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. 2013 Oct 7;8(10):e74388. doi: 10.1371/journal.pone.0074388

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© 2013 Comoglio et al

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited.

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(A) A total of n identical objects (in gray, n = 50 in this example) is homogeneously dispersed in a finite volume V. A fraction r of V, having volume rV, is sampled (dashed red rectangle) and the number of object therein, denoted with k (k = 4 in this example) is determined. Given the measurement, the posterior distribution of n is a negative binomial probability distribution (bottom) computed from a binomial likelihood (right) and a discrete uniform prior P(n) (left). (B) Generalization of (A) to m measurements. Fractions of volume are sampled the number of objects therein () determined as before. However, when m>1 two cases can be distinguished: i) the fractions are replaced (sampling with replacement); ii) the fractions are removed from V (sampling without replacement). In both cases, we derived a formula for the posterior distribution which is reported in the text as equation 6 for case i and equation 10 for case ii.

Inline graphic — (A) A total of n identical objects (in gray, n = 50 in this example) is homogeneously dispersed in a finite volume V. A fraction r of V, having volume rV, is sampled (dashed red rectangle) and the number of object therein, denoted with k (k = 4 in this example) is determined. Given the measurement, the posterior distribution of n is a negative binomial probability distribution (bottom) computed from a binomial likelihood (right) and a discrete uniform prior P(n) (left). (B) Generalization of (A) to m measurements. Fractions of volume are sampled the number of objects therein () determined as before. However, when m>1 two cases can be distinguished: i) the fractions are replaced (sampling with replacement); ii) the fractions are removed from V (sampling without replacement). In both cases, we derived a formula for the posterior distribution which is reported in the text as equation 6 for case i and equation 10 for case ii.