Philosophy of Science
The Problem of Old Evidence
Why can't known facts confirm new theories? A paradox that strikes at the heart of Bayesian reasoning.
In 1915, Albert Einstein completed his General Theory of Relativity. One of its first triumphs was explaining why Mercury's orbit precesses slightly more than Newtonian mechanics predicts.
This was celebrated as a stunning confirmation of Einstein's theory. But there's a problem.
Mercury's orbital anomaly had been known since 1859.
How can known facts confirm new theories?
According to Bayesian confirmation theory, evidence E confirms hypothesis H if and only if:
The probability of H given E is greater than the prior probability of H
But here's the catch: if you already know E is true, then P(E) = 1. And when P(E) = 1, Bayes' theorem gives us:
No confirmation! The posterior equals the prior.
This means old evidence can never confirm new theories.
But scientists routinely use known facts to support new theories. Something has gone wrong.
The Bayesian Update Calculator
Explore how Bayesian confirmation works. Toggle between new and old evidence to see why known facts fail to confirm hypotheses.
Bayes' Theorem
Prior P(H)
10.0%
Posterior P(H|E)
30.0%
E confirms H (posterior > prior)
When evidence is new (P(E) < 1), surprising successful predictions boost the probability of a theory. But when evidence is old (P(E) = 1), the math forces the posterior to equal the prior.
The Mercury Perihelion
The most famous example of the old evidence problem comes from Einstein's General Relativity and its explanation of Mercury's anomalous orbit.
Le Verrier discovers anomaly
Mercury's orbit precesses 43 arcseconds per century more than Newtonian mechanics predicts.
The Paradox
By 1915, the Mercury anomaly was common knowledge among physicists. When Einstein showed that General Relativity explained it, this was celebrated as a major confirmation of the theory.
But according to Bayesian confirmation theory: since P(Mercury anomaly) = 1 for Einstein and his contemporaries, the anomaly should not have confirmed GR at all!
“This discovery was, I believe, by far the strongest emotional experience in Einstein's scientific life... He told me that for several days he was beside himself with joyous excitement.”
- Abraham Pais, Einstein biographer
Einstein himself considered this explanation to be one of the most powerful confirmations of his theory. Yet according to standard Bayesianism, it should have provided zero confirmation.
Old vs. New Evidence
Compare how Bayesian confirmation treats established facts versus novel predictions. The asymmetry reveals the paradox in stark terms.
New Evidence (Novel Prediction)
Gravitational lensing during 1919 eclipse
P(E) before test
0.1
Uncertain - could go either way
Confirmatory power
Strong confirmation: P(H|E) >> P(H)
Before the 1919 Eddington expedition, no one knew if starlight would bend around the sun. This was a genuine test.
1919 Eclipse Expedition
Mercury Perihelion
Both pieces of evidence seem equally important for confirming GR. But standard Bayesianism says only the eclipse confirms it!
Intuitively, both the Mercury perihelion and the 1919 eclipse observation should confirm General Relativity. But Bayesianism privileges the eclipse because its outcome was uncertain beforehand.
The Core Tension
Bayesian confirmation theory is supposed to capture how evidence supports theories. But it implies that the timing of when you learned a fact matters more than the fact itself. This seems to confuse psychology with epistemology.
How P(E) Affects Confirmation
The more surprising the evidence, the stronger the confirmation. Watch how confirmation strength drops to zero as P(E) approaches 1.
P(E|H) = 0.95 (fixed)
Watch how confirmation strength varies with P(E):
Notice how confirmation strength decreases as P(E) increases. When P(E) = 1 (old evidence), the posterior exactly equals the prior: no confirmation at all.
This is sometimes called the “predictive advantage” of surprising evidence. The less expected the evidence, the more it confirms a theory that predicts it.
This visualization shows why Bayesianism values risky predictions. A theory that predicts something surprising (low P(E)) gets a bigger boost when the prediction succeeds than a theory that “predicts” something we already knew.
The Counterfactual Solution
One natural response: imagine you didn't know E. What probability would you assign? Use that hypothetical probability to calculate confirmation.
The Counterfactual Move
Imagine you didn't know E. What would your probabilities be?
P(E)
???
P(H|E)
???
This approach is intuitive but philosophically problematic. It requires us to reason about hypothetical probability assignments that we never actually held, and there's no principled way to determine what those should be.
Proposed Solutions
Philosophers have proposed various solutions to the problem. Each captures something important, but none is universally accepted. Explore the major approaches:
Garber's Solution
Daniel Garber, 1983
The Idea
What confirms GR is not the Mercury anomaly itself, but learning that GR explains the anomaly. The new evidence is: "GR implies E".
The Problem
This changes the subject. We wanted to know if E confirms H, not if "H implies E" confirms H. Also requires a strange kind of logical learning.
Each proposed solution captures something important, but none fully resolves the paradox. The problem of old evidence remains one of the deepest challenges to Bayesian epistemology.
What the solutions share
All solutions recognize that there's something right about using old evidence to support theories. They try to modify Bayesianism to accommodate this intuition.
Why none fully succeeds
Each solution either changes what we mean by “confirmation,” introduces problematic counterfactuals, or abandons core Bayesian principles.
Why This Matters
The problem of old evidence isn't just a technical puzzle for philosophers. It reveals deep tensions in how we think about scientific reasoning.
Scientific Practice
Scientists regularly cite known phenomena as evidence for new theories. If Bayesianism can't account for this, it fails to describe actual scientific reasoning.
Theory Choice
How should we choose between competing theories? If explanatory success with old evidence doesn't count, we lose a major criterion for theory evaluation.
Machine Learning
Bayesian methods in AI face similar issues. How should models update on data that was used in their training? The old evidence problem has practical implications.
Rational Belief
If our best formal theory of confirmation fails in basic cases, what does this say about the nature of rational belief? Maybe confirmation is more complex than probability calculus suggests.
The Status of the Problem
The problem of old evidence, first identified by Clark Glymour in 1980, remains unsolved. It stands as one of the most significant challenges to Bayesian epistemology, the dominant framework for understanding confirmation in philosophy of science.
Perhaps confirmation is not purely about probability.
Explanation, unification, and logical relations may play roles that Bayes' theorem alone cannot capture.
Explore More Philosophy of Science
The problem of old evidence connects to deep questions about confirmation, explanation, and scientific reasoning. Discover more puzzles that challenge our understanding of knowledge.
Reference: Glymour, C. (1980). Theory and Evidence