April 25, 2024
Frequentists interpret probability as the number of times an outcome belongs to an event divided by the number of times it is sampled. This assumes it is possible to repeatedly sample identical situations. The law of large numbers says the frequency converges to the probablity of the event. This is fine when applied to flipping coins, rolling dice, or running other physical experiments where conditions don’t change, but it limits the use of probability.
Bayesians interpret probability as a subjective degree of belief based on available information. As more information becomes available it can be used to update individual degrees of belief. They are delayed satisfaction frequentists who believe subjective degrees of belief will converge for all individuals given sufficient common information over repeated trials.
A probability space is a set Ω of possible outcomes and a probability measure P that takes subsets of Ω to a number between 0 and 1. Measures satisfy P(E\cup F) = P(E) + P(F) - P(E\cap F) and P(\emptyset) = 0, E,F\subseteq Ω. Measures don’t count things twice and the measure of nothing is zero. Probability measures are non-negative and P(Ω) = 1. Subsets of a sample space are events. If the sample space is finite we only need to specify the probability of each outcome. The probability of each outcome must be between 0 and 1 and sum to 1. The probability of an event is P(E) = \sum_{ω \in E} P(\{ω\}).
The conditional probability of A given B is defined by P(A\mid B) = P(A\cap B)/P(B). It satifies P(B\mid B) = 1 and A\mapsto P(A\mid B) is a probability measure. Since P(A\mid B) = P(A\cap B)/P(B) and P(B\mid A) = P(B\cap A)/P(A) it follows P(A\mid B) = P(A\cap B)/P(B) = P(B\cap A)/P(B) = P(B\mid A)P(A)/P(B). P(A\mid B) = P(B\mid A)P(A)/P(B) This formula is the basis of Baysian probabilty. It shows how to update probabilities given new information. Bayesians interpret probability as a subjective degree of belief instead of a physical frequency of (potential?) occurance. Frequentists can flip a coin to their heart’s content but have nothing to say about the probability you will get hit by a car while crossing a particular steet at some specific time.
Bayesians like to belive that as more infomation becomes available subjective probabilites will converge to the same value. They haven’t come up with a general theory ensuring this, but that seems to frequently occur.
A coin is fair if heads and tails occur with equal probability when flipped1. Suppose a coin might be fair or might have two heads, but we cannot examine the coin directly. The only information we will be given is the outcomes of a series of flips. If we ever see tails we know the coin is not double-headed but if every flip we see is heads then that provides evidence the coin is double-headed.
Suppose the first flip of the coin is heads, H_1. A naive application of Bayes theorem is P(C|H_1) = P(H_1|C)P(C)/P(H_1), where C indicates the coin is fair. For a fair coin P(H_1|C) = 1/2 and for a double-headed coin P(H_1|D) = 1, where D indicates the coin is double-headed. The other probabilities require some assumptions.
Step one in probability theory is to specify the sample space of possible outcomes and the probability of events (subsets of the sample space). If the sample space is finite it is sufficient to specify the probability of each outcome.
In our case the sample space has two elements Ω = \{C,D\} indicating the coin is either fair or double-headed. Assume the probability that the coin is fair equals p so the probability of the coin being double-headed is 1 - p and P(H_1) = p/2 + (1 - p) = (1 - p)/2. Using Bayes formula, p_1 = P(F|H_1) = (1/2)p/((1 - p)/2) = p/(1 - p) < p if 0 < p < 1. Seeing a head flipped on the first try provides evidence against the coin being fair.
If you had trouble believing P(H_1) = p/2 + (1 - p) then you have good intuition. We were sloppy about specifying the sample space. (This happens quite often in the literature, for example, the Monte Hall problem.) The sample space should also include the outcome of the first toss Ω = \{(C,H), (C,T), (D,H), (D,T)\} = \{C,D\}\times \{H,T\}; the coin can be fair or double-headed and the first toss can be either heads or tails. The probability of each outcome is P(\{(C,H)\} = p/2, P(\{(C,T)\} = p/2, P(\{(D,H)\} = 1 - p, P(\{(D,T)\} = 0, where 0 < p < 1. Flipping heads on the first toss is the subset H_1 = \{(C,H),(D,H)\} so P(H_1) = P(\{(C,H),(D,H)\}) = P(\{(C,H)\}) + P(\{(D,H)\}) = p/2 + (1 - p).
If we want to model flipping the coin n times the sample space becomes Ω = \{C,D\}\times\{H,T\}^n and the probabilites are P(\{C,ω)\} = p/2^n for ω\in\{H,T\}^n, and P(\{D\}, ω) = 0 unless ω is all heads, denoted H_n. Since P(H_n) = p/2^n + (1 - p) Bayes Theorem gives p_n = P(C\mid H_n) = p/2^n/(p/2^n + (1 - p)) = p/(p + 2^n(1 - p)). As the number of consecutive heads goes to infinity the probability of the coin being fair approaches zero no matter the initial guess p > 0 of the coin being fair.
