Convergence Rates of Gaussian ODE Filters

A recently-introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution $x$ and its first $q$ derivatives \emph{a priori} as a Gauss--Markov process $\boldsymbol{X}$, which is then iteratively conditioned on information about $\dot{x}$. This article establishes worst-case local convergence rates of order $q+1$ for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order $q$ in the case of $q=1$ and an integrated Brownian motion prior, and analyses how inaccurate information on $\dot{x}$ coming from approximate evaluations of $f$ affects these rates. Moreover, we show that, in the globally convergent case, the posterior credible intervals are well calibrated in the sense that they globally contract at the same rate as the truncation error. We illustrate these theoretical results by numerical experiments which might indicate their generalizability to $q \in \{2,3,\dots\}$.Comment: 26 pages, 5 figure

Kindergarten Enrollment and the Intergenerational Transmission of Education

We use Swiss data to test whether intergenerational educational mobility is affected by the age at which children enroll in kindergarten. Taking advantage of heterogeneity across cantons we find that early kindergarten enrollment significantly increases educational mobility.Kindergarten, pre-school enrollment, educational mobility, intergenerational transmission of education

Age at School Entry and Intergenerational Educational Mobility

We use Swiss data to test whether intergenerational educational mobility is affected by the age at which children first enter (primary) school. Early age at school entry significantly affects mobility and reduces the relative advantage of children of better educated parents.educational mobility, intergenerational transmission of education, age at entry

Age at School Entry and Intergenerational Educational Mobility

We use Swiss data to test whether intergenerational educational mobility is affected by the age at which children first enter (primary) school. Early age at school entry significantly affects mobility and reduces the relative advantage of children of better educated parents.age at entry, intergenerational transmission of education, educational mobility