree of belief (though, since it is probability, it should obey axioms of probability, so, for example, it should be between 0 and 1; usually it also should not be 1 or 0; if P(T) = 0, automatically P(T|E) = 0 and thus no evidence can support the theory; if P(T) = 1, every other alternative theories should have probability 0). Such arbitrariness may seem to be against our intuition about the objectivity of scientific theories. However, even if we start from a variety of prior probabilities, we can get very close posterior probabilities by accumulating evidence (as likewise can be proven from Bayes's theorem). This result also agrees with our intuition of scientific practices.

In spite of these advantages, Bayesianism has also several difficulties. One of the major difficulties is put forward by Clark Glymour (1980, 85-93), and called "old evidence" problem. In the history of science, we can find many examples of a theory seemingly being confirmed by evidence already known before the theory was proposed. For example, the anomalous advance of the perihelion of Mercury is usually considered as strong support for the general relativity theory. When Einstein finally established his general relativity theory in November 1915, this phenomenon was a well established fact (Glymour 1980,88). Several attempts were made to explain this phenomenon by Newtonian physics or the special relativity theory, but they were not successful (Earman 1992, 132). In fact, one of Einstein's motivations in constructing the general relativity theory was to explain this anomaly (ibid., 123). He was not sure if his final version of field equations could completely explain the phenomenon in November 1915 (ibid., 115), and this relation was established later. The general relativity theory has other factual supports such as the phenomena of light bending by gravity (and this is really a novel fact that the general relativity theory predicted), but the perihelion phenomenon is considered as the strongest support for the theory (Brush 1989). There are similar relationships between Newton's law of gravity and Kepler's laws, or the special relativity theory and Michelson-Morley experiment. In fact, these old evidences are regarded as crucial for the acceptance of the theories.
Surprisingly enough, this common sense relation is hard to explain if we take (BT) literally. First, we should assign probability 1 to an old evidence E (P(E)=1), because we know that it happened already; and the probability of E under any theory T also should be 1 (P(E|T)=1). (This last requirement needs a little explanation. As a theorem of probability, there is a relation P(E)=P(T)P(E|T)+P(notT)P(E|notT) Here no probabilities can be greater than 1, and P(T)+P(notT)=1. If P(E)=1, to meet all these conditions both P(E|T) and P(E|notT) should be 1.) If we put these values into (BT), the posterior probability of T and prior probability of T become the same (P(T|E)=P(T)). This means that the old evidence has nothing to do with the authenticity of the theory and therefore cannot confirm it. Thus, against the judgments by most scientists, the perihelion phenomenon cannot support the general relativity theory. This seems ridiculous. The result is surprising because Bayesianism is supposed to explain the relationship between theory and evidence very well.

There are several answers from Bayesians, and most typical answers are Garber's (1983) and Howson's (1985,1991).
Garber's answer admits that old evidence will not confirm a new theory. But when we formulate a new theory, we get new knowledge also, that is, knowledge that the theory (together with other assumptions) implies the evidence. This relation of implication (or explanation) is derivable from the knowledge we already have, so if we were logically omniscient, it would not be new to us. Garber says that the logical omniscience assumption is false and unnecessary. If we admit that we learn this kind of logical facts, the support from old evidence can be explained by such learning. There is a supportive observation for this answer. If we deliberately make the new theory to explain the evidence, i. e. when we already know the logical relation between the theory and the evidence, it is natural that the evidence does not improve the plausibility of the theory.
Howson, on the other hand, suggests a revision of Bayesianism. He says that when we evaluate whether an old evidence E confirm a new theory T, we should reassign the prior probability of T and the expectedness of E as if we did not know the evidence. We should imagine, for example, if we did not know the result of the Michelson-Morley experiment, what probabilities should be assigned to the result and to the special relativity theory. And if we put these values into (BT), we will find that this experiment strongly confirms the special relativity theory.
Both answers pose several problems. One objection to Garber concerns consistency with the axioms of probability (Chihara 1987). Bayesianism requires that our subjective probabilities obey axioms of probability. For example, one of the axioms states that if A is a logical truth, then P(A)= 1. If we are not logically omniscient, we cannot employ this axiom. Another objection maintains that there are cases in which we already know the logical relation, and at the same time the evidence support the theory. Even if Einstein had known that the general relativity theory entails the perihelion phenomenon, this phenomenon nevertheless would have increased the authenticity of the general relativity theory. The situation is clearer when we take other people into account. As Earman points out (Earman 1989, 333), for most of us, the relation between the perihelion phenomenon and the general relativity theory is the first thing we learn about that theory. And nevertheless it plays an important role in our acceptance of the theory.
Howson's answer also encounters several problems. One is already pointed out by Glymour himself (Glymour 1980, Chihara 1987). There is something wrong with the idea of "counterfactual degree of belief". How can we find such a degree? If we simply subtract E from our knowledge, that is not enough, for we can derive E from other related knowledge. Imagine that we do not know the result of the Michelson- Morley experiment, but we know the reaction from scientists after the experiment; we can easily guess what happened in the experiment. We should subtract these related knowledge, but to what extent? It is not easy to answer this question, and Howson does not show a convincing answer. In addition to this formal problem, there is a further problem with the applicability of Howson's answer to historical cases (Garber 1983, 103). When we explain a historical case such as Einstein's, we are dealing with actual degrees of belief of Einstein and other scientists. With these actual degrees (including, of course, the knowledge of the old evidence), how did they reach their decisions on the authenticity of the general relativity theory?

