 Hey guys, I'm Nico, and I'm finally going to explain what a probabilistic ODE solver is. Some people say that probabilistic ODE solvers are just Kalman filters, and others say that this is a drastic oversimplification. While I'll do my best to not take sides in this argument, let me at least explain where the Kalman filter comes into play. But first, recall the setup. The task is to solve an ODE, which, since in most interesting applications has a non-linear vector field F, needs to be done with a numerical algorithm. This raises two questions. First of all, how much error does this algorithm introduce, and second of all, how well does this algorithm interact with other pieces of a statistical model? We'll try to answer both questions by using probabilistic inference to solve ODE's, and since we're replicating numerical algorithms, we'll call it probabilistic numerics. We need a prior, a likelihood, and we need to compute a posterior. As a prior, Gauss-Marckoff processes are a good choice. They solve linear, time-invarying SDEs, which is the reason why we can compute the transition densities from the state at some time point to the state at another time point easily. And this is one of the reasons why the probabilistic ODE solver is as fast as non-probabilistic methods. In this plot here, you can see an integrated Vina process prior. Here in Tubingen, we like integrated Vina processes, but you can use any Gauss-Marckoff prior. The measurement model is what turns this algorithm into an ODE solver. I'm glad you're asking. So assume that the prior process was an ODE solution. Then it would satisfy y dot minus f of y equals zero. So we'll just turn this into our likelihood, we'll measure the discrepancy between y dot and f of y at a bunch of time points, and if this discrepancy is large, then we have a terrible approximation of our ODE solution. But if it's small, then we have a great approximation. So let's only keep it small. Right? We'll use an excellent Kalman filter to iterate forward through time, grid point to grid point, and at each grid point, we'll update the current belief on the fact that this discrepancy shall be as small as possible. And by the time we reach the end of our time domain, we have recovered something that looks like the solution of the logistic ODE. We're only one backward pass away from having a full posterior with uncertainty intervals that we can sample from or do all sorts of other things with. So let me recap. In some sense, probabilistic ODE solvers are, in fact, excellent Kalman filters. And this opens up a few fun directions. First of all, we can use any Bayesian filter as a probabilistic ODE solver. And if we're using them inside inverse problems, then there's no need to treat the ODE solver as a black box anymore. Or while we're at it, we might as well merge them with probabilistic models all together. In practice, there are some things that need to be taken into account though, but that's a story for a different day. All in all, next time you're talking to a probabilistic numeric person and they are trying to impress you with their ODE solver knowledge, then just let them know how you're well aware this is just a Kalman filter and see what happens.