The differential of probabilistic entailment

doi:10.1016/j.apal.2021.102945

Annals of Pure and Applied Logic

Volume 172, Issue 6, June 2021, 102945

To Ettore Casari, in memoriam

https://doi.org/10.1016/j.apal.2021.102945 Get rights and content

Abstract

Suppose elements $a_{1}, \dots, a_{k}$ of a boolean algebra A are assigned probabilities $π_{1}, \dots π_{k}$ . Boole asked to determine the possible probabilities of some other element $a \in A$ , given the assignment $a_{i} \mapsto π_{i}, (i = 1, \dots, k)$ . De Finetti's solution of Boole's problem yields a closed interval $I_{a} \subseteq [0, 1]$ such that the set of possible probabilities of a coincides with $I_{a}$ . $I_{a}$ is nonempty iff the assignment is consistent in de Finetti's sense. Now suppose the probability of $a_{k}$ undergoes a small perturbation $π_{k} \mapsto π_{k} + d π_{k}$ . We study the resulting modification of $I_{a}$ . For instance, we prove that the one-sided derivatives $\partial length (I_{a}) / \partial π_{k}^{\pm}$ always have a rational, (generally non-integer) value. In the particular case when each probability $π_{i}$ is rational and A is presented as a Lindenbaum algebra, all these derivatives are Turing computable. Their existence domain is decidable by the Tarski-Seidenberg algorithm for polynomial equations and inequalities in real closed fields. Our results build on de Finetti's consistency notion and his solution of Boole's problem, and extend Hailperin's polyhedral methods for combining bounds on probabilities.

Section snippets

Introduction and statement of the main results

Let $a_{1}, \dots, a_{k}$ be elements of a boolean algebra A, $π_{1}, \dots, π_{k} \in [0, 1]$ , and $a_{1} \mapsto π_{1}, \dots, a_{k} \mapsto π_{k}$ a $[0, 1]$ -valued map. Interpreting each $π_{i}$ as a probability, Boole asked to determine the probability of some other event $a \in A$ given the assignment $a_{i} \mapsto π_{i}, i = 1, \dots, k$ . De Finetti's solution of Boole's problem, Theorem 2.7, yields a closed interval $I_{a} \subseteq [0, 1]$ such that the set of probabilities of a compatible with the assignment $a_{i} \mapsto π_{i}$ coincides with $I_{a}$ . The interval $I_{a}$ is nonempty iff the assignment is consistent in the sense

Preliminaries: Boole's extension problem and de Finetti's solution

Boole [2, Chapter XVI, 4, p. 246] writes:

the object of the theory of probabilities might be thus defined. Given the probabilities of any events, of whatever kind, to find the probability of some other event connected with them.

Let us call this Boole's extension problem. This section is devoted to giving a short proof of de Finetti's solution of the problem. His result and the preliminary material for the proof will find repeated use below.

So let us first assume “events” $a_{1}, \dots, a_{l - 1}$ are assigned

Minimal paths in the state space $S_{A}$ induced by a consistent book

Let A be a finite boolean algebra together with elements $a_{1}, \dots, a_{k}, a$ and a consistent book $β : a_{1} \mapsto π_{1}, \dots, a_{k - 1} \mapsto π_{k - 1}$ . Theorem 2.7 yields the de Finetti interval $I_{a_{k}} = [π_{\min}, π_{\max}]$ of consistent probabilities for $a_{k}$ given β. We may write where ${\bar{a}}_{k} : S_{A} \to [0, 1]$ is as defined in (3). For fixed β, our aim in this section is to study the dependence of $I_{a}$ on the probability $x \in I_{a_{k}}$ .

