https://grdm.io/posts/Recent content in Posts on Gerardo Duran-MartinHugo -- gohugo.ioen-usThu, 06 Mar 2025 00:00:00 +0000https://grdm.io/posts/filtering-notes-ii/Thu, 06 Mar 2025 00:00:00 +0000https://grdm.io/posts/filtering-notes-ii/Part II of the filtering notes series: Part I: signal plus noise models Part II: state-space-models* Introduction Whether modelling the trajectory of a rocket, forecasting economic trends, or powering large language models, discrete linear state-space models (SSMs) provide a unifying framework for estimation, prediction, and learning. At their core, a linear SSM models observations as projections of an unknown, time-dependent latent process.1 A classic example is object tracking, where noisy measurements—like the angle between a rocket and two stars—are used to reconstruct the rocket’s trajectory.https://grdm.io/posts/filtering-notes-i/Mon, 06 Jan 2025 00:00:00 +0000https://grdm.io/posts/filtering-notes-i/Part I of the filtering notes series: Part I: signal plus noise models* Part II: state-space-models Introduction From tracking epidemics and predicting financial markets to controlling complex systems like rockets and personalising online experiences, many real-world problems rely on understanding and processing sequential data. In machine learning, this includes classes of problems such as online (continual) learning, one-step-ahead (prequential) forecasting, contextual bandits, and reinforcement learning. Filtering techniques, such as the Kalman Filter, have a long and successful history of processing sequential data in fields like aerospace engineering and signal processing.https://grdm.io/posts/bone-slides/Wed, 04 Dec 2024 00:00:00 +0000https://grdm.io/posts/bone-slides/(B)ayesian (O)nline learning in (N)on-stationary (E)nvironments Paper: arxiv/2411.10153 Slides: bone/talk Repo: gerdm/BONE What is BONE? Framework that encompasses many Bayesian models that tackle online learning in non-stationary environments. By Bayesian we mean using Bayes’ rule to update beliefs. By online we mean observing a stream of datapoints and making a decision about the next datapoint. By non-stationary we mean that the data-generating process (DGP) changes through time. Online learning and prediction Observe sequence of features $x_t$ and observations $y_t$: $$ {\cal D}_{1:t} = \{(x_1, y_1), \ldots, (x_t, y_t)\}.https://grdm.io/posts/bocd-coin-tosses/Wed, 09 Oct 2024 00:00:00 +0000https://grdm.io/posts/bocd-coin-tosses/Introduction In this post, we introduce the Bayesian Online Changepoint Detection (BOCD) model and its application for estimating the probability of heads in a sequence of coin tosses where the underlying probability changes over time. Flipping coins with fixed probability Suppose the following sequence of coin tosses arrive in a stream: 1 2 3 4 5 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0 Here, 1 corresponds to heads and 0 corresponds to tails.https://grdm.io/posts/wolf-ewma/Sat, 13 Jul 2024 00:00:00 +0000https://grdm.io/posts/wolf-ewma/Our weighted-observation likelihood filter (WolF) got accepted at ICML 2024. Our method robustifies the Kalman filter (KF) by replacing the classical Gaussian likelihood assumption with a loss function. As we show, this modification results in a provably-robust and easy-to-implement variant of the KF. Despite its simplicity, the terminology behind WoLF (similar to the terminology of the KF), might not be familiar to everyone and the method might seem a bit abstract for newcomers.https://grdm.io/posts/kalman-filter-in-four-steps/Wed, 22 Feb 2023 10:23:00 +0530https://grdm.io/posts/kalman-filter-in-four-steps/The Kalman filter Suppose $\forall t=1,\ldots,T.{\bf x}_{t}\in\mathbb{R}^n, {\bf y}_t\in\mathbb{R}^m$, $$ \begin{aligned} p({\bf x}_t \vert {\bf x}_{t-1}) &= {\cal N}({\bf x}_t \vert {\bf A}_{t-1}{\bf x}_{t-1}, {\bf Q}_{t-1})\\ p({\bf y}_t \vert {\bf x}_t) &= {\cal N}({\bf y}_t \vert {\bf H}_t{\bf x}_t, {\bf R}_t) \end{aligned} $$ then, the Kalman filter equations are given by $$ \begin{aligned} p({\bf x}_t \vert {\bf y}_{1:t-1}) &= {\cal N}({\bf x}_t \vert \bar{\bf m}_t,\bar{\bf P}_t) &\quad \text{(Predict)}\\ p({\bf x}_t \vert {\bf y}_{1:t}) &= {\cal N}({\bf x}_t \vert {\bf m}_t,{\bf P}_t) &\quad \text{(Update)} \end{aligned} $$ where the prediction-step equations are given byhttps://grdm.io/posts/einsums-in-the-wild/Thu, 25 Nov 2021 00:00:00 +0000https://grdm.io/posts/einsums-in-the-wild/Do you know what inm,kij,jnm->knm is all about? For an interactive version of this post, see this Colab notebook. Introduction Linear combinations are ubiquitous in machine learning and statistics. Many algorithms and models in the statistics and machine learning literature can be written (or approximated) as a matrix-vector multiplication. Einsums are a way of representing the linear interaction among vectors, matrices and higher-order dimensional arrays. In this post, I lay out examples that make use of Einsums.https://grdm.io/posts/gsoc2021/Mon, 16 Aug 2021 00:00:00 +0000https://grdm.io/posts/gsoc2021/Introduction In this post I report my work during the Google Summer of Code (GSoC) 2021 program. During the summer I contributed to Volume 2 of the Probabilistic Machine Learning book by Kevin P. Murphy. The GSoC 2021 TF/Pyprobml program was divided into in two Github repositories: a public repo probml/pyprobml where the final code was pushed and a private repo pyprobml/hermes where discussions and code reviews took place. Highlights I spent 394.https://grdm.io/posts/journey-prml/Thu, 10 Sep 2020 00:00:00 +0000https://grdm.io/posts/journey-prml/On July 10, 2018 I committed myself to read and do all the exercises of the book Pattern Recognition and Machine Learning from Christopher Bishop. After seven hundred and ninety two days, I can finally say I accomplished my goal. One of the main reasons why I decided to study the book from cover to cover was to demystify machine learning (ML). Reading through the book taught me to question the underlying assumptions behind every model, to understand from a mathematical point of view why some models fail, and to not fret the mathematics behind ML.