De Finetti Research Articles

Convex geometry of finite exchangeable laws and de Finetti style representation with universal correlated corrections

We present a novel analogue for finite exchangeable sequences of the de Finetti, Hewitt and Savage theorem and investigate its implications for multi-marginal optimal transport (MMOT) and Bayesian statistics. If \((Z_1, \ldots ,Z_N)\) is a finitely exchangeable sequence of N random variables taking values in some Polish space X, we show that the law \(\mu _k\) of the first k components has a representation of the form $$\begin{aligned} \mu _k=\int _{\mathcal{P}_{\frac{1}{N}}(X)} F_{N,k}(\lambda ) \, \text{ d } \alpha (\lambda ) \end{aligned}$$for some probability measure \(\alpha \) on the set of \(\frac{1}{N}\)-quantized probability measures on X and certain universal polynomials \(F_{N,k}\). The latter consist of a leading term \(N^{k-1}\! /\prod _{j=1}^{k-1}(N\! -\! j) \lambda ^{\otimes k}\) and a finite, exponentially decaying series of correlated corrections of order \(N^{-j}\) (\(j=1, \ldots ,k\)). The \(F_{N,k}(\lambda )\) are precisely the extremal such laws, expressed via an explicit polynomial formula in terms of their one-point marginals \(\lambda \). Applications include novel approximations of MMOT via polynomial convexification and the identification of the remainder which is estimated in the celebrated error bound of Diaconis and Freedman (Ann Probab 8(4):745–764, 1980) between finite and infinite exchangeable laws.

Partial Exchangeability for Contingency Tables

A parameter free version of classical models for contingency tables is developed along the lines of de Finetti’s notions of partial exchangeability.

The notion of event in probability and causality: Situating myself relative to Bruno de Finetti

As Bruno de Finetti taught us, the notion of event in a theory of probability is fundamental, perhaps determinative. In this paper, I compare the notion of event in de Finetti's subjective theory of probability with the more situated notion of event that underlies the theory of probability and causality that I developed in the 1990s.This text was prepared for talks in Pisa, March 14, 2001, and Bologna, March 15, 2001.

Invariant measures on products and on the space of linear orders

Let $M$ be an $\aleph_0$-categorical structure and assume that $M$ has no algebraicity and has weak elimination of imaginaries. Generalizing classical theorems of de Finetti and Ryll-Nardzewski, we show that any ergodic, $\operatorname{Aut}(M)$-invariant measure on $[0, 1]^M$ is a product measure. We also investigate the action of $\operatorname{Aut}(M)$ on the compact space $\mathrm{LO}(M)$ of linear orders on $M$. If we assume moreover that the action $\operatorname{Aut}(M) \curvearrowright M$ is transitive, we prove that the action $\operatorname{Aut}(M) \curvearrowright \mathrm{LO}(M)$ either has a fixed point or is uniquely ergodic.

Characterization of quasi-arithmetic means without regularity condition

We show that bisymmetry, which is an algebraic property, has a regularity improving feature. More precisely, we prove that every bisymmetric, partially strictly monotonic, reflexive and symmetric function \(F \colon I ^2\to I\) is continuous. As a consequence, we obtain a finer characterization of quasi-arithmetic means than the classical results of Aczél [1], Kolmogoroff [18], Nagumo [20] and de Finetti [11].

Permutation Invariant Strong Law of Large Numbers for Exchangeable Sequences

We provide a permutation invariant version of the strong law of large numbers for exchangeable sequences of random variables. The proof consists of a combination of the Komlós–Berkes theorem, the usual strong law of large numbers for exchangeable sequences, and de Finetti’s theorem.

De Finetti’s Theorem in Categorical Probability

We present a novel proof of de Finetti's Theorem characterizing permutation-invariant probability measures of infinite sequences of variables, so-called exchangeable measures. The proof is phrased in the language of Markov categories, which provide an abstract categorical framework for probability and information flow. The diagrammatic and abstract nature of the arguments makes the proof intuitive and easy to follow. We also show how the usual measure-theoretic version of de Finetti's Theorem for standard Borel spaces is an instance of this result.

