Bayesian Object Identification: Variants

We present a Bayesian theory of object identification. Here, identifying an object means selecting a particular observation from a group of observations (variants), this observation (the regular variant) being characterized by a distributional model. In this sense, object identification means assigning a given model to one of several observations. Often, it is the statistical model of the regular variant, only, that is known. We study an estimator which relies essentially on this model and not on the characteristics of the “irregular” variants. In particular, we investigate under what conditions this variant selector is optimal. It turns out that there is a close relationship with exchangeability and Markovian reversibility. We finally apply our theory to the case of irregular variants generated from the regular variant by a Gaussian linear model.     

Synthesis and properties of amine‐containing poly(arylene ether sulfone) as an anion‐exchange matrix

Poly(arylene ether sulfone) (PSF), showing good thermal stability and excellent mechanical properties, was synthesized as an anion‐exchange matrix. It was synthesized by the condensation polymerization between bisphenol A and 4,4′‐dichlorodiphenylsulfone. 1°‐Amine‐containing poly(arylene ether sulfone) (1°‐APSF) was synthesized by the reduction reaction of a nitrated PSF. Then, it was transferred to 3°‐amine‐containing poly(arylene ether sulfone) (3°‐APSF) by the alkylation of the amine of 1°‐APSF. The properties of PSF, 1°‐APSF, and 3°‐APSF were investigated by Fourier transform infrared, ¹H NMR spectroscopy, differential scanning calorimetry, and thermogravimetric analysis. The introduction of the 3°‐amine group into PSF increased the glass‐transition temperature but decreased thermooxidative stability. The ion‐exchange capacities of 1°‐APSF and 3°‐APSF were shown to be 2.24 and 2.86 mequiv/g, respectively. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 40: 4281–4287, 2002     

A characterization for mixtures of semi-Markov processes

Mixtures of recurrent semi-Markov processes are characterized through a partial exchangeability condition of the array of successor states and holding times. A stronger invariance condition on the joint law of successor states and holding times leads to mixtures of Markov laws.     

On mixtures of distributions of Markov chains

Let X be a chain with discrete state space I, and V be the matrix of entries V_i,n, where V_i,n denotes the position of the process immediately after the nth visit to i. We prove that the law of X is a mixture of laws of Markov chains if and only if the distribution of V is invariant under finite permutations within rows (i.e., the V_i,n's are partially exchangeable in the sense of de Finetti). We also prove that an analogous statement holds true for mixtures of laws of Markov chains with a general state space and atomic kernels. Going back to the discrete case, we analyze the relationships between partial exchangeability of V and Markov exchangeability in the sense of Diaconis and Freedman. The main statement is that the former is stronger than the latter, but the two are equivalent under the assumption of recurrence. Combination of this equivalence with the aforesaid representation theorem gives the Diaconis and Freedman basic result for mixtures of Markov chains.     

Extendibility of spherical matrix distributions

We investigate the structure of distributions for matrices which can be embedded in arbitrarily large matrices whose distributions have properties of invariance under orthogonal rotations.     

Remarks on the role of conditional probability in data exploration

A careful discussion of the concept of conditional event leads to a sensible use of frequency data as conditional probabilities: as a by-product, the well-known ‘paradoxes’ arising from the so-called confounding effect are avoided.     

On a notion of partially conditionally identically distributed sequences

A notion of conditionally identically distributed (c.i.d.) sequences has been studied as a form of stochastic dependence weaker than exchangeability, but equivalent to it in the presence of stationarity. We extend such notion to families of sequences. Paralleling the extension from exchangeability to partial exchangeability in the sense of de Finetti, we propose a notion of partially c.i.d. dependence, which is shown to be equivalent to partial exchangeability for stationary processes. Partially c.i.d. families of sequences preserve attractive limit properties of partial exchangeability, and are asymptotically partially exchangeable. Moreover, we provide strong laws of large numbers and two central limit theorems. Our focus is on the asymptotic agreement of predictions and empirical means, which lies at the foundations of Bayesian statistics. Natural examples of partially c.i.d. constructions are interacting randomly reinforced processes satisfying certain conditions on the reinforcement.     

Representation theorems for probability functions satisfying spectrum exchangeability in inductive logic

We prove de Finetti style representation theorems covering the class of all probability functions satisfying spectrum exchangeability in polyadic inductive logic and give an application by characterizing those probability functions satisfying spectrum exchangeability which can be extended to a language with equality whilst still satisfying that property.     

An Invariance Principle for Triangular Arrays

Let A _{n, i} be a triangular array of sign-symmetric exchangeable random variables satisfying nE(A ² _{n, i} )1, nE(A ⁴ _{n, i} )0, n ² E(A ² _{n, 1} A ² _{n, 2})1. We show that ^[nt] _i=1 A _ni, 0t1, converges to Brownian motion. This is applied to show that if A is chosen from the uniform distribution on the orthogonal group O _n and X _n(t)=^[nt] _i=1 A _ii, then X _n converges to Brownian motion. Similar results hold for the unitary group.     

On Waiting Time for Reversed Patterns in Random Sequences

By using a combinatorial method it is shown that for every finite pattern, the distribution of the waiting time for the reversed pattern coincides with that of the waiting time for the original pattern in a multi-state dependent sequence with a certain type of exchangeability. The number of the typical sequences until the occurrence of a given pattern and that of the typical sequences until the occurrence of the reversed pattern are shown to be equal. Further, the corresponding results for the waiting time for the r-th occurrence of the pattern, and for the number of occurrences of a specified pattern in n trials are also studied. Illustrative examples based on urn models are also given.