Geodesic
Asymptotic behaviour of a BIPF algorithm with an improper target
Kybernetika, Tome 45 (2009) no. 2, pp. 169-188 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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The BIPF algorithm is a Markovian algorithm with the purpose of simulating certain probability distributions supported by contingency tables belonging to hierarchical log-linear models. The updating steps of the algorithm depend only on the required expected marginal tables over the maximal terms of the hierarchical model. Usually these tables are marginals of a positive joint table, in which case it is well known that the algorithm is a blocking Gibbs Sampler. But the algorithm makes sense even when these marginals do not come from a joint table. In this case the target distribution of the algorithm is necessarily improper. In this paper we investigate the simplest non trivial case, i. e. the $2\times2\times2$ hierarchical interaction. Our result is that the algorithm is asymptotically attracted by a limit cycle in law.
The BIPF algorithm is a Markovian algorithm with the purpose of simulating certain probability distributions supported by contingency tables belonging to hierarchical log-linear models. The updating steps of the algorithm depend only on the required expected marginal tables over the maximal terms of the hierarchical model. Usually these tables are marginals of a positive joint table, in which case it is well known that the algorithm is a blocking Gibbs Sampler. But the algorithm makes sense even when these marginals do not come from a joint table. In this case the target distribution of the algorithm is necessarily improper. In this paper we investigate the simplest non trivial case, i. e. the $2\times2\times2$ hierarchical interaction. Our result is that the algorithm is asymptotically attracted by a limit cycle in law.
Classification : 60J05, 60J22, 62F15, 62H17, 65C40
Keywords: log-linear models; marginal problem; null Markov chains 
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     title = {Asymptotic behaviour of a {BIPF} algorithm with an improper target},
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Asci, Claudio; Piccioni, Mauro. Asymptotic behaviour of a BIPF algorithm with an improper target. Kybernetika, Tome 45 (2009) no. 2, pp. 169-188. http://geodesic.mathdoc.fr/item/KYB_2009_45_2_a0/

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