An Application of Quantum Logic to Experimental Behavioral Science

In 1933, Kolmogorov synthesized the basic concepts of probability that were in general use at the time into concepts and deductions from a simple set of axioms that said probability was a σ-additive function from a boolean algebra of events into [0, 1]. In 1932, von Neumann realized that the use of probability in quantum mechanics required a different concept that he formulated as a σ-additive function from the closed subspaces of a Hilbert space onto [0,1]. In 1935, Birkhoff & von Neumann replaced Hilbert space with an algebraic generalization. Today, a slight modification of the Birkhoff-von Neumann generalization is called “quantum logic”. A central problem in the philosophy of probability is the justification of the definition of probability used in a given application. This is usually done by arguing for the rationality of that approach to the situation under consideration. A version of the Dutch book argument given by de Finetti in 1972 is often used to justify the Kolmogorov theory, especially in scientific applications. As von Neumann in 1955 noted, and his criticisms still hold, there is no acceptable foundation for quantum logic. While it is not argued here that a rational approach has been carried out for quantum physics, it is argued that (1) for many important situations found in behavioral science that quantum probability theory is a reasonable choice, and (2) that it has an arguably rational foundation to certain areas of behavioral science, for example, the behavioral paradigm of Between Subjects experiments.

Particle Representation for the Solution of the Filtering Problem. Application to the Error Expansion of Filtering Discretizations

We introduce a weighted particle representation for the solution of the filtering problem based on a suitably chosen variation of the classical de Finetti theorem. This representation has important theoretical and numerical applications. In this paper, we explore some of its theoretical consequences. The first is to deduce the equations satisfied by the solution of the filtering problem in three different frameworks: the signal independent Brownian measurement noise model, the spatial observations with additive white noise model and the cluster detection model in spatial point processes. Secondly we use the representation to show that a suitably chosen filtering discretisation converges to the filtering solution. Thirdly we study the leading error coefficient for the discretisation. We show that it satisfies a stochastic partial differential equation by exploiting the weighted particle representation for both the approximation and the limiting filtering solution.

Lockean Beliefs, Dutch Books, and Scoring Systems

On the Lockean thesis one ought to believe a proposition if and only if one assigns it a credence at or above a threshold (Foley in Am Philos Q 29(2):111–124, 1992). The Lockean thesis, thus, provides a way of characterizing sets of all-or-nothing beliefs. Here we give two independent characterizations of the sets of beliefs satisfying the Lockean thesis. One is in terms of betting dispositions associated with full beliefs and one is in terms of an accuracy scoring system for full beliefs. These characterizations are parallel to, but not merely derivative from, the more familiar Dutch Book (de Finetti in Theory of probability, vol 1, Wiley, London, 1974) and accuracy (Joyce in Philos Sci 65(4):575–603, 1998) arguments for probabilism.

Sequentially Forecasting Economic Indices Using Mixture Linear Combinations of EP Distributions

This article displays an application of the statistical method motivated by Bruno de Finetti’s operational subjective theory of probability. We use exchangeable forecasting distributions based on mixtures of linear combinations of exponential power (EP) distributions to forecast the sequence of daily rates of return from the Dow-Jones index of stock prices over a 20 year period. The operational subjective statistical method for comparing distributions is quite different from that commonly used in data analysis, because it rejects the basic tenets underlying the practice of hypothesis testing. In its place, proper scoring rules for forecast distributions are used to assess the values of various forecasting strategies. Using a logarithmic scoring rule, we find that a mixture linear combination of EP distributions scores markedly better than does a simple mixture over the EP family, which scores much better than does a simple Normal mixture. Surprisingly, a mixture over a linear combination of three Normal distributions also makes a substantial improvement over a simple Normal mixture, although it does not quite match the performance of even the simple EP mixture. All substantive forecasting improvements become most marked after extreme tail phenomena were actually observed in the sequence, in particular after the abrupt drop in market prices in October, 1987. However, the improvements continue to be apparent over the long haul of 1985-2006 which has seen a number of extreme price changes. This result is supported by an analysis of the Negentropies embedded in the forecasting distributions, and a proper scoring analysis of these Negentropies as well.

Schur\u2013Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations

Schur–Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the t-fold tensor powers U^{otimes t} of all unitaries Uin U(d) is spanned by the permutations of the t tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries. The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of fault-tolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications:We resolve an open problem in quantum property testing by showing that “stabilizerness” is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group.We find that tensor powers of stabilizer states have an increased symmetry group. Conversely, we provide corresponding de Finetti theorems, showing that the reductions of arbitrary states with this symmetry are well-approximated by mixtures of stabilizer tensor powers (in some cases, exponentially well).We show that the distance of a pure state to the set of stabilizers can be lower-bounded in terms of the sum-negativity of its Wigner function. This gives a new quantitative meaning to the sum-negativity (and the related mana) – a measure relevant to fault-tolerant quantum computation. The result constitutes a robust generalization of the discrete Hudson theorem.We show that complex projective designs of arbitrary order can be obtained from a finite number (independent of the number of qudits) of Clifford orbits. To prove this result, we give explicit formulas for arbitrary moments of random stabilizer states.

