The goal of sequential learning is to draw inference from data that is gathered gradually through time. This is a typical situation in many applications, including finance. A sequential inference procedure is `anytime-valid’ if the decision to stop or continue an experiment can depend on anything that has been observed so far, without compromising statistical error guarantees. A recent approach to anytime-valid inference views a test statistic as a bet against the null hypothesis. These bets are constrained to be supermartingales - hence unprofitable - under the null, but designed to be profitable under the relevant alternative hypotheses. This perspective opens the door to tools from financial mathematics. In this talk I will discuss how notions such as supermartingale measures, log-optimality, and the optional decomposition theorem shed new light on anytime-valid sequential learning. (This talk is based on joint work with Wouter Koolen (CWI), Aaditya Ramdas (CMU) and Johannes Ruf (LSE).)
 

We characterize which Borel functions are decomposable into
a countable union of functions which are piecewise continuous on
$\Pi^0_n$ domains, assuming projective determinacy. One ingredient of
our proof is a new characterization of what Borel sets are $\Sigma^0_n$
complete. Another important ingredient is a theorem of Harrington that
there is no projective sequence of length $\omega_1$ of distinct Borel
sets of bounded rank, assuming projective determinacy. This is joint
work with Adam Day.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.