Before starting my reply, I should modify the question. Actually we have no reason to assume that the probability of old evidence is 1. It is possible that our memories are incorrect, or we made systematic mistake in measuring, or there is even a possibility of collective hallucination. By counting these possibilities in, the probability of old evidence E will become, say, P(E)=0.995. By this modification, technically the "old evidence" problem vanishes (Earman 1992, 121). However, as Earman argues, this is not so helpful in itself (ibid.). To the degree that the P(E) is very high, E cannot give strong support to T (one can check the reason by putting the value into (BT) and calculating. Even if one assumes P(E|T) = 1, the prior and posterior probabilities remain almost the same). Now, the question is this; with a very high probability for E, how can we get a strong support to T? This small modification is nevertheless a necessary part of my argument. The reason becomes clear later.
My own answer is a revision of Garber's answer by way of introducing another factor. First of all, the old evidence situation is not a single situation. At least we can distinguish four different situations as set forth below. (Hereafter I discuss the situation in which theory T entails E, i. e., P(E|T)=1. This is for simplicity; the results can be easily expanded to more general cases, i. e., P(E|T)<1.)

(1) we already knew that T entails E before the theory formation, and E seems to confirm T.
(2) we already knew that T entails E before the theory formation, and E does not seem to confirm T.
(3) we learned that T entails E after the theory formation, and E seems to confirm T.
(4) we learned that T entails E after the theory formation, and E does not seem to confirm T.

First, consider the difference between situations (1) and (2). In the case of the general relativity theory, no other theory was available to explain the perihelion phenomenon. Thus, even if Einstein had known that his theory entails that phenomenon, the theory would have had an advantage relative to other theories. This is the situation envisaged in (1). On the other hand, the general relativity theory explained not only the perihelion phenomenon but also classic phenomena such that planetary orbits are almost ellipse. These phenomena do not seem to confirm the general relativity theory because the Newtonian physics can also explain them very well. This is the situation in (2). Garber does not recognize situation (1). What is the difference between (1) and (2)? My answer is the difference depends on the existence of other available theories.
This same difference exists between situations (3) and (4). We have already seen many examples of (3). To know when (4) applies, suppose that there was a phenomenon which was well known but was not explained when the general relativity theory was established; and suppose that later it was found that it can be explained by Newtonian physics, and thus also explained by the general relativity theory. This phenomenon will not confirm general relativity theory at all, for the same reason as situation (2). This is the situation in (4).
Is there any difference between the confirmations in (1) and those in (3)? Yes. Generally speaking, when we do not expect the entailment, the discovery of it increases the authenticity of the theory remarkably. So the confirmatory power is stronger in (3) than in (1).
Can we deal with all these results in terms of Bayesianism? I think we can.
First, the difference between (1) and (3) is already explained by Garber. The difference does not come from the evidence E itself, but rather from other evidence about the relation between T and E. Among the objections I considered above, the last one (there are cases where E seems to confirm T and, nevertheless, we already know the relation ) can be answered if I can give (1) an independent support. I present the argument later. How about the logical omniscience problem? Eells gives an answer to it (Eells 1990, 215-216). When we use subjective probabilities, it is not our logical competency that we should change, but rather the interpretation of the axioms of probability. Take the axiom "if A is a logical truth, then P(A)= 1". This should be interpreted as follows; "if I know A is a logical truth, then I should assign P(A)=1". Here there is no need for logical omniscience. I think this answer is reasonable. Secondly, when we compare (1) and (3) with (2) and (4), actually the "old evidence" problem disappear. To explain this I would like to introduce a new concept, "relative confirmation". Usually in Bayesianism we think that the evidence confirms the theory iff P(T|E)>P(T). This is not necessarily the only path to confirmation; at least this is not a part of the "hard core" of Bayesianism. My "relative confirmation" is defined as follows:

