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2016 | OriginalPaper | Buchkapitel

On the Foundations of Multinomial Sequence Based Estimation

verfasst von : B. John Oommen, Sang-Woon Kim

Erschienen in: Computational Collective Intelligence

Verlag: Springer International Publishing

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Abstract

This paper deals with the relatively new field of sequence-based estimation which involves utilizing both the information in the observations and in their sequence of appearance. Our intention is to obtain Maximum Likelihood estimates by “extracting” the information contained in the observations when perceived as a sequence rather than as a set. The results of [15] introduced the concepts of Sequence Based Estimation (SBE) for the Binomial distribution. This current paper generalizes these results for the multinomial “two-at-a-time” scenario. We invoke a novel phenomenon called “Occlusion” that can be described as follows: By “concealing” certain observations, we map the estimation problem onto a lower-dimensional binomial space. Once these occluded SBEs have been computed, we demonstrate how the overall Multinomial SBE (MSBE) can be obtained by mapping several lower-dimensional estimates onto the original higher-dimensional space. We formally prove and experimentally demonstrate the convergence of the corresponding estimates.

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This information is, of course, traditionally used when we want to consider dependence information, as in the case of Markov models and n-gram statistics.

The paper lists numerous theorems whose proofs are found in [16]. The results for longer subsequences (i.e., three-at-a-time, four-at-a-time etc.) are also found in [16].

We apologize for this cumbersome notation, but this is unavoidable considering the complexity of the problem and the ensuing analysis.

For the present, we consider non-overlapping subsequences. We shall later extend this to overlapping sequences when we report the experimental results.

The reader must take pains to differentiate between the q’s and the s’s, because the former refer to the BSBEs and the latter to the MSBEs.

How BSBEs are obtained for specific instantiations of \(\pi (a,b)\) is discussed later.

The fact that c is a dummy variable will not be repeated in future invocations.

This, of course, makes sense only if \(\forall c, \widehat{q}_{a}\Big |_ {{\pi (a,c)}} ^{ac} \ne 0\).

Observe that it would be statistically advantageous (since the number of occurrences obtained would be almost doubled) if all the overlapping \(N_{ab}-1\) subsequences of length 2 were considered. The computational consequences of this are given in [16].

In the tables, values of unity/zero represent the cases when the roots are complex or when the number of occurrences of the event concerned are zero.

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Titel: On the Foundations of Multinomial Sequence Based Estimation
verfasst von: B. John Oommen
Sang-Woon Kim
Verlag: Springer International Publishing
Buch: Computational Collective Intelligence
Print ISBN: 978-3-319-45242-5

Electronic ISBN: 978-3-319-45243-2

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-319-45243-2_20

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