Random Parameters **new** 2003/08/22 15:44:14.769 GMT-5 2003/08/25 13:52:40.779 GMT-5 Don Johnson dhj@rice.edu Charlet Reedstrom charlet@rice.edu Don Johnson dhj@rice.edu unknowns When we know the density of θ , the likelihood function can be expressed as p r ℳ i r θ p r ℳ i θ r p θ θ and the likelihood ratio in the random parameter case becomes Λ r θ p r ℳ i θ r p θ θ θ p r ℳ 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. 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 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 statistically independent observations are made. ℳ 0 : r 0 σ 2 I ℳ 1 : r m σ 2 I m m … m 12 m … m 12 The numerator of the likelihood ratio is the sum of two Gaussian densities weighted by 12 (the a priori probability values), one having a positive mean, the other negative. The likelihood ratio, after simple cancellation of common terms, becomes 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 m σ 2 l 0 L 1 r l ≷ ℳ 0 ℳ 1 η L m 2 2 σ 2 where x is the hyperbolic cosine given simply as 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 0 L 1 r l ≷ ℳ 0 ℳ 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 , 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.