Expected information gain (EIG) is a crucial quantity in Bayesian optimal experimental design (BOED), quantifying how useful an experiment is by the amount we expect the posterior to differ from the prior. However, evaluating the EIG can be computationally expensive since it generally requires estimating the posterior normalizing constant. In this work, we leverage two idiosyncrasies of BOED to improve efficiency of EIG estimation via sequential Monte Carlo (SMC). First, in BOED we simulate the data and thus know the true underlying parameters. Second, we ultimately care about the EIG, not the individual normalizing constants. Often we observe that the Monte Carlo variance of standard SMC estimators for the normalizing constant of a single dataset are significantly lower than the variance of the normalizing constants across datasets; the latter thus contributes the majority of the variance for EIG estimates. This suggests the potential to slightly increase variance while drastically decreasing computation time by reducing the SMC population size, which leads us to an EIG-specific SMC estimator that starts with a only a single sample from the posterior and tempers \textit{backwards} towards the prior. Using this single-sample estimator, which we call reverse-annealed SMC (RA-SMC), we show that it is possible to estimate EIG with orders of magnitude fewer likelihood evaluations in three models: a four-dimensional spring-mass, a six-dimensional Johnson-Cook model and a four-dimensional source-finding problem.

Section 16(b) of the Securities Exchange Act of 1934 allows for the recovery of profits realized by certain insiders from trading in a corporation's stock within a period of less than six months. For more than seventy years, U.S. courts and corporate attorneys have calculated this liability following the greedy algorithm described in Smolowe v. Delendo Corp. (2nd Cir. 1943), which the Securities and Exchange Commission proposed as a method of maximizing the recovery in that case. Even though Dantzig's simplex algorithm (1947) subsequently provided a more accurate method for calculating the maximum recovery as the solution to a linear programming problem, the legal community to date has resisted its adoption. This paper provides (1) a brief introduction to Section 16(b) and the Smolowe algorithm; (2) a review of the caselaw that has enshrined the Smolowe algorithm in legal precedent; (3) a proof that the Smolowe algorithm's worst case error is 50%; (4) a description of a new Web-based liability calculator for the legal community's use; and (5) a historically important case where the new calculator yields a larger recovery than the amount actually sought and obtained by the plaintiffs.