Most Likely Maximum Entropy for Population Analysis: a case study in decompression sickness prevention

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Most Likely Maximum Entropy for Population Analysis: a case study in decompression sickness prevention


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        <identifier identifierType="DOI">10.23723/9603/11332</identifier><creators><creator><creatorName>Youssef Bennani</creatorName></creator><creator><creatorName>Luc Pronzato</creatorName></creator><creator><creatorName>Maria-Joao Rendas</creatorName></creator></creators><titles>
            <title>Most Likely Maximum Entropy for Population Analysis: a case study in decompression sickness prevention</title></titles>
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	    <date dateType="Created">Sat 30 Aug 2014</date>
	    <date dateType="Updated">Mon 2 Oct 2017</date>
            <date dateType="Submitted">Thu 15 Mar 2018</date>
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            <description descriptionType="Abstract"></description>

Most Likely Maximum Entropy for Population Analysis: a case study in decompression sickness prevention Youssef Bennani, Luc Pronzato & Maria Jo˜ao Rendas∗ Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271 06900 Sophia Antipolis, France Abstract The paper proposes a new estimator for non-parametric density estimation from region censored observations in the context of population studies, where stan- dard Maximum Likelihood is known to be affected by over-fitting and non- uniqueness problems, combining the Maximum Entropy and Maximum Likeli- hood criteria. By combining the two criteria we propose a novel density es- timator that is able to overcome the singularities of the Maximum Likelihood estimator while maintaining a good fit to the observed data, and illustrate its behavior in real data (hyperbaric diving). The density estimation problem is motivated by a problem of population analysis: we are interested in the distribution πθ of the biophysical parameters θ of a mathematical model [1] for the instantaneous volume of micro-bubbles flowing through the right ventricle of a diver’s heart when executing a decom- pression profile P: (θ, {P(u)}u≤t) → B(t, θ, {P(u)}u≤t)). The instantaneous gas volume B is observed only through measurements of bubble grades G, strongly quantified versions of the peak volume b(θ, P) = maxt B(t, θ, {P(u)}u≤t): G(θ, P) = l ⇔ b(θ, P) ∈ [τl, τl+1[ , l ∈ {0, . . . , L}. (1) The problem is thus region-censored: all parameter values in the regions Ri Ri ≡ {θ ∈ Θ : b(θ, Pi) ∈ [τGi , τGi+1[} . yield the same observed grade Gi for profile Pi. ∗This work has been partially supported by DGA (France), through contract SAFEDIVE. 1 Several facts are known about the NPMLE for interval-censored observa- tions: (i) its support is confined to a finite number of disjoint intervals (the so called “elementary regions”); (ii) all distributions that put the same probabil- ity mass in these intervals have the same likelihood; (iii) there is in general no unique assignment of probabilities to the elementary regions that maximises the likelihood. Together, these facts imply that the NPMLE will frequently exhibit a sin- gular behaviour, in the sense that its mass is concentrated in a subset of Θ of small Lebesgue measure. This may lead to dangerous biases in the context of risk assessment, by not taking into consideration the presence of individuals for which risk can be large. In this paper, we remove ambiguity in the estimation of πθ by relying in the principle of Maximum Entropy (maxent, for short) [2], that finds the most un-informative density that can match the observed data. In our case, the constraints defining the Maximum Entropy solution are the empirical grade dis- tributions observed in the set of executed diving profiles. Note that these con- straints are not observed simultaneously, but derived from independent samples of the population. While maxent has been frequently used for density estimation from joint observation of empirical moments of a set of features, its use for region-censored data arising from strongly quantified data from independent observations is, as far as we know, novel. In particular we show that the equivalence between regularised maxent and penalised Maximum Likelihood in the exponential fam- ily [3,4] is lost in our case. This leads us to formulate a new estimation criterion, finding the most likely constrained maxent density. We compare the proposed estimator to the NPMLE and to the best fitting maxent solutions in real data from hyperbaric diving, showing that the resulting distribution is a best candi- date than NPMLE or maxent alone for the distribution of biological parameters in a given population. References [1] Julien Hugon, Vers une mod´elisation biophysique de la d´ecompression, Ph.D. thesis, Aix Marseille 2, 2010. [2] E. T. Jaynes, “Information theory and statistical mechanics,” Physics Reviews, vol. 106, pp. 620630, 1957. [3] M. Dudik, Maximum entropy density estimation and modeling geographic distri- butions of species, Ph.D. thesis, Princeton University, Department of Computer Science, 2007. [4] M. Dudik, M. Phillips, and R. Schapire, “Performance guarantees for regularised maximum entropy density estimation,” Proc. 7th Annual Conf. on Comp. Learn- ing Theory, 2004. 2