The Annals of Probability

Trivial intersection of σ-fields and Gibbs sampling

Patrizia Berti, Luca Pratelli, and Pietro Rigo

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Abstract

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathcal{N}$ the class of those $F\in\mathcal{F}$ satisfying P(F)∈{0, 1}. For each $\mathcal{G}\subset\mathcal{F}$, define $\overline{\mathcal{G}}=\sigma(\mathcal{G}\cup\mathcal {N})$. Necessary and sufficient conditions for $\overline{\mathcal{A}}\cap\overline{\mathcal{B}}=\overline {\mathcal {A}\cap\mathcal{B}}$, where $\mathcal{A},\mathcal{B}\subset\mathcal{F}$ are sub-σ-fields, are given. These conditions are then applied to the (two-component) Gibbs sampler. Suppose X and Y are the coordinate projections on $(\Omega,\mathcal{F})=(\mathcal{X}\times\mathcal{Y},\mathcal {U}\otimes \mathcal{V})$ where $(\mathcal{X},\mathcal{U})$ and $(\mathcal{Y},\mathcal{V})$ are measurable spaces. Let (Xn, Yn)n≥0 be the Gibbs chain for P. Then, the SLLN holds for (Xn, Yn) if and only if $\overline{\sigma(X)}\cap\overline{\sigma(Y)}=\mathcal{N}$, or equivalently if and only if P(XU)P(YV)=0 whenever $U\in\mathcal{U}$, $V\in\mathcal{V}$ and P(U×V)=P(Uc×Vc)=0. The latter condition is also equivalent to ergodicity of (Xn, Yn), on a certain subset S0⊂Ω, in case $\mathcal{F}=\mathcal{U}\otimes\mathcal{V}$ is countably generated and P absolutely continuous with respect to a product measure.

Article information

Source
Ann. Probab., Volume 36, Number 6 (2008), 2215-2234.

Dates
First available in Project Euclid: 19 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1229696601

Digital Object Identifier
doi:10.1214/07-AOP387

Mathematical Reviews number (MathSciNet)
MR2478681

Zentralblatt MATH identifier
1159.60007

Subjects
Primary: 60A05: Axioms; other general questions 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx} 60J22: Computational methods in Markov chains [See also 65C40] 65C05: Monte Carlo methods

Keywords
Ergodicity Gibbs sampler iterated conditional expectation Markov chain strong law of large numbers sufficiency

Citation

Berti, Patrizia; Pratelli, Luca; Rigo, Pietro. Trivial intersection of σ -fields and Gibbs sampling. Ann. Probab. 36 (2008), no. 6, 2215--2234. doi:10.1214/07-AOP387. https://projecteuclid.org/euclid.aop/1229696601


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References

  • [1] Berti, P., Pratelli, L. and Rigo, P. (2007). Skorohod representation on a given probability space. Probab. Theory Related Fields 137 277–288.
  • [2] Burkholder, D. L. and Chow, Y. S. (1961). Iterates of conditional expectation operators. Proc. Amer. Math. Soc. 12 490–495.
  • [3] Burkholder, D. L. (1961). Sufficiency in the undominated case. Ann. Math. Statist. 32 1191–1200.
  • [4] Burkholder, D. L. (1962). Successive conditional expectations of an integrable function. Ann. Math. Statist. 33 887–893.
  • [5] Diaconis, P., Khare, K. and Saloff-Coste, L. (2008). Gibbs sampling, exponential families and orthogonal polynomials. Statist. Sci. 23 151–178.
  • [6] Diaconis, P., Freedman, D., Khare, K. and Saloff-Coste, L. (2007). Stochastic alternating projections. Preprint, Dept. Statistics, Stanford Univ. Currently available at http://www-stat.stanford.edu/~cgates/PERSI/papers/altproject-2.pdf.
  • [7] Jones, G. L. and Hobert, J. P. (2001). Honest exploration of intractable probability distributions via Markov chain Monte Carlo. Statist. Sci. 16 312–334.
  • [8] Liu, J. S. (2001). Monte Carlo Strategies in Scientific Computing. Springer Series in Statistics. Springer, New York.
  • [9] Meyn, S. P. and Tweedie, R. L. (1996). Markov Chains and Stochastic Stability. Springer, New York.
  • [10] Tierney, L. (1994). Markov chains for exploring posterior distributions. Ann. Statist. 22 1701–1762. With discussion and a rejoinder by the author.