Under a partition of the set N we shall mean a sequence (b1, b2, . . .) of subsets of N such that (i) the sets bj are disjoint, (ii) ∪jbj = N, (iii) if bk = ∅ then also bk+1 = ∅ and (iv) if bk+1 6= ∅ then min bk < min bk+1. Condition (iv) says that the sequence of minimal elements of the blocks is increasing. One can think of partition as a mapping which sends a generic element j ∈ N to one of ...

These notes were written to accompany a sequence of workshop lectures at the Indian Institute of Science, Bangalore in January 2013. My lectures had two purposes. The first was to introduce some of the main ideas of exchangeability theory at a level suitable for graduate students (assuming familiarity with measure-theoretic probability). The second was to show the key rôle exchangeability has p...

If Equation (1) be satisfied, then all sets S are equally likely under the null hypothesis. To obtain a permutation test that is both unbiased and most powerful, one need only select the set S so that its probability under the alternative >0 is a maximum. This is accomplished in most cases by rejecting the hypothesis for all values of the test statistic that lie in the upper tail of the permuta...

Soil salinization and sodification, caused by inadequate land management, is one of the main threats to semiarid agroecosystems. It essential investigate saline levels under different land-use conditions maintain sustainability agricultural production. The objective this study was diagnose salinity status soil in region Northeast Brazil. samples were collected surface layer (0–5 cm) three condi...

BACKGROUND Exchangeable neck stems, defined as those with a dual taper (that is, a modular junction between the femoral head and the femoral neck and an additional junction between the neck and the stem body), were introduced in THA to improve restoration of joint biomechanics (restoring anteversion, offset, and limb length) and reduce the risk of dislocation. However exchangeable necks have be...

The deposition of high levels of reactive nitrogen (N) and sulphur (S), or the legacy of that deposition, remain among the world's most important environmental problems. Although regional impacts of acid deposition in aquatic ecosystems have been well documented, quantitative evidence of wide-scale impacts on terrestrial ecosystems is not common. In this study we analysed surface and subsoil ch...

Nonparametric Bayesian models are often based on the assumption that the objects being modeled are exchangeable. While appropriate in some applications (e.g., bag-ofwords models for documents), exchangeability is sometimes assumed simply for computational reasons; non-exchangeable models might be a better choice for applications based on subject matter. Drawing on ideas from graphical models an...

Exchangeable models for vertex labeled graphs cannot replicate the large sample behaviors of sparsity and power law degree distributions observed in many network datasets. Out of this mathematical impossibility emerges the question of how network data can be modeled in a way that reflects known empirical behaviors and respects basic statistical principles. We address this question by observing ...

In addition to A1 in the crystal lattice of clay minerals, A1 in soils can be divided into two major groups viz., exchangeable and non-exchangeable. The A1 equilibrium in soils is controlled by the solubility of the solid phase A1 compounds on the one hand and the adsorption characteristics of the adsorption phase on the other. Thus for a given soil the amount of exchangeable A1 will depend on ...