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Detection in the Presence of Unknowns

Module by: Don Johnson. E-mail the author

We assumed in the previous sections that we have a few well-specified models (hypotheses) for a set of observations. These models were probabilistic; to apply the techniques of statistical hypothesis testing, the models take the form of conditional probability densities. In many interesting circumstances, the exact nature of these densities may not be known. For example, we may know a priori that the mean is either zero or some constant (as in the Gaussian example). However, the variance of the observations may not be known or the value of the non-zero mean may be in doubt. In an array processing context, these respective situations could occur when the background noise level is unknown (a likely possibility in applications) or when the signal amplitude is not known because of far-field range uncertainties (the further the source of propagating energy, the smaller its received energy at each sensor). In an extreme case, we can question the exact nature of the probability densities (everything is not necessarily Gaussian!). The model evaluation problem can still be posed for these situations; we classify the "unknown" aspects of a model testing problem as either parametric (the variance is not known, for example) or nonparametric (the formula for the density is in doubt). The former situation has a relatively long history compared to the latter; many techniques can be used to approach parametric problems while the latter is a subject of current research (Gibson and Melsa). We concentrate on parametric problems here.

We describe the dependence of the conditional density on a set of parameters by incorporating the parameter vector θ θ as part of the condition. We write the likelihood function as p r | ℳ i θ r p r ℳ i θ r for the parametric problem. In statistics, this situation is said to be a composite hypothesis (Cramér). Such situations can be further categorized according to whether the parameters are random or nonrandom. For a parameter to be random, we have an expression for its a priori density, which could depend on the particular model. As stated many times, a specification of a density usually expresses some knowledge about the range of values a parameter may assume and the relative probability of those values. Saying that a parameter has a uniform distribution implies that the values it assumes are equally likely, not that we have no idea what the values might be and express this ignorance by a uniform distribution. If we are ignorant of the underlying probability distribution that describes the values of a parameter, we will characterize them simply as being unknown (not random). Once we have considered the random parameter case, nonrandom but unknown parameters will be discussed.

References

  1. J.D. Gibson and J.L. Melsa. (1975). Introduction to Non-Parametric Detection with Applications. New York: Academic Press.
  2. H. Cramér. (1946). Mathematical Methods of Statistics. Princeton, NJ: Princeton University Press.

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