A class of distribution function processes which have derivatives

Abstract

In the .author and van Eeden considered, as prior distributions for the cumulative, F, of the bio-assay problem, processes whose sample functions are, with probability one, distribution functions. The example we considered there had the undesirable property that its mean, E(F), was singular with respect to Lebesgue measure. In fact, Dubins and Freedman have shown that a class of such processes, which includes the example we considered, has sample functions F which are, with probability one, singular.

Bayesian nonparametric model selection and model testing

“…3 Many examples of such priors include Dirichlet Processes (Ferguson, 1973), mixtures of Dirichlet Processes (Antoniak, 1974;Lo, 1984), Po´lya Trees (Ferguson, 1974;Kraft, 1964;Lavine, 1992Lavine, , 1994Mauldin, Sudderth, & Williams, 1992), generalized exponential Gaussian process priors (Lenk, 1988(Lenk, , 1991, Bernstein polynomial priors (Petrone, 1999a,b;Petrone & Wasserman, 2002), neutral priors (Doksum, 1974;Hjort, 1990;Walker & Muliere, 1997), Dirichlet diffusion-tree priors (Neal, 2003), quantile pyramids (Hjort & Walker, 2004) and other priors (Dubins & Freedman, 1967;Nieto-Barajas, Pru¨nster, & Walker, 2004). For excellent reviews of nonparametric priors, see Sinha and Dey (1997), Dey, Mu¨ller, and Sinha (1998), Walker, Damien, Laud, and Smith (1999), MacEachern and Mu¨ller (2000), Ghosh and Ramamoorthi (2003), Hjort (2003), and Mu¨ller and Quintana (2004).…”

Random Bernstein Polynomials

“…The random measure obtained with the scheme of Dubins & Freedman (1967) is almost surely singular (see also Mauldin & Monticino, 1995). Metivier (1971) provides conditions for obtaining a measure absolutely continuous with respect to Lebesgue measure, in this sense generalizing the results of Kraft (1964). The Dirichlet process of Ferguson (1973Ferguson ( , 1974 assigns probability one to the set of discrete probability measures on the sample space.…”

“…To our knowledge, in the literature there are few proposals of priors which satisfy requirements (i) and (ii). Early proposals of priors on Ä[0, 1] are due to Kraft (1964), Kraft & van Eeden (1964), and Dubins & Freedman (1967). The random measure obtained with the scheme of Dubins & Freedman (1967) is almost surely singular (see also Mauldin & Monticino, 1995).…”

Latent features in similarity judgments: A nonparametric Bayesian approach

“…This implied prior can be useful for data clustering purposes (e.g., Navarro, Griffiths, Steyvers, & Lee, 2006), particularly since samples from this prior can be generated using a simple stochastic process known as the Chinese restaurant process 1 (Blackwell & MacQueen, 1973;Aldous, 1985;Pitman, 1996). In a similar manner, it is possible to generate infinite latent hierarchies using other priors, such as the Pólya tree (Ferguson, 1974;Kraft, 1964) and Dirichlet diffusion tree (Neal, 2003) distributions. The key insight in all cases is to separate the prior over the structure (e.g., partition, tree, etc) from the prior over the other parameters associated with that structure.…”