Abstract
We consider the problem of computing tail probabilities - that is, probabilities of regions with low density - for high-dimensional Gaussian mixtures. We consider three approaches: the first is a bound based on the central and non-central ?2 distributions; the second uses Pearson curves with the first three moments of the criterion random variable U; the third embeds the distribution of U in an exponential family, and uses exponential tilting, which in turn suggests an importance sampling distribution. We illustrate each method with examples and assess their relative merits.
Highlights
INTRODUCTIONSuppose that X has the following finite Gaussian mixture probability density function (pdf) in Rd: c (1)
Suppose that X has the following finite Gaussian mixture probability density function in Rd: c (1)f (x) = γiφ(x|μi, Σi), i=1 where φ(x|μ, Σ) = φd(x|μ, Σ) = |2πΣ|−1/2 exp− 1 (x − μ) Σ−1(x − μ) 22010 Mathematics Subject Classification
Having observed [X = a], we address the problem of estimating the tail probability pt = P [U = f (X) ≤ f (a) = t]
Summary
Suppose that X has the following finite Gaussian mixture probability density function (pdf) in Rd: c (1). We call U the criterion (random) variable, and denote its pdf and cdf as g(u) and G(u), respectively Such problems arise in several contexts: for example, see [3, 7] for genetic and psychiatric applications of mixture distributions. Where Qij = (X − μj) Σ−j 1(X − μj) when [M = i] and X is a N (μi, Σi) random vector. This because if the average of positive numbers is less than t/c, at least one of them must be less than t/c. Due to the poor performance of both (3) and (6) bounds, we argue for other methods that can provide better approximations, to which we turn
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