We construct an integrated probabilistic model to capture interactions between atoms of a nanocomponent. We then use this model to assess reliabilities of nanocomponents with different structures. Several properties of our proposed model are also described under a sparseness condition. The model is an extension of our previous model based on Markovian random field theory. The proposed integrated model is flexible in that pairwise relationship information among atoms as well as features of individual atoms can be easily incorporated. An important feature that distinguishes the integrated probabilistic model from our previous model is that the integrated approach uses all available sources of information with different weights for different types of interaction. In this paper we consider the nanocomponent at a fixed moment of time, say the present moment, and we assume that the present state of the nanocomponent depends only on the present states of its atoms.

A learning matrix is defined by a set of input and output pattern vectors. The entries in these vectors are zeros and ones. The matrix is the maximum of the outer products of the input and output pattern vectors. The entries in the matrix are also zeros and ones. The product of this matrix with a selected input pattern vector defines an activity vector. It is shown that when the patterns are taken to be random, then there are central limit and large deviation theorems for the activity vector. They give conditions for when the activity vector may be used to reconstruct the output pattern vector corresponding to the selected input pattern vector.

A continuum structure function γ is a non-decreasing mapping from the unit hypercube to the unit interval. Block and Savits (1984) use the sets and to determine bounds on the distribution of γ (X) when X is a vector of associated random variables. It is shown that, if γ admits of a modular decomposition, improved bounds may be obtained.

A continuum structure function γ is a non-decreasing mapping from the unit hypercube to the unit interval. Minimal path (cut) sets of upper (lower) simple continuum structure functions are introduced and are used to determine bounds on the distribution of γ (Χ) when X is a vector of associated random variables and when γ is right (left)-continuous. It is shown that, if γ admits of a modular decomposition, improved bounds may be obtained.