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Estimation of distributions involving unobservable events: the case of optimal search with unknown Target Distributions

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Abstract

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.

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Notes

  1. It is possible to get very good estimates of θ if one is provided with random occurrences of a known function of X. Thus, instead of receiving {X i }, if we are provided with {Y i }, where, for example, if each Y i  = X 2 i , an MLE can be easily devised to estimate θ by observing the Y i ’s.

  2. Even though the review is extensive, we have tried to keep it crisp and “to the point”.

  3. Typically, the value of p 0 should satisfy 0.5  ≤  p 0  ≤  1.

  4. We are not aware of anyone who has any real-life data on this. While this is generally a long-term goal for learning from the web, we believe that, for the most part, this is an unsolved problem.

  5. These results are based on the joint of work of the second author and his former student, Mr. Amr Ellaithy. The details of these results and other simulations (which are still being compiled) are to be included in a forthcoming paper jointly co-authored by Mr. Ellaithy and the second author.

  6. In the simulation, this essentially meant assigning the binary value of that slot to unity.

  7. For example, consider the scenario of searching for a missing climber in a mountain. Suppose a person was known to be at point x 0 at time 0. Then, when the search begins at time t, the target is distributed in an area that is centered at x 0, and has a radius vt, where v is the maximum speed at which the man can move. Due to the life-threatening risk in question, and due to the fact that we do not have time to repeat the search process to estimate and compare the detection probability, intelligent search techniques akin to what we have explained can be extremely crucial to the search endeavour.

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Authors and Affiliations

  1. School of Computer Science and Engineering, University of Electronics Science and Technology of China, Chengdu, China

    Qingxin Zhu

  2. School of Computer Science, Carleton University, Ottawa, Canada, K1S 5B6

    Qingxin Zhu & B. John Oommen

  3. University of Agder, Grimstad, Norway

    B. John Oommen

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  1. Qingxin Zhu
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  2. B. John Oommen
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Correspondence to B. John Oommen.

Additional information

B. J. Oommen is Fellow of the IEEE and IAPR. The work of B. J. Oommen was partially supported by the Natural Sciences and Engineering Research Council of Canada.

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Zhu, Q., Oommen, B.J. Estimation of distributions involving unobservable events: the case of optimal search with unknown Target Distributions. Pattern Anal Applic 12, 37–53 (2009). https://doi.org/10.1007/s10044-007-0095-5

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