Suppose A has u atoms. Given affinely independent points $ϵ_{1}, \dots, ϵ_{u} \in R^{u - 1}$ a convenient geometric representation of the state space $S_{A}$ is

The dynamics of Boole's extension problem

In this section we consider the totality of consistent books on a fixed but otherwise arbitrary k-tuple $X = (a_{1}, \dots, a_{k})$ of elements of a finite boolean algebra A. Our aim is to investigate the computability of the differential version of Boole's extension problem. We let $Π_{X} = {(x_{1}, \dots, x_{k}) \in {[0, 1]}^{k} | the book a_{1} \mapsto x_{1}, \dots, a_{k} \mapsto x_{k} is consistent} .$ In Proposition 4.2 we will provide a concrete realization of the set $Π_{X} \subseteq {[0, 1]}^{k}$ . To this purpose we prepare the following easy lemma, whose proof we have not been able to

Examples

Example 5.1

The faces of a tetrahedral biased dice are denoted $o_{1}, \dots, o_{4}$ and are identified with the atoms of the free two-generator boolean algebra $F_{2}$ . The outcomes $o_{1} \lor o_{2}, o_{2} \lor o_{3}$ have equal probability 1/3.

Problem: How does the probability of rolling $o_{4}$ depend on the probability of rolling $o_{3} \lor o_{1}$ ?

Fig. 1 gives a geometric representation of the problem and its solution. The boolean algebra $F_{2}$ is presented as the powerset of the set $Ω = {ϵ_{1}, \dots, ϵ_{4}}$ , the latter being in one-one correspondence with both $hom (F_{2})$ and $at ($

Related work and final remarks

In [10, §§ 1-5] Hailperin considered the possible probabilities of a boolean formula ϕ given a $[0, 1]$ -valued map $A_{i} \mapsto a_{i}$ defined on the free generating set ${A_{1}, \dots, A_{n}}$ of the free n-generator boolean algebra $F_{n}$ . In Hailperin's terminology, “ $A_{1}, \dots, A_{n}$ are arbitrary events of which nothing is known except that their respective probabilities are $a_{1}, \dots, a_{n}$ .” An effective procedure is described, based on Linear Programming, by which, given any formula $ϕ = ϕ (A_{1}, \dots, A_{n})$ one can obtain the best possible upper and

Acknowledgements

The author thanks the two Referees for their competent and careful reading of this paper and their valuable comments and suggestions for improvement.

References (26)

G. Georgakopoulos et al.
Probabilistic satisfiability
J. Complex.
(1988)
P. Hansen et al.
Boole's conditions of possible experience and reasoning under uncertainty
Discrete Appl. Math.
(1995)
N.J. Nilsson
Probabilistic logic
Artif. Intell.
(1986)
S. Basu et al.
Algorithms in Real Algebraic Geometry
(2003)
G. Boole
An Investigation of the Laws of Thought
(1854)
B. de Finetti
Sul significato soggettivo della probabilità
Fundam. Math.
(1931)
B. de Finetti
La prévision: ses lois logiques, ses sources subjectives
Ann. Inst. Henri Poincaré
(1937)
(1980)
B. de Finetti
Opere Scelte (Selected Works), Vol. 1
(2006)
B. de Finetti
Theory of Probability. A Critical Introductory Treatment
(2017)
H. Edelsbrunner
Geometric algorithms

K.R. Goodearl

Partially Ordered Abelian Groups with Interpolation

(1986)

T. Hailperin

Best possible inequalities for the probability of a logical function of events

Am. Math. Mon.

(1965)

T. Hailperin

Boole's Logic and Probability: A Critical Exposition from the Standpoint of Contemporary Algebra, Logic and Probability Theory

(1986)

Cited by (0)

View full text

The differential of probabilistic entailment

Abstract

Section snippets

Introduction and statement of the main results

Preliminaries: Boole's extension problem and de Finetti's solution

Minimal paths in the state space SA induced by a consistent book

The dynamics of Boole's extension problem

Examples

Related work and final remarks

Acknowledgements

J. Complex.

Discrete Appl. Math.

Artif. Intell.

Algorithms in Real Algebraic Geometry

An Investigation of the Laws of Thought

Sul significato soggettivo della probabilità

Fundam. Math.

La prévision: ses lois logiques, ses sources subjectives

Ann. Inst. Henri Poincaré

Opere Scelte (Selected Works), Vol. 1

Theory of Probability. A Critical Introductory Treatment

Geometric algorithms

Partially Ordered Abelian Groups with Interpolation

Best possible inequalities for the probability of a logical function of events

Am. Math. Mon.

Boole's Logic and Probability: A Critical Exposition from the Standpoint of Contemporary Algebra, Logic and Probability Theory

Minimal paths in the state space $S_{A}$ induced by a consistent book