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Random Parameters

Module by: Don Johnson. E-mail the author

When we know the density of θ θ, the likelihood function can be expressed as p r | i r=p r | i θ rp θ θdθ p r i r θ p r i θ r p θ θ and the likelihood ratio in the random parameter case becomes Λr=p r | i θ rp θ θdθp r | i θ rp θ θdθ Λ r θ p r i θ r p θ θ θ p r i θ r p θ θ Unfortunately, there are many examples where either the integrals involved are intractable or the sufficient statistic is virtually the same as the likelihood ratio, which can be difficult to compute.

Example 1

A simple, but interesting, example that results in a computable answer occurs when the mean of Gaussian random variables is either zero (model 0) or is ±m ± m with equal probability (hypothesis 1). The second hypothesis means that a non-zero mean is present, but its sign is not known. We are therefore stating that if hypothesis one is in fact valid, the mean has fixed sign for each observation - what is random is its a priori value. As before, L L statistically independent observations are made. 0 : r𝒩0σ2I 0 : r 0 σ 2 I 1 : r𝒩mσ2I 1 : r m σ 2 I m=mmProb1/2mmProb1/2 m m m 12 m m 12 The numerator of the likelihood ratio is the sum of two Gaussian densities weighted by 1/2 12 (the a priori probability values), one having a positive mean, the other negative. The likelihood ratio, after simple cancellation of common terms, becomes Λr=12e2ml=0L1 r l Lm22σ2+12e-2ml=0L1 r l Lm22σ2 Λ r 1 2 2 m l 0 L 1 r l L m 2 2 σ 2 1 2 -2 m l 0 L 1 r l L m 2 2 σ 2 and the decision rule takes the form coshmσ2l=0L1 r l 0 1 (ηeLm22σ2) m σ 2 l 0 L 1 r l 0 1 η L m 2 2 σ 2 where coshx x is the hyperbolic cosine given simply as ex+ex2 x x 2 . As this quantity is an even function, the sign of its argument has no effect on the result. The decision rule can be written more simply as |l=0L1 r l | 0 1 (σ2|m|arccoshηeLm22σ2) l 0 L 1 r l 0 1 σ 2 m η L m 2 2 σ 2 The sufficient statistic equals the magnitude of the sum of the observations in this case. While the right side of this expression, which equals γ γ, is complicated, it need only be computed once. Calculation of the performance probabilities can be complicated; in this case, the false-alarm probability is easy to find and the others more difficult.

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