Stochastic Processes

There is a beautiful connection between probability theory and partial differential equations (PDEs), given by the famous Feynman–Kac theorem. This result allows us to translate between finite-dimensional stochastic problems and infinite-dimensional deterministic problems. It appears throughout finance, physics, control theory, and machine learning (generative modeling and reinforcement learning). What makes this connection especially useful is that it works in both directions. The solution to a high-dimensional PDE (which can be very expensive to compute numerically) can be evaluated at a single point simply by simulating a random process. Conversely, a difficult question about a stochastic process can be transformed into a deterministic PDE and tackled using standard techniques. ...

Gaussian Process Regression is one of the most elegant and theoretically rich algorithms in machine learning. With this post, I want to celebrate the mathematical beauty underlying Gaussian Processes. I will divide this post into two sections: theory and practice, accompanied by code examples. One of the key advantages of Gaussian Processes compared to Deep Learning methods is that they inherently provide interpretability (through confidence intervals and uncertainty estimation). They also offer excellent extrapolation properties, as we will see, and a way to incorporate knowledge about the structure of the data into the model. However, these benefits come at a cost. The algorithm has a wide variety of hyperparameters that are difficult to configure; for instance, kernel selection alone is challenging. Understanding and having a good intuition for the inner workings of this algorithm (and the data) is key to making the most of it. ...