Sunrise problem
Encyclopedia
The sunrise problem can be expressed as follows: "What is the probability that the sun will rise tomorrow?"
The sunrise problem illustrates the difficulty of using probability theory
when evaluating the plausibility of statements or beliefs.
According to the Bayesian
interpretation of probability
, probability theory can be used to evaluate the plausibility of the statement, "The sun will rise tomorrow." We just need a hypothetical random process that determines whether the sun will rise tomorrow or not. Based on past observations, we can infer
the parameters of this random process, and from there evaluate the probability that the sun will rise tomorrow.
, who treated it by means of his rule of succession
. Let p be the long-run frequency of sunrises, i.e., the sun rises on 100 × p% of days. Prior to knowing of any sunrises, one is completely ignorant of the value of p. Laplace represented this prior ignorance by means of a uniform probability distribution
on p. Thus the probability that p is between 20% and 50% is just 30%. This must not be interpreted to mean that in 30% of all cases, p is between 20% and 50%; that would be a frequentist approach to applied probability. Rather, it means that one's state of knowledge (or ignorance) justifies one in being 30% sure that the sun rises between 20% of the time and 50% of the time. Given the value of p, and no other information relevant to the question of whether the sun will rise tomorrow, the probability that the sun will rise tomorrow is p. But we are not "given the value of p". What we are given is the observed data: the sun has risen every day on record. Laplace inferred the number of days by saying that the universe was created about 6000 years ago, based on a literal reading of the Bible
. To find the conditional probability
distribution of p given the data, one uses Bayes theorem, which some call the Bayes-Laplace rule. Having found the conditional probability distribution of p given the data, one may then calculate the conditional probability, given the data, that the sun will rise tomorrow. That conditional probability is given by the rule of succession
. The plausibility that the sun will rise tomorrow increases with the number of days on which the sun has risen so far.
Laplace, however, recognised this to be a misapplication of the rule of succession through not taking into account all the prior information available immediately after deriving the result:
It is noted by Jaynes & Bretthorst (2003) that Laplace's warning had gone unheeded by workers in the field.
A reference class problem
arises: the plausibility inferred will depend on whether we take the past experience of one person, of humanity, or of the earth. A consequence is that each referent would hold different plausibility of the statement. In Bayesianism, any probability is a conditional probability
given what one knows. That varies from one person to another.
s every day, being the star that one sees in the morning. The plausibility of the "sun will rise tomorrow" (i.e., the probability of that being true) will then be the proportion of stars that do not "die", e.g., by becoming nova
e, and so failing to "rise" on their planets (those that still exist, irrespective of the probability that there may then be none, or that there may then be no observers).
One faces a similar reference class problem: which sample of stars should one use. All the stars? The stars with the same age as the sun? The same size?
Mankind's knowledge of star formations will naturally lead one to select the stars of same age and size, and so on, to resolve this problem. In other cases, one's lack of knowledge of the underlying random process then makes the use of Bayesian reasoning less useful. Less accurate, if the knowledge of the possibilities is very unstructured, thereby necessarily having more nearly uniform prior probabilities (by the principle of indifference
). Less certain too, if there are effectively few subjective prior observations, and thereby a more nearly minimal total of pseudocount
s, giving fewer effective observations, and so a greater estimated variance in expected value, and probably a less accurate estimate of that value.
The sunrise problem illustrates the difficulty of using probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
when evaluating the plausibility of statements or beliefs.
According to the Bayesian
Bayesian probability
Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...
interpretation of probability
Probability interpretations
The word probability has been used in a variety of ways since it was first coined in relation to games of chance. Does probability measure the real, physical tendency of something to occur, or is it just a measure of how strongly one believes it will occur? In answering such questions, we...
, probability theory can be used to evaluate the plausibility of the statement, "The sun will rise tomorrow." We just need a hypothetical random process that determines whether the sun will rise tomorrow or not. Based on past observations, we can infer
Statistical inference
In statistics, statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation...
the parameters of this random process, and from there evaluate the probability that the sun will rise tomorrow.
One sun, many days
The sunrise problem was first introduced in the 18th century by Pierre-Simon LaplacePierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...
, who treated it by means of his rule of succession
Rule of succession
In probability theory, the rule of succession is a formula introduced in the 18th century by Pierre-Simon Laplace in the course of treating the sunrise problem....