This is an example of Bayesian reasoning showing subjective probabilities converge given sufficient information. It is important to note that this depends on the model. A different model could allow for the possibility the coin might also be two-tailed. Every model makes assumptions about what information is available so all probabilities are conditional.
If you had trouble believing P(A\mid B) = P(A\cap B)/P(B) then you have good intuition. Mathematics hijacks standard vocabulary to define mathematical statements. Descartes and Spinoza seem to be the first scholars to attempt rigourous definitions of words beyond the limited vocabulary of and, or, not, and implies used in Aristotelian logic. Prior to his invention of a logic involving only true or false statements (propositions) the main effort was directed at determining valid methods of reasoning to arrive at plausible statements. This most likely originated early in our prehistory when shamans could fool people into giving them bear skins and arrow heads in exchange for empty promises.
Probability theory is an extension of logic. Instead of statements that are either false or true it assigns a probability between 0 and 1 to events. From its sordid beginnings in of games of chance it achieved mathematical respectability when Kolmogorov axiomatized it in 1933. (cite?) Reputable mathematicians could now slot Probability Theory into the existing Measure Theory they were comfortable with as positive measures having mass 1.
Mathematics is following your nose and thinking rigourously, which immediately leads to difficulties most people, rightfully so, think are Much Ado About Nothing. For example, probability 1 does not correspond to true and must be replaced with the more subtle notion of almost surely.
Richard Threlkeld Cox put conditional probabilty on firmer philosophical foundations by axiomatizing the notion of likelihood. Staying true to the earliest foundations, he considered statements instead of propositions. He denoted the likelihood of statement A given statement B by A\mid B and required consistency with standard symbolic logic.
If the shaman tells you he can make it rain tomorrow if you give him a basket of grain and you come back from a long day in the field to find him scurrying out of your hut and your wife with her hair mussed and won’t look you in the eye, how likely is it that he will deliver on his promise given this information? If history is any guide, he will likely tell you he now needs two baskets of grain.
Cox assumed likelihood is a real number. Real numbers are totally ordered so this is a big assumption. He was also vague on exactly what statements constitute information. His notation for the likelihood of statement A given statement/information B is A\mid B. He assumes the likelihood of statements A and B, denoted AB, given information C is AB\mid C = p(A\mid B, B\mid AC) for some function p. It can be shown that consistency with symbolic logic requires p(p(x,y),z) = p(x,p(y,z)) for any real numbers x, y, and z.
Cox wanted to show P(ST) = P(S)P(T\mid S) is a consequence of of more general assumptions. His original derivation was not mathematically correct, but it inspired others to improve his assumptions that utimately led to a precise formulation.
This was really an aside to his life’s work. He was an experimental physicist who, among other results, demontrated a parity violation for double scattering of β rays from radium that could not be explained by existing theory. Eventually theory caught up and proved him correct.
Keynes A Treatise on Probability - probability is not a total order.
Is our expectation of rain, when we start out for a walk, always more likely than not, or less likely than not, or as likely as not? I am prepared to argue that on some occasions none of these alternatives hold, and that it will be an arbitrary matter to decide for or against the umbrella. If the barometer is high, but the clouds are black, it is not always rational that one should prevail over the other in our minds, or even that we should balance them, though it will be rational to allow caprice to determine us and to waste no time on the debate.
maxplus algebras
[ { “id”: “http://zotero.org/users/6482136/items/H8KQMTYV”, “type”: “article-journal”, “abstract”: “By basing Bayesian probability theory on five axioms, we can give a trivial proof of Cox’s Theorem on the product rule and sum rule for conditional plausibility without assuming continuity or differentiablity of plausibility. Instead, we extend the notion of plausibility to apply to unknowns, giving them plausible values. Thus, we combine the best aspects of two approaches to Bayesian probability theory, namely the Cox-Jaynes theory and the de Finetti theory.”, “container-title”: “Bayesian Analysis”, “DOI”: “10.1214/09-BA422”, “ISSN”: “1936-0975”, “issue”: “3”, “journalAbbreviation”: “Bayesian Anal.”, “language”: “en”, “source”: “DOI.org (Crossref)”, “title”: “New axioms for rigorous Bayesian probability”, “URL”: “https://projecteuclid.org/journals/bayesian-analysis/volume-4/issue-3/New-axioms-for-rigorous-Bayesian-probability/10.1214/09-BA422.full”, “volume”: “4”, “author”: [ { “family”: “Dupré”, “given”: “Maurice J.” }, { “family”: “Tipler”, “given”: “Frank J.” } ], “issued”: { “date-parts”: [ [ “2009”, 9, 1 ] ] } }]
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