Architecture between heteronomy and self-generation

Architecture between heteronomy and self-generation

Deciding Koopman's qualitative probability

In their recent paper in this journal, Delgrande, Renne and Sack study qualitative probability and discuss its role in Artificial Intelligence. Building on related work by de Finetti, Scott, Segerberg and others, the authors provide general axioms for qualitative probability. In this paper we investigate Koopman's conditional qualitative probability from the computational viewpoint. An important part of his work, published in the Annals of Mathematics in 1940-1941, deals with finite conjunctions K of statements of the form “the probability of a given h does not exceed the probability of b given k”, with a,b,h,k elements of a boolean algebra. Upon coding these elements by boolean formulas, we provide a decision procedure to check the consistency of any such K. As an immediate consequence, also inferences in Koopman's probability theory are shown to be computable. These problems of qualitative probability theory significantly generalize Boole's (typically quantitative) problem of estimating the possible probabilities of a new event given the probabilities of other events. Boole's classical problem today is known as the optimization version of the probabilistic satisfiability problem PSAT. In 1986 Nilsson published an influential paper on this subject in this journal. The scope of our results is much larger than that of PSAT, because Koopman's conjunctions K also formalize the key notion of independence. Some familiarity with boolean logic and finite boolean algebras is the only prerequisite for this paper.

Equity Cost Induced Dichotomy for Optimal Dividends with Capital Injections in the Cramér-Lundberg Model

We investigate a control problem leading to the optimal payment of dividends in a Cramér-Lundberg-type insurance model in which capital injections incur proportional cost, and may be used or not, the latter resulting in bankruptcy. For general claims, we provide verification results, using the absolute continuity of super-solutions of a convenient Hamilton-Jacobi variational inequality. As a by-product, for exponential claims, we prove the optimality of bounded buffer capital injections (−a,0,b) policies. These policies consist in stopping at the first time when the size of the overshoot below 0 exceeds a certain limit a, and only pay dividends when the reserve reaches an upper barrier b. An exhaustive and explicit characterization of optimal couples buffer/barrier is given via comprehensive structure equations. The optimal buffer is shown never to be of de Finetti (a=0) or Shreve-Lehoczy-Gaver (a=∞) type. The study results in a dichotomy between cheap and expensive equity, based on the cost-of-borrowing parameter, thus providing a non-trivial generalization of the Lokka-Zervos phase-transition Løkka-Zervos (2008). In the first case, companies start paying dividends at the barrier b*=0, while in the second they must wait for reserves to build up to some (fully determined) b*>0 before paying dividends.

Semidefinite programming hierarchies for constrained bilinear optimization

We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form {mathrm {Tr}}big [H(Dotimes E)big ], maximized with respect to semidefinite constraints on D and E. Applied to the problem of approximate error correction in quantum information theory, this gives hierarchies of efficiently computable outer bounds on the success probability of approximate quantum error correction codes in any dimension. The first level of our hierarchies corresponds to a previously studied relaxation (Leung and Matthews in IEEE Trans Inf Theory 61(8):4486, 2015) and positive partial transpose constraints can be added to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our sum-of-squares hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo’s asymptotic de Finetti theorem for quantum channels. Finally, our proof methods answer a question of Brandão and Harrow (Proceedings of the forty-fourth annual ACM symposium on theory of computing, STOC’12, p 307, 2012) by improving the approximation factor of de Finetti theorems with no symmetry from O(d^{k/2}) to {mathrm {poly}}(d,k), where d denotes local dimension and k the number of copies.

LEONARD SAVAGE, THE ELLSBERG PARADOX, AND THE DEBATE ON SUBJECTIVE PROBABILITIES: EVIDENCE FROM THE ARCHIVES

This paper explores archival material concerning the reception of Leonard J. Savage’s foundational work of rational choice theory in its subjective-Bayesian form. The focus is on the criticism raised in the early 1960s by Daniel Ellsberg, William Fellner, and Cedric Smith, who were supporters of the newly developed subjective approach but could not understand Savage’s insistence on the strict version he shared with Bruno de Finetti. The episode is well known, thanks to the so-called Ellsberg Paradox and the extensive reference made to it in current decision theory. But Savage’s reaction to his critics has never been examined. Although Savage never really engaged with the issue in his published writings, the private exchange with Ellsberg and Fellner, and with de Finetti about how to deal with Smith, shows that Savage’s attention to the generalization advocated by his correspondents was substantive. In particular, Savage’s defense of the normative value of rational choice theory against counterexamples such as Ellsberg’s did not prevent him from admitting that he would give careful consideration to a more realistic axiomatic system, should the critics be able to provide one.

