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Age estimation by assessment of pulp chamber volume: a Bayesian network for the evaluation of dental evidence

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Abstract

Purpose

The present study aimed to investigate the performance of a Bayesian method in the evaluation of dental age-related evidence collected by means of a geometrical approximation procedure of the pulp chamber volume. Measurement of this volume was based on three-dimensional cone beam computed tomography images.

Methods

The Bayesian method was applied by means of a probabilistic graphical model, namely a Bayesian network. Performance of that method was investigated in terms of accuracy and bias of the decisional outcomes. Influence of an informed elicitation of the prior belief of chronological age was also studied by means of a sensitivity analysis.

Results

Outcomes in terms of accuracy were adequate with standard requirements for forensic adult age estimation. Findings also indicated that the Bayesian method does not show a particular tendency towards under- or overestimation of the age variable. Outcomes of the sensitivity analysis showed that results on estimation are improved with a ration elicitation of the prior probabilities of age.

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Notes

  1. We consider here the Bayesian (subjective) definition of probability, which states that the latter is a measure of the degree of a personal belief [37]. Such vision of probability offers a logical and coherent framework to handle uncertainty in forensic science, as widely discussed, for example, by Taroni et al. [38] or Biedermann [39].

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Acknowledgements

The authors wish to thank Rachel Irlam (King’s College London, UK) for proof-reading the document as well as Lorenzo Gaborini (University of Lausanne, Switzerland) for its valuable contribution in the R Code redaction and all users who tested it. Many acknowledgements are also addressed to the anonymous reviewers for their valuable comments on the manuscript.

Funding

This work has been kindly supported by the Swiss National Science Foundation (grant no. P2LAP1_164912).

Author information

Authors and Affiliations

  1. School of Criminal Justice, Building Batochime, University of Lausanne, 1015, Lausanne-Dorigny, Switzerland

    Emanuele Sironi & Franco Taroni

  2. Section of Forensic Medical Sciences, Department of Health Sciences, University of Florence, Largo Brambilla 3, 50134, Florence, Italy

    Claudio Baldinotti, Gian-Aristide Norelli & Vilma Pinchi

  3. Radiodiagnostic Unit n. 2, Department of Experimental and Clinical Biomedical Sciences, University of Florence, Largo Brambilla 3, 50134, Florence, Italy

    Cosimo Nardi

  4. Division of Analytical and Environmental Sciences, Faculty of Life Science & Medicine, King’s College London, 150 Stamford Street, London, SE1 9NH, UK

    Matteo Gallidabino

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  1. Emanuele Sironi
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Addendum

Addendum

From a general point of view, the Bayesian theorem applied to the assessment of continuous age-related characteristics for age estimation purposes take the following form:

$$ f\left(\left.a\right|z,s,\boldsymbol{t},\boldsymbol{g},\boldsymbol{v}\right)=\frac{f\left(\left.z\right|s,a,\boldsymbol{t},\boldsymbol{g},\boldsymbol{v}\right)\times f(a)}{\int_Af\left(\left.z\right|s,a,\boldsymbol{t},\boldsymbol{g},\boldsymbol{v}\right)\times f(a) da} $$
(5)

where a is the age of the examined individual, s his/her sex and z the value of the considered age-related characteristics (e.g., the Z-ratio or its logarithm, lnZ) computed during the individual’s examination. t, g and v are vectors, which refer respectively to the age, sex and measurements of the considered age-related characteristics of all the subjects in the reference sample (background data). Thus, the probability distribution associated with the likelihood f(z| s, a, t, g, v) models the probability of observing a specific value for z on the examined individual, given his/her age and sex and the background data. Eq. (3) in Section 3 is essentially a notation-simplified form of (5), where variables concerning background data are intentionally omitted from notation.

In a parametric framework, the characteristics of the probability distribution for f(z| s, a, t, g, v) can be defined based on a set of parameters θ, which can be estimated from the characteristics of the reference sample (i.e., the t, g, v vectors). The likelihood function in Eq. (5) is therefore a posterior predictive distribution, which can be further defined as follows [71]:

$$ f\left(\left.z\right|s,a,\boldsymbol{t},\boldsymbol{g},\boldsymbol{v}\right)={\int}_{\varTheta }f\left(\left.z\right|a,s,\boldsymbol{\theta} \right)\times \pi \left(\left.\boldsymbol{\theta} \right|\boldsymbol{t},\boldsymbol{g},\boldsymbol{v}\right)d\boldsymbol{\theta} $$
(6)

where

$$ \pi \left(\left.\boldsymbol{\theta} \right|\boldsymbol{t},\boldsymbol{g},\boldsymbol{v}\right)=\frac{f\left(\left.\boldsymbol{v}\right|\boldsymbol{t},\boldsymbol{g},\boldsymbol{\theta} \right)\times \pi \left(\boldsymbol{\theta} \right)}{\int_{\varTheta }f\left(\left.\boldsymbol{v}\right|\boldsymbol{t},\boldsymbol{g},\boldsymbol{\theta} \right)\times \pi \left(\boldsymbol{\theta} \right)d\boldsymbol{\theta}} $$
(7)

is the posterior probability distribution of the parameters, also obtained through the application of the Bayesian theorem. π(θ) is the prior probability distribution of the parameters and f(v| t, g, θ) models, in probabilistic terms, the relationship between the considered age-related characteristics of the subjects in the reference sample with their chronological age and sexes. This relationship is defined as a function of the parameters θ.

In this study, the likelihood function has been modelled by means of a Normal distribution, with parameters μ and σ. The former describes the relationship between lnZ and the age and sex of an individual. Since this relationship has been found to be linear [5], a multinomial linear regression model can be used for quantifying the value of lnZ. Formally:

$$ {\mu}_{a,s}={\beta}_{\mathsf{0}}+{\beta}_{\mathsf{1}}\times a+{\beta}_{\mathsf{2}}\times s, $$
(8)

where β 0 is the intercept parameter, and β 1 and β 2 are the slope parameters related to chronological age and sex. The regression analysis involves uncertainty, which has to be integrated in the model. It can therefore be assumed that the observed lnZ value results from a combination of a systematic part and a random part that takes into account such uncertainty [45]. Formally:

$$ lnZ={\mu}_{a,s}+\varepsilon, $$
(9)

where the error term ε is assumed to be Normally distributed, i.e., ε~Ɲ(0, σ2). The variance term σ2 is assumed constant in the whole age range, since the logarithmic transformation reduces the heteroscedasticity of the data. Therefore, from Eq. (9) it can be inferred that lnZ~Ɲ(μ a, s, σ2). Furthermore, θ = {β 0, β 1, β 2, σ} and can be estimated from the reference sample by means of the Eq. (7) presented above.

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Sironi, E., Taroni, F., Baldinotti, C. et al. Age estimation by assessment of pulp chamber volume: a Bayesian network for the evaluation of dental evidence. Int J Legal Med 132, 1125–1138 (2018). https://doi.org/10.1007/s00414-017-1733-0

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