. Let p be the long-run frequency of sunrises, i.e., the sun rises on 100 × p% of days. Prior to knowing of any sunrises, one is completely ignorant of the value of p. Laplace represented this prior ignorance by means of a uniform probability distribution
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...
on p. Thus the probability that p is between 20% and 50% is just 30%. This must not be interpreted to mean that in 30% of all cases, p is between 20% and 50%; that would be a frequentist approach to applied probability. Rather, it means that one's state of knowledge (or ignorance) justifies one in being 30% sure that the sun rises between 20% of the time and 50% of the time. Given the value of p, and no other information relevant to the question of whether the sun will rise tomorrow, the probability that the sun will rise tomorrow is p. But we are not "given the value of p". What we are given is the observed data: the sun has risen every day on record. Laplace inferred the number of days by saying that the universe was created about 6000 years ago, based on a literal reading of the Bible
Bible
The Bible refers to any one of the collections of the primary religious texts of Judaism and Christianity. There is no common version of the Bible, as the individual books , their contents and their order vary among denominations...
. To find the conditional probability
Conditional probability
In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...
distribution of p given the data, one uses Bayes theorem, which some call the Bayes-Laplace rule. Having found the conditional probability distribution of p given the data, one may then calculate the conditional probability, given the data, that the sun will rise tomorrow. That conditional probability is given by the rule of succession
Rule of succession
In probability theory, the rule of succession is a formula introduced in the 18th century by Pierre-Simon Laplace in the course of treating the sunrise problem....
. The plausibility that the sun will rise tomorrow increases with the number of days on which the sun has risen so far.
Laplace, however, recognised this to be a misapplication of the rule of succession through not taking into account all the prior information available immediately after deriving the result:
It is noted by Jaynes & Bretthorst (2003) that Laplace's warning had gone unheeded by workers in the field.
A reference class problem
Reference class problem
In statistics, the reference class problem is the problem of deciding what class to use when calculating the probability applicable to a particular case...
arises: the plausibility inferred will depend on whether we take the past experience of one person, of humanity, or of the earth. A consequence is that each referent would hold different plausibility of the statement. In Bayesianism, any probability is a conditional probability
Conditional probability
In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...
given what one knows. That varies from one person to another.
One day, many suns
Alternatively, one could say that a sun is selected from all the possible starStar
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...
s every day, being the star that one sees in the morning. The plausibility of the "sun will rise tomorrow" (i.e., the probability of that being true) will then be the proportion of stars that do not "die", e.g., by becoming nova
Nova
A nova is a cataclysmic nuclear explosion in a star caused by the accretion of hydrogen on to the surface of a white dwarf star, which ignites and starts nuclear fusion in a runaway manner...
e, and so failing to "rise" on their planets (those that still exist, irrespective of the probability that there may then be none, or that there may then be no observers).
One faces a similar reference class problem: which sample of stars should one use. All the stars? The stars with the same age as the sun? The same size?
Mankind's knowledge of star formations will naturally lead one to select the stars of same age and size, and so on, to resolve this problem. In other cases, one's lack of knowledge of the underlying random process then makes the use of Bayesian reasoning less useful. Less accurate, if the knowledge of the possibilities is very unstructured, thereby necessarily having more nearly uniform prior probabilities (by the principle of indifference
Principle of indifference
The principle of indifference is a rule for assigning epistemic probabilities.Suppose that there are n > 1 mutually exclusive and collectively exhaustive possibilities....
). Less certain too, if there are effectively few subjective prior observations, and thereby a more nearly minimal total of pseudocount
Pseudocount
A pseudocount is an amount added to the number of observed cases in order to change the expected probability in a model of those data, when not known to be zero. Depending on the prior knowledge, which is sometimes a subjective value, a pseudocount may have any non-negative finite value...
s, giving fewer effective observations, and so a greater estimated variance in expected value, and probably a less accurate estimate of that value.
See also
- Doomsday argumentDoomsday argumentThe Doomsday argument is a probabilistic argument that claims to predict the number of future members of the human species given only an estimate of the total number of humans born so far...
: a similar problem that raises intense philosophical debate - Problem of inductionProblem of inductionThe problem of induction is the philosophical question of whether inductive reasoning leads to knowledge. That is, what is the justification for either:...
- Unsolved problems in statisticsUnsolved problems in statisticsThere are many longstanding unsolved problems in mathematics for which a solution has still not yet been found. The unsolved problems in statistics are generally of a different flavor; according to John Tukey, "difficulties in identifying problems have delayed statistics far more than difficulties...
Further reading
- Howie, David. (2002). Interpreting probability: controversies and developments in the early twentieth century. Cambridge University Press. pp. 24. ISBN 978-0521812511
- Ribera, David. (2004). Laplace's Probability of Sunrise (PDF)