What Bayesian reasoning is
Bayesian reasoning is the formal mathematical framework for updating beliefs in response to evidence. At its simplest, Bayes’s theorem says: the probability of a hypothesis given the evidence equals the prior probability of the hypothesis, multiplied by the likelihood of the evidence under the hypothesis, divided by the total probability of the evidence.
In criminal-trial form: start with a prior probability of guilt before seeing evidence (typically very low, because most people are not guilty of a given crime); multiply by how likely the observed evidence is if the defendant is guilty, versus how likely it is if the defendant is innocent; update to a posterior probability of guilt.
The three components applied to Letby
1. The prior probability
The prior probability of any given nurse committing a specific unprecedented cluster of fatal neonatal air embolism acts is extraordinarily low. Deliberate neonatal air embolism as a method of homicide is vanishingly rare in the international medical-legal literature. A cluster of seven fatal acts plus several attempts in eighteen months on one small unit has no international precedent. See our air-embolism base-rate analysis.
Quantitatively, this means the prior is on the order of at most one in millions. It could be higher if the unit had other specific indicators of criminal activity; it does not. The prior is therefore extremely low to begin with.
2. The likelihood under the prosecution hypothesis
If Letby had deliberately committed the alleged acts, what is the probability of observing the evidence the Crown adduced? Fenton’s analysis asks specifically: how likely are the observed skin signs under deliberate air embolism, how likely is the Roche Cobas insulin pattern under deliberate insulin administration, how likely is the shift-rota overlap pattern under single-actor causation?
On each strand, the likelihood is moderate-to-high but not definitive: the skin signs do not meet the Lee 1989 paper’s specificity criteria; the Roche Cobas result is a screening assay with false-positive rates; the shift-rota pattern is a function of selection as much as underlying pattern.
3. The likelihood under natural-cause alternatives
If the deaths and collapses were natural (sepsis, NEC, thrombosis, the documented outbreak conditions), how likely is the observed evidence? The answer, on independent expert review, is: high. Each piece of evidence the Crown adduced is explicable under the natural-cause framework with high likelihood.
The posterior probability
Bayes’s theorem produces:
Posterior probability of guilt = (very low prior) × (moderate likelihood under prosecution) / (moderate likelihood under prosecution + high likelihood under innocence) ≈ low posterior probability.
The posterior probability of guilt, calculated on Fenton’s framework with the available evidence, does not approach the criminal-law beyond-reasonable-doubt threshold. This is not advocacy; it is what a formal Bayesian analysis produces on the numbers the evidence supports.
Why this analysis is not speculative
Every component of the Bayesian analysis is grounded in the evidential record:
- The prior is grounded in the international forensic-pathology and criminological literature on rare homicide methods.
- The prosecution-hypothesis likelihood is grounded in the specific claims the Crown made at trial.
- The natural-cause likelihood is grounded in the Shoo Lee Panel’s case-by-case review and the documented outbreak, staffing and infrastructure conditions.
None of the inputs is speculative. The output — low posterior probability of guilt — is the mathematical consequence of the inputs.
The statistical-community consensus
Fenton’s Bayesian analysis is not a minority view within the statistical community. Prof. Richard Gill’s de-Berk-precedent analysis, Prof. Peter Green’s RSS-framework commentary, Sir David Spiegelhalter’s public communication work, and Prof. Leila Schneps’s Math on Trial framework each converge on the same conclusion: the statistical evidence the Crown adduced does not survive Bayesian scrutiny.
No body of peer-reviewed statistical work has emerged defending the Crown’s statistical framing against the Bayesian critique. The statistical-community consensus is settled. That is what the Cannings principle addresses, and what the CCRC application operationalises.