A probabilistic model for murder weapon identification using stab-marks in human ribs

Abstract

The aim of this article is to provide a scientific and statistical basis to identify the murder weapon in stabbing cases from the geometric characteristics of the stab-marks left on human ribs. For this purpose, a quantitative predictive model is developed, based on geometric measurements of the stab-mark and its location along the rib. A general method based on Bayesian inference and probabilities is used for the model development, rather than a deterministic model given its inability in certain occasions to identify the murder weapon. Following the process explained in this article to collect the stab-mark information required, the complete probabilistic model exposed attained a high accuracy in the identification of the murder weapon between two macroscopically identical blades with a microscopic alteration in one of them (more than 90% of correct identification is achieved).

Appendix

We proceed to develop the computation of the probability that a weapon is the cause of a rib stab-mark, given some geometrical conditions of the stab-mark obtained.

Recalling Eq. 2 considering that the set of measurable conditions $\mathcal {C}$ is first given only by the geometrical features $\mathcal {C}=G$ and introducing the conditions mentioned in the manuscript:

$$ P_{W_{i}}=\mathbb{P}(W_{i}|G)=\cfrac{\mathbb{P}(G|W_{i})}{\mathbb{P}(G|W_{0})+\mathbb{P}(G|W_{1})} $$

(8)

Considering a geometry G = (ℓ_l, ℓ_w, η) and each measure being in the interval (ℓ_i, ℓ_i + Δℓ_i), the probability $\hat {P}_{i}:=\mathbb {P}(G|W_{i})$ is:

$$ \begin{array}{@{}rcl@{}} \hat{P}_{i}({\Delta}\ell_{l}, {\Delta}\ell_{w}, {\Delta}\eta) & = &\mathbb{P}(\ell_{l}<u<\ell_{l}+{\Delta}\ell_{l}, \ell_{w}<v<\ell_{w}\\ &&+{\Delta}\ell_{w}, \eta<w<\eta+{\Delta}\eta | W_{i}) \\ & = & \!\displaystyle{\int}_{\ell_{l}}^{\ell_{l}+{\Delta}\ell_{l}} \!\!{\int}_{\ell_{w}}^{\ell_{w}+{\Delta}\ell_{w}} \!\!{\int}_{\eta}^{\eta+{\Delta}\eta}\! f_{W_{i}}(u, v, w)\\ && \text{d}u\ \text{d}v\ \text{d}w \end{array} $$

(9)

And replacing in Eq. 8:

$$ P_{W_{i}} = \! \lim\limits_{\Delta\ell_{l}, {\Delta}\ell_{w},{\Delta}\eta \to 0}\! \frac{\hat{P}_{i}({\Delta}\ell_{l}, {\Delta}\ell_{w} {\Delta}\eta)}{\hat{P}_{1}\!({\Delta}\ell_{l}, {\Delta}\ell_{w}, {\Delta}\eta) + \hat{P}_{2}({\Delta}\ell_{l}, {\Delta}\ell_{w}, {\Delta}\eta)} $$

(10)

See that, for the previous limit, the probability that a measurement takes exactly a numerical value ℓ_i is zero. Therefore, a limit of the type 0/0 is obtained and thus the L’Hôpital’s rule should be applied:

$$ \begin{array}{@{}rcl@{}}{ll} P_{W_{i}} = {\lim}_{{\Delta}\ell_{l}, {\Delta}\ell_{w}, {\Delta}\eta \to 0}\frac{ \partial_{1} \hat{P}_{i}}{\partial_{1} \hat{P}_{1} + \partial_{1} \hat{P}_{2}} \\ &= {\lim}_{{\Delta}\ell_{l}, {\Delta}\ell_{w}, {\Delta}\eta \to 0} \frac{ \partial_{1}\partial_{2}\partial_{3}\hat{P}_{i}}{\partial_{1}\partial_{2}\partial_{3} \hat{P}_{1} + \partial_{1}\partial_{2}\partial_{3} \hat{P}_{2}}\\ &= \cfrac{f_{W_{i}}(\ell_{l}, \ell_{w}, \eta)}{f_{W_{1}}(\ell_{l}, \ell_{w}, \eta)+f_{W_{2}}(\ell_{l}, \ell_{w}, \eta)} \end{array} $$

(11)

obtaining Eq. 3, for which the distributions $f_{W_{i}}(\ell _{l},\ell _{w},\eta )$ can be directly computed. Furthermore, being the PC_i independent random variables by definition, the probability density functions (PDF) can be factored as follows:

$$ f_{W_{i}}(\ell_{l},\ell_{w},\eta) = \phi_{i}(\ell_{l})\cdot \theta_{i}(\ell_{w})\cdot \psi_{i}(\eta) $$

(12)

The Bayesian model proposed, which now includes the quantitative variables related to the geometry G of the stab-mark, can be extended by considering the anatomical region R_k of the rib, which is a qualitative variable R_k = (R_ant, R_lat, R_post):

$$ \tilde{P}_{W_{i}} := \mathbb{P}(W_{i}|\ell_{l}, \ell_{w}, \eta;R_{k}) = \frac{f_{W_{i},R_{k}}(\ell_{l}, \ell_{w}, \eta)}{f_{W_{0},R_{k}}(\ell_{l}, \ell_{w}, \eta)+f_{W_{1},R_{k}}(\ell_{l}, \ell_{w}, \eta)} $$

corresponding to Eq. 3 where, again, the functions ϕ_ik(ℓ_l), 𝜃_ik(ℓ_w), and ψ_ik(η) for each dimension or PC, each weapon i, and each anatomical region k will be obtained on the basis of the N = 150 stab-marks performed in the experimental tests.

Finally, the model can be improved to include the absence or occurrence of the central elliptical region Q ∈{0, 1} in the stab-mark; considering p_i,Q as the probability of obtaining or not the elliptical region with the weapon i, then the probability $P_{W_{i}}$ is:

$$ P_{W_{i}} := \mathbb{P}(W_{i}|\ell_{l}, \ell_{w}, \eta;R_{k};Q) = \frac{\tilde{P}_{W_{i}}\ p_{i,Q}}{\tilde{P}_{W_{0}}\ p_{0,Q} + \tilde{P}_{W_{1}}\ p_{1,Q}} $$

which is the final purposed model (4), where the probability is computed as p_i,Q = N_i,1/N_i, and $\tilde {P}_{W_{i}}$ is obtained using Eq. 3.