Conditional probability distribution (CPD) method in temperature based death time estimation: Error propagation analysis

Abstract

Bayesian estimation applied to temperature based death time estimation was recently introduced as conditional probability distribution or CPD-method by Biermann and Potente. The CPD-method is useful, if there is external information that sets the boundaries of the true death time interval (victim last seen alive and found dead). CPD allows computation of probabilities for small time intervals of interest (e.g. no-alibi intervals of suspects) within the large true death time interval. In the light of the importance of the CPD for conviction or acquittal of suspects the present study identifies a potential error source. Deviations in death time estimates will cause errors in the CPD-computed probabilities. We derive formulae to quantify the CPD error as a function of input error. Moreover we observed the paradox, that in cases, in which the small no-alibi time interval is located at the boundary of the true death time interval, adjacent to the erroneous death time estimate, CPD-computed probabilities for that small no-alibi interval will increase with increasing input deviation, else the CPD-computed probabilities will decrease. We therefore advise not to use CPD if there is an indication of an error or a contra-empirical deviation in the death time estimates, that is especially, if the death time estimates fall out of the true death time interval, even if the 95%-confidence intervals of the estimate still overlap the true death time interval.

Introduction

In their recent paper [2] Biermann and Potente applied Bayesian estimation to death time determination by the nomogram method (NM) and the compound method (CM) introducing this application as conditional probability distribution (CPD) method. The authors describe the computation of conditional probabilities and demonstrate the power of the method in cases from a field study by Henssge [3]. This study was originally intended to emphasize the value of the CM in reducing the 95%-confidence interval width from the NM. By integrating external information when the deceased was last seen alive and found dead the CPD-method was able to further narrow the width of the 95%-confidence intervals. Biermann and Potente [2] conclude that “the CPD-method … will always lead to a correct 95.45% interval for use in court and investigations. This interval will always be smaller than the initial interval determined by the NM algorithm and in some cases smaller than the interval after application of CM.”

While CPD is in principle a promising tool for a more accurate death time estimation, Biermann and Potente [2] do not discuss potential error sources of the method. But the CPD-method does not only allow integration of external information that sets reliable limits to the possible death time interval but also computation of probabilities for small time intervals of interest (e.g. no-alibi time intervals of suspects). In the light of the importance of the CPD-method as court evidence we identify and complement potential error sources.

The knowledge of the correct probability distribution of the death time estimator t^ from temperature based death time estimation (TDE) represents the central presupposition of CPD. CPD initially estimates this probability distribution. If this estimated probability distribution is biased – e.g. by a systematic model error in TDE or by random variation of the estimated TDE value from the true value – the results of CPD may be severely distorted. We deduce a formula for the errors in CPD results caused by deviations of the estimator t^ from the true value and provide asymptotic approximations of these errors for large t^-deviations. The study also highlights a paradox of CPD in certain case scenarios, in which an increasing TDE-induced bias produces increasing CPD-probability values (up to 100%) for small time intervals. We demonstrate the results using a synthetic homicide case example.