Predictive recursion (PR) is a fast stochastic algorithm for nonparametric
estimation of mixing distributions in mixture models. It is known that the PR
estimates of both the mixing and mixture densities are consistent under fairly
mild conditions, but currently very little is known about the rate of
convergence. In this note we investigate asymptotic convergence properties of
the PR estimate under model mis-specification in the special case of finite
mixtures with known support. Tools from stochastic approximation are used to
prove that the PR estimates converge at a nearly root-n rate.

Predictive recursion is an accurate and computationally efficient algorithm
for nonparametric estimation of mixing densities in mixture models. In
semiparametric mixture models, however, the algorithm fails to account for any
uncertainty in the additional unknown structural parameter. As an alternative
to existing profile likelihood methods, we treat predictive recursion as a
filter approximation to fitting a fully Bayes model, whereby an approximate
marginal likelihood of the structural parameter emerges and can be used for
inference.

The editing of a combinatorial object is the alteration of some of its
elements such that the resulting object satisfies a certain fixed property. The
edit problem for graphs, when the edges are added or deleted, was first studied
independently by the authors and K\'ezdy [J. Graph Theory (2008), 58(2),
123--138] and by Alon and Stav [Random Structures Algorithms (2008), 33(1),
87--104].

The Dempster--Shafer (DS) theory is a powerful tool for probabilistic
reasoning based on a formal calculus for combining evidence. DS theory has been
widely used in computer science and engineering applications, but has yet to
reach the statistical mainstream, perhaps because the DS belief functions do
not satisfy long-run frequency properties. Recently, two of the authors
proposed an extension of DS, called the weak belief (WB) approach, that can
incorporate desirable frequency properties into the DS framework by
systematically enlarging the focal elements.

For a family $\mathcal{F}$ of subsets of [n]=\{1, 2, ..., n} ordered by
inclusion, and a partially ordered set P, we say that $\mathcal{F}$ is P-free
if it does not contain a subposet isomorphic to P. Let $ex(n, P)$ be the
largest size of a P-free family of subsets of [n]. Let $Q_2$ be the poset with
distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean
lattice. We show that $2N -o(N) \leq ex(n, Q_2)\leq 2.283261N +o(N), $ where $N
= \binom{n}{\lfloor n/2 \rfloor}$.