Pierre-Simon Laplace posed the sunrise problem in his Essai philosophique sur les probabilites (1814) as a test case for inductive reasoning. The question seems almost absurd — of course the sun will rise tomorrow — yet it encapsulates one of the deepest problems in the philosophy of science: how do repeated observations justify predictions about the future? David Hume had argued in 1739 that induction has no logical foundation. Laplace's Bayesian answer was an attempt to put induction on a precise mathematical footing.
The problem has been debated for over two centuries, generating insights about the choice of prior, the meaning of probability, and the relationship between mathematical models and physical reality. It remains one of the most discussed thought experiments in Bayesian statistics.
Laplace's Rule of Succession
Laplace modeled the situation as follows. Let theta be the unknown probability that the sun rises on any given day. Assume a uniform prior p(theta) = 1 on [0,1], representing ignorance about theta. After observing n consecutive sunrises (and no failures), the posterior distribution for theta is Beta(n+1, 1), and the predictive probability of a sunrise on day n+1 is:
Derivation Prior: θ ~ Beta(1, 1) = Uniform(0, 1)
Likelihood: n successes in n trials → L(θ) = θⁿ
Posterior: θ | data ~ Beta(n+1, 1)
Predictive: P(next = 1 | data) = E[θ | data] = (n+1)/(n+2)
If the sun has risen every day for 5,000 years of recorded history — roughly n = 1,826,250 days — the probability it will rise tomorrow is 1,826,251 / 1,826,252, or about 0.999999452. Not certainty, but very close.
The Bayesian answer is never exactly 1 (unless we observe infinitely many sunrises), because the posterior always retains some weight on the possibility that theta is less than 1. This embodies Cromwell's rule: never assign probability 0 or 1 to any empirical proposition, because no finite amount of evidence can rule out alternatives with absolute certainty. The tiny residual uncertainty is a mathematical expression of humility in the face of induction.
The Problem with the Uniform Prior
The uniform prior on theta treats "the sun rises with probability 0.01" as just as plausible a priori as "the sun rises with probability 0.99." This is the weakest point of Laplace's analysis. In reality, we know a great deal about the sun — it is a main-sequence star with billions of years of fuel remaining — and a prior that concentrates near theta = 1 would be far more appropriate.
Posterior: θ | data ~ Beta(a + n, b)
Predictive: P(next sunrise) = (a + n) / (a + n + b)
Example: a = 10⁶, b = 1 P(next sunrise | n days) = (10⁶ + n) / (10⁶ + n + 1) ≈ 1 − 1/(10⁶ + n)
With an informative prior, the probability of sunrise is even closer to 1 and converges faster. But the choice of prior is precisely the point of the thought experiment: it forces us to confront what we know, what we assume, and how our assumptions affect our conclusions.
Philosophical Significance
Hume's Problem of Induction
David Hume argued that there is no logical basis for inferring that the future will resemble the past. The sun rising n times does not logically guarantee it will rise again. Laplace's rule of succession provides a probabilistic — not logical — bridge. It does not prove the sun will rise; it quantifies the degree of belief that it will, given the evidence and the prior. Whether this resolves Hume's challenge or merely restates it is debated.
Exchangeability and De Finetti
The sunrise problem's Bayesian solution assumes the daily outcomes are exchangeable. This is a substantive assumption: it means the order in which sunrises are observed does not matter. In a cosmological model where the sun has a finite lifespan, the outcomes are not exchangeable — later days are less likely to see a sunrise than earlier ones. The exchangeability assumption is not wrong, but it is a modeling choice, and different choices lead to different answers.
"We may judge that the probability of the sun rising tomorrow, for anyone ignorant of the fact that it is a star governed by physics, exceeds (n+1)/(n+2). For someone who knows the relevant physics, the calculation is unnecessary — but as a calibration of pure inductive inference from observation alone, it is unimprovable." — Pierre-Simon Laplace, Essai philosophique sur les probabilites (1814), paraphrased
Generalizations
Arbitrary Prior Parameters
The rule of succession (n+1)/(n+2) arises specifically from the uniform prior Beta(1,1). The general rule, for a Beta(a,b) prior after n successes and f failures, is (a+n)/(a+b+n+f). The uniform prior is one member of a family, and the sensitivity of the answer to the prior is itself informative about how much the data dominate inference.
Multiple Categories
Laplace also considered a generalized version with k possible outcomes. If one category has been observed n times out of n trials, the predicted probability of that category on the next trial is (n+1)/(n+k) under a symmetric Dirichlet prior. This generalization underpins additive (Laplace) smoothing in natural language processing and machine learning.
Where nⱼ → Number of times category j has been observed
n → Total number of observations
k → Number of categories
Laplace smoothing — the direct descendant of the sunrise problem's mathematics — is used daily in billions of computations. When a naive Bayes spam filter encounters a word it has never seen before, Laplace smoothing prevents the probability from being exactly zero. The rule of succession, conceived to address an 18th-century philosophical puzzle, turns out to be essential plumbing for 21st-century information technology.
The Limits of the Model
The sunrise problem also teaches a lesson about model limitations. The Beta-Bernoulli model treats each day as an independent coin flip with fixed probability theta. But the real question — will the sun rise? — involves astrophysics, not coin flips. The model is a mathematical abstraction of inductive reasoning, not a serious physical prediction. Laplace himself was well aware of this distinction: he used the problem to illustrate the logic of induction, not to make astronomical forecasts.
This tension — between the mathematical elegance of the model and the complexity of the real world — is a recurring theme in Bayesian statistics. The sunrise problem reminds us that models are tools for reasoning, and their conclusions are only as good as the assumptions that go into them.