Pairwise Event Probability Research Articles

Linear programming bounds on the union probability

Lower and upper bounds on the union probability for N events are derived in terms of the individual and pairwise event probabilities by solving a linear program with variables. The bounds, which can be efficiently determined, are shown to be optimal when and are always sharper than recent optimal bounds which use slightly less information. Their competitive sharpness is also illustrated via numerical comparisons with state-of-the-art bounds in the literature.

On bounding the union probability using partial weighted information

Lower bounds on the finite union probability are established in terms of the individual event probabilities and a weighted sum of the pairwise event probabilities. The lower bounds have at most pseudo-polynomial computational complexity and generalize recent analytical bounds.

Lower Bounds on the Probability of a Finite Union of Events

In this paper, lower bounds on the probability of a finite union of events are considered, i.e., $P(\bigcup_{i=1}^N A_i)$, in terms of the individual event probabilities $\{P(A_i), i=1,\ldots,N\}$ and the sums of the pairwise event probabilities, i.e., $\{\sum_{j:j\neq i} P(A_i\cap A_j), i=1,\ldots,N\}$. The contribution of this paper includes the following: (i) in the class of all lower bounds that are established in terms of only the $P(A_i)$'s and $\sum_{j:j\neq i} P(A_i\cap A_j)$'s, the optimal lower bound is given numerically by solving a linear programming (LP) problem with $N^2-N+1$ variables; (ii) a new analytical lower bound is proposed based on a relaxed LP problem, which is at least as good as the bound due to Kuai, Alajaji, and Takahara [Discrete Math., 215 (2000), pp. 147--158]; (iii) numerical examples are provided to illustrate the performance of the bounds.

A lower bound on the probability of a finite union of events

A new lower bound on the probability P(A 1∪⋯∪A N) is established in terms of only the individual event probabilities P(A i) 's and the pairwise event probabilities P(A i∩A j) 's. This bound is shown to be always at least as good as two similar lower bounds: one by de Caen (1997) and the other by Dawson and Sankoff (1967). Numerical examples for the computation of this inequality are also provided.

Tight error bounds for nonuniform signaling over AWGN channels

We consider a Bonferroni-type lower bound due to Kounias (1968) on the probability of a finite union. The bound is expressed in terms of only the individual and pairwise event probabilities; however, it suffers from requiring an exponentially complex search for its direct implementation. We address this problem by presenting a practical algorithm for its evaluation. This bound is applied together with two other bounds, a recent lower bound (the KAT bound) and a greedy algorithm implementation of an upper bound due to Hunter (1976), to examine the symbol error (P/sub a/) and bit error (P/sub b/) probabilities of an uncoded communication system used in conjunction with M-ary phase-shift keying (PSK)/quadrature amplitude (QAM) (PSK/QAM) modulations and maximum a posteriori (MAP) decoding over additive white Gaussian noise (AWGN) channels. It is shown that the bounds-which can be efficiently computed-provide an excellent estimate of the error probabilities over the entire range of the signal-to-noise ratio (SNR) E/sub b//N/sub 0/. The new algorithmic bound and the greedy bound are particularly impressive as they agree with the simulation results even during very severe channel conditions.

Analysis of the pairwise error probability of noninterleaved codes on the Rayleigh-fading channel

The majority of previous analytical studies of signal-space coding techniques (includes trellis and block codes) on the Rayleigh-fading channel have assumed ideal interleaving. The effect of finite interleaving on the performance of different coding schemes has been studied only by simulation In this paper we first derive a maximum likelihood (ML) decoder for codewords transmitted over a noninterleaved Rayleigh flat fading channel, followed by an exact expression for the pairwise error event probability of such a decoder. It includes phase shift keying (PSK), quadrature amplitude modulation (QAM) signal sets, trellis coded modulation (TCM) and block coded modulation (BCM) schemes, as well as coherent (ideal channel state information) and partially coherent (e.g., differential, pilot tone, etc.) detection. We derive an exact expression for the pairwise event probability in the case of very slow fading-i.e., the fading experienced by all the symbols of the codeword is highly correlated. We also show that the interleaving depth required to optimize code performance for a particular minimum fading bandwidth can be approximated by the first zero of the fading channel's auto-correlation function.