Finitely additive mixtures of probability measures

Let D be a linear space of real bounded functions and P:D→R a coherent functional. Also, let Q be a collection of coherent functionals on D. Under mild conditions, there is a finitely additive probability Π on the power set of Q such that P(f)=∫QQ(f)Π(dQ) for each f∈D. This fact has various consequences and such consequences are investigated in this paper. Three types of results are provided: (i) Existence of common extensions satisfying certain properties, (ii) Finitely additive mixtures of extreme points, (iii) Countably additive mixtures. Among other things, we obtain new versions of Kolmogorov's consistency theorem and de Finetti's representation theorem.

What Was E. Borel Referring to in His 1939 Book when He Describes '…the Beautiful Work of Mr. J.m. Keynes…' in the A Treatise on Probability? The Answer Is the Belated Recognition on Borel’s Part of the Extremely Important Part Ii of the A Treatise on Probability on Imprecision and Inexact Measurement

E. Borel, in his 1924 review of the A Treatise on Probability, did not read Part II. He skipped Part II, although he did apologize to Keynes and Russell for doing so in his review, acknowledging that this was the most important part of Keynes’s A Treatise on Probability .Borel was certainly correct .He does not use the word “beautiful” to describe Keynes’s work at any place in his 1924 review. Now Parts I, III, IV, and V of the A Treatise on Probability are well done and make some breakthroughs, as acknowledged by Edgeworth in 1922 in his two reviews. However, statistics is not the type of mathematics where one mathematician would use the highest form of compliment possible among mathematicians, ”beautiful”, to describe the work of another mathematician. Borel recognized that Keynes was certainly a mathematician. By a process of elimination a la Sherlock Holmes, Borel can only be talking about Part II of the A Treatise on Probability. Part II is the part that Borel skipped in his review in 1924.Fifteen years later, Borel had to have finally been able to figure out what Keynes was doing ,with the help of Bertrand Russell and William Ernest Johnson, in Part II, much like Edwin Bidwell Wilson had finally been able to grasp Keynes’s points in Part II in 1934, some eleven years after his 1923 review of the A Treatise on Probability, although Wilson argued that uncertainty ,in Keynes’s sense, is really not important for a statistician. Borel must have been referring implicitly to Keynes’s truly impressive work on non additivity and interval valued probability in Part II of the A Treatise on Probability, which Keynes had based on G. Boole’s 1854 The Laws of Thought, in his footnotes in Chapter Two of his 1939 French book,the title of which is translated variously into English as Probability:Its Philosophy and Practical Value or Practical and Philosophical Value of Probabilities or Practical Value and Philosophy of Probabilities. Unfortunately, B. de Finetti ‘s review of Keynes’s A Treatise on Probability took place in 1938 and so he could not make use of Borel’s revised opinion of A Treatise on Probability that appeared in 1939 in Borel’s Practical and Philosophical Value of probabilities. The strange and incomprehensible failure of, especially ,philosophers and economists, but ,in fact ,all academics who have written on Keynes’s A Treatise on Probability, to cover Part II of the A Treatise on Probability means that theories of imprecise probability had to be created from scratch all over again, using Koopman’s 1940 works as a foundation, instead of building on Adam Smith, George Boole and J M Keynes with one,and only one, exception-the American mathematician, Theodore Hailperin. A careful reader of the contributions made by Adam Smith, G.Boole,J. M. Keynes, and T. Hailperin to imprecise probability ,with their emphasis on interval valued probability,non additivity and partial ordering, has no need to spend an inordinate amount of time reading the Koopman, C A B Smith ,Dempster, I J Good, H. Kyburg, I.Levi, i.e. SIPTA,etc. literature on imprecise probability because the latter literature is simply grossly ignorant about the former, which is basically involved in an exercise in reinventing the wheel . Borel needed to have made an explicit reference to Part II of the A Treatise on Probability in chapter 2 of his 1939 book. If he had, then, possibly, the history of decision science would have been very different and Ramsey’s error filled reviews in 1922 and 1926 would have been ignored.

Nested conditionals and genericity in the de Finetti semantics

Nested conditionals and genericity in the de Finetti semantics