(RC) An evidence E relatively confirms a theory T iff P(T|E) is not less than P(T) and for any alternative theory Ti, P(Ti|E)<P(Ti).

Suppose we consider only three theories (T1, T2, T3); and suppose that T1 is relatively confirmed as compared to T2 and T3. This means that the probability of T1 remains the same as before, and probabilities of T2 and T3 have decreased because of E. In this case, even if the degree of belief in T1 does not increase, we will sense that it has increased. This can be put in another way: the proportion of the degree of belief in T1 has increased relative to other theories. Bayesians can admit that this is a kind of support. For we now have stronger reason than before to choose T1 instead of T2 or T3.
Old evidence can give this kind of relative support to a theory. Suppose T1 is the general relativity theory and T2 is Newtonian physics. T1 entails E (here, the perihelion phenomenon), so P(E|T1)=1. With T2 it is hard to explain E, but since there may be some overlooked factors which help T2, so let us estimate P(E|T2)=0.25. P(E) is almost 1; say, P(E)=0.995 ( by the way, if P(E)=1, it is impossible that P(E|T2)=0.25, as I showed above; this is why I modified the question at the biginning of this section). With these values, we can calculate that P(T1|E) is approximately 1.005 P(T1), and P(T2|E) is approximately 0.251 P(T2). This means the probability of T2 reduces, while the probability of T1 remains almost the same as before. The same thing happens between the general relativity theory and the special relativity theory. Only these three theories were seriously considered at the time. This remarkable increase of relative authenticity of the general relativity theory is the reason scientists accepted it. Such support can be very strong, as we saw above.
To summarize, my answer proposes to supplement Garber's weak point by introducing the notion of "relative confirmation". I think this answer is also an approximation of the correct account, but hopefully this supplement strikes out for the right direction.

Brush, S. G. (1989) "Prediction and theory evaluation: the case of light bending", Science 246, 1124-1129.
Chihara, C. S. (1987) "Some problems for Bayesian confirmation theory", The British Journal for the Philosophy of Science 38, pp. 551-60.
Earman, J. (1989) " Old evidence, new theories: two unresolved problems in Bayesian confirmation theory", Pacific philosophical Quarterly 70, 323-340.
--(1992) Bayes or Bust? : a critical examination of Bayesian confirmation theory, The MIT Press.
Eells, E. (1990) "Bayesian problems of old evidence", in C. Wade Savage (ed). Scientific theories, Minnesota Studies in the Philosophy of Science, Vol. X, Minneapolis, University of Minnesota Press, 205-223.
Garber, D. (1983) "Old evidence and logical omniscience in Bayesian Confirmation theory", in J. Earman (ed). Testing Scientific Theories, Minnesota Studies in the Philosophy of Science, Vol. X, Minneapolis, University of Minnesota Press.
Glymour, C. (1980) Theory and Evidence, Princeton, Princeton UP.
Howson, C. (1985) "Some recent objection to the Bayesian theory of support", The British Journal for the Philosophy of Science 36, pp. 305-9.
--(1990) "Fitting your theory to the facts: probably not such a bad thing after all", in C. Wade Savage (ed). Scientific theories, Minnesota Studies in the Philosophy of Science, Vol. X, Minneapolis, University of Minnesota Press.
--(1991) "The 'old evidence' problem", The British Journal for the Philosophy of Science 42, pp. 547-55.
Jeffrey, R. C. (1983) "Bayesian with a human face", in J. Earman (ed). Testing Scientific Theories, Minnesota Studies in the Philosophy of Science, Vol. X, Minneapolis, University of Minnesota Press.
Mayo, D. G. (1991) "Novel evidence and severe tests" Philosophy of Science 58, 523-552.
Niiniluto, I. (1983) "Novel facts and Bayesianism", The British Journal for the Philosophy of Science 34, pp. 375-9.

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