  Tabulation audits for an election provide statistical evidence that a
reported contest outcome is "correct" (meaning that the tabulation of votes was
properly performed), or else the tabulation audit determines the correct
outcome.
  Stark proposed risk-limiting tabulation audits for this purpose; such audits
are effective and are beginning to be used in practice.
  We expand the study of election audits based on Bayesian methods, first
introduced by Rivest and Shen in 2012. (The risk-limiting audits proposed by
Stark are "frequentist" rather than Bayesian in character.)
  We first provide a simplified presentation of Bayesian tabulation audits. A
Bayesian tabulation audit begins by drawing a random sample of the votes in
that contest, and tallying those votes. It then considers what effect
statistical variations of this tally have on the contest outcome. If such
variations almost always yield the previously-reported outcome, the audit
terminates, accepting the reported outcome. Otherwise the audit is repeated
with an enlarged sample.
  Bayesian audits are attractive because they work with any method for
determining the winner (such as ranked-choice voting).
  We then show how Bayesian audits may be extended to handle more complex
situations, such as auditing contests that \emph{span multiple jurisdictions},
or are otherwise "stratified."
  We highlight the auditing of such multiple-jurisdiction contests where some
of the jurisdictions have an electronic cast vote record (CVR) for each cast
paper vote, while the others do not. Complex situations such as this may arise
naturally when some counties in a state have upgraded to new equipment, while
others have not. Bayesian audits are able to handle such situations in a
straightforward manner.
  We also discuss the benefits and relevant considerations for using Bayesian
audits in practice.
