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We consider the problem of estimating the parameters of a distribution when the underlying events are themselves unobservable. The aim of the exercise is to perform a task (for example, search a web-site or query a distributed database) based on a distribution involving the state of nature, except that we are not allowed to observe the various “states of nature” involved in this phenomenon. In particular, we concentrate on the task of searching for an object in a set of N locations (or bins) {C 1, C 2,…, C N }, in which the probability of the object being in the location C i is p i , where P = [p 1, p 2,…, p N ]T is called the Target Distribution. Also, the probability of locating the object in the bin within a specified time, given that it is in the bin, is given by a function called the Detection function, which, in its most common instantiation, is typically, specified by an exponential function. The intention is to allocate the available resources so as to maximize the probability of locating the object. The handicap, however, is that the time allowed is limited, and thus the fact that the object is not located in bin C i within a specified time does not necessarily imply that the object is not in C i . This problem has applications in searching large databases, distributed databases, and the world-wide web, where the location of the files sought for are unknown, and in developing various military and strategic policies. All of the research done in this area has assumed the knowledge of the {p i }. In this paper we consider the problem of obtaining error bounds, estimating the Target Distribution, and allocating the search times when the {p i } are unknown. To the best of our knowledge, these results are of a pioneering sort - they are the first available results in this area, and are particularly interesting because, as mentioned earlier, the events concerning the Target Distribution, in themselves, are unobservable.