Bayesian methods in the courts

Using the Bayesian method, the prior probability that Sally Clark murdered her two sons can be estimated from rates of infanticide in the United Kingdom (Hill, 2004). Most parents do not murder their children, so the rate of murdering two children is very low (the probability is approximately one in 2.7 million). Thus, we have the prior probability that Sally Clark murdered her two sons (0.00000037), and the probabilities of her two sons dying given the two hypotheses (1 in 300000 if she did not murder them and 1 if she did), so we can calculate the posterior probability:

Pr(Sally Clark murdered her sons given the data) = 3.7 x 10-7 x 1/[3.7 x 10-7 x 1 +(1 - 3.7 x 10-7)x 1/300000] = 0.1.

Therefore, it is approximately ten times more likely than not (given the two deaths) that Sally Clark is innocent (Hill, 2004; see also Bondi, 2004 and Joyce, 2002). Of course, other evidence could be brought to bear on this case. Firstly, the rate of infanticide among parents is much lower than the figure used if those parents do not have a history of violence towards their children, as is the case with Sally Clark. At the same time, the rate of cot death is also lower for such families (Hill, 2004). Secondly, medical evidence, some of which only came to light on appeal, increases the likelihood of death by natural causes. The first son was found to have a respiratory infection and the second a bacterial infection, both of which were likely causes of death. Thankfully, given the evidence, Sally Clark appealed her conviction and is now free after spending more than three years in jail. However, ecologists continue to fall into the same trap that appears to have contributed to the conviction of Sally Clark, which is known as the prosecutor's fallacy. This is the mistaken belief that a low probability of obtaining data given a hypothesis means that the alternative hypothesis is likely to be true.

for the effect of lead for model 3, which included the interaction term, is almost twice that of model 1 (Table 2.1).

By using model averaging, it is possible to calculate an effect of lead that accounts for uncertainty in the choice of the best model. Model averaging weights the estimated effect by the Akaike weights. Additionally, the standard error of the model-averaged predictions is a function of the within-model variation (i.e. the standard error of the prediction for each model), the between-model variation (i.e. the differences in the predictions among the different models) and the Akaike weights (see Burnham and Anderson, 2002 for details). In this example, the model-averaged prediction is that lead reduces survival by 0.335 with a standard error of 0.125.

The ability to consider more than one model when making inferences is one of the strengths of information theoretic methods. The chief advantage is recognizing that there is usually some uncertainty about which of the candidate models best describes the data. It is risky to put all one's eggs in one basket (a single model) when other plausible models might make different predictions. More detail on using multimodel inference in ecology can be found in Burnham and Anderson (2002).

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