@misc{Riv18a,
  author =        { Ronald L. Rivest },
  title =         { Bayesian Tabulation Audits: Explained and Extended },
  date =          { 2018-01-01 },
  OPTyear =       { 2018 },
  OPTmonth =      { January 1, },
  urla =          { <a href="https://arxiv.org/abs/1801.00528">arXiv</a> },
  abstract = {%
  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.
  \par
  Stark
  proposed \textbf{risk-limiting tabulation audits} for this purpose;
  such audits are effective and are beginning to be used in
  practice in Colorado and other states.
  \par
  We expand the study of election audits based on \textbf{Bayesian}
  methods.  Such Bayesian audits
  use a slightly different approach first introduced by Rivest and Shen in
  2012.  (The risk-limiting audits
  proposed by Stark are ``frequentist'' rather than Bayesian in
  character.)
  \par
  We first provide a simplified presentation of
  Bayesian tabulation audits.  
  Suppose an election has been run and the tabulation of votes
  reports a given outcome.
  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.
  \par
  Bayesian audits are attractive because they work with \textbf{any}
  method for determining the winner (such as ranked-choice voting).
  \par
  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.''
  \par
  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.
  \par
  We also discuss the benefits and relevant considerations for using
  Bayesian audits in practice.
  }                  
}