A risk-limiting post-election audit is a procedure for confirming the outcome of an election that has a known, pre-specified minimum chance of correcting an incorrect electoral outcome. Statistical methods have been developed for risk-limiting ballot comparison and ballot-polling audits of plurality contests and elections with other relatively simple social choice functions, including D’Hondt, approval voting, and Borda count. Complicated statistical likelihoods make it difficult to derive risk-limiting audits for more complex electoral systems. Examples include elections where precincts use heterogeneous voting technology and elections with complex social choice functions. Rivest and Shen (2012) proposed Bayesian audits as a solution. Bayesian audits use simulation, rather than analytical expressions, to estimate the posterior probability that the reported outcome is incorrect, given the observed audit sample. This auditing method is agnostic to the social choice function and can work for both ballot-polling and ballot comparison audits. In general, Bayesian audits do not control the risk of failing to correct an incorrect electoral outcome. We may exploit the relationship between the posterior probabilities derived from Bayesian audits and the frequentist risk to derive new risk-limiting audits for complex elections.
We address two questions here:
- When is the Bayesian audit method of Rivest and Shen risk-limiting?
- Can we exploit the Bayes-minimax duality to derive new risk-limiting audits based on existing Bayesian ones?