inference: fundamental equations

lectures_beamer/S1E2_Inference.tex

+% !TEX program = xelatex
+\PassOptionsToPackage{usenames,dvipsnames}{xcolor}
+\documentclass[aspectratio=169,20pt, USenglish]{beamer}
+
+\usetheme{ost}
+\input{include/beamer_basics.tex} % PACKAGES
+\input{include/mathsymbols.tex}   % SYMBOLS
+
+% lower case math
+\usepackage[cal=boondoxo]{mathalfa}
+
+% Change math fonts
+\usefonttheme[onlymath]{serif}
+
+% Show notes
+\usepackage{pgfpages}
+%\setbeameroption{show notes on second screen=bottom}
+\setbeamertemplate{note page}[plain]
+\makeatletter
+\def\beamer@framenotesbegin{% at beginning of slide
+     \usebeamercolor[fg]{normal text}
+      \gdef\beamer@noteitems{}%
+      \gdef\beamer@notes{}%
+}
+\makeatother
+% do not color links
+\hypersetup{colorlinks=false}
+
+% To show videos
+\usepackage{multimedia}
+
+\graphicspath{{../static/img/}}
+
+\title{Introduction to recursive probabilistic learning}
+\subtitle{Inference}
+\date{08.10.2021}
+\author{JuanPi Carbajal}
+\institute{Institute for Energy Technology}
+
+\begin{document}
+
+    \begin{frame}
+        \titlepage
+    \end{frame}
+
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    % 90 min
+    \section{Recap and Q\&A}
+    \interludeframe{\textbf{\large Recap and Q\&A}}
+    % TODO: Propagate errors to measurement
+
+
+    \section{Conditional probability}
+    \interludeframe{\textbf{\large Conditional probability}}
+    % Plausible reasoning
+    % Probability as logic
+    % Contingency table
+    % Probability
+    % Conditional probability
+    % Factorization of the joint probability
+    % TODO: Propagation errors to measurement as marginalization
+    % \input{include/probability.tex}
+
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    % \interludeframe{\textbf{\large Coffee break: 20 minutes}}
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    % 100 min
+
+    \section{Gaussian Distribution}
+    \interludeframe{\textbf{\large Gaussian (Normal) distribution}}
+    % Distribution of Jointly Gaussian variables
+    % Conditional distribution of Gaussian variables
+    \input{include/mvnormal_distribution_conditioning.tex}
+
+    \section{Update step}
+    \interludeframe{\textbf{\large Update step}}
+    % Noisy iterated map with measurement
+    % Conditional distribution of Gaussian variables
+    % Update step
+    \input{include/update_step.tex}
+
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    %\interludeframe{\textbf{\large Lunch: 1 hour 30 minutes}}
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    % 60 min
+
+    \section{Data driven models}
+    \interludeframe{\textbf{\large Data driven models}}
+    % What's data driven?
+    % recursive linear regression
+    % recursive delayed regression
+    % Polynomial regression
+    % Recursive extreme learning
+    % \input{include/data_driven_models.tex}
+
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    %\interludeframe{\textbf{\large Coffee break: 30 minutes}}
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    % 60 min
+    \section{Parameter estimation}
+    \interludeframe{\textbf{\large Parameter estimation}}
+    % Online parameter update
+    % Augmented state
+    % Maximum likelihood
+    % Batch regression of historical data: functional analysis, etc.
+    % \input{include/parameter_estimation.tex}
+
+    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+    % 60 min
+    \section{Wrap-up}
+    \interludeframe{\textbf{\large Wrap-up}}
+
+\end{document}

lectures_beamer/include/mathsymbols.tex

@@ -35,4 +35,3 @@
 \DeclareMathOperator*{\GP}{\mathcal{G}\!\mathcal{P}} % Gaussian process
 \newcommand{\ccov}[1]{{\color{red}k}\left(#1\right)}
 \newcommand{\cmean}[1]{{\color{blue}m}\left(#1\right)}
-

lectures_beamer/include/mvnormal_distribution_conditioning.tex

+\begin{frame}{Distribution of Jointly Gaussian variables}
+    \[
+        \gaussian(\bm{x}| \bm{m}, \tens{P}) = \frac{1}{\sqrt{(2\pi)^n\det{\tens{P}}}}
+        \exp{-\frac{1}{2}\left(\bm{x} - \bm{m}\right)^\trp \tens{P}^{-1}\left(\bm{x} - \bm{m}\right)}
+    \]
+    \begin{columns}[t]
+        \column{0.5\textwidth}
+            \[
+                    \begin{bmatrix} \bm{x} \\ \bm{y}\end{bmatrix}
+                    \sim \gaussian\left(\begin{bmatrix}\bm{a} \\ \bm{b}\end{bmatrix},
+                    \begin{bmatrix}\tens{A} & \tens{C} \\ \tens{C}^\trp & \tens{B}\end{bmatrix}\right)
+            \]
+            \begin{align*}
+                    \bm{x} &\sim \gaussian(\bm{a},\tens{A})\\
+                    \bm{y} &\sim \gaussian(\bm{b},\tens{B})\\
+                    \bm{x} | \bm{y} &\sim \gaussian(\bm{a} + \tens{C}\tens{B}^{-1}(\bm{y} - \bm{b}),
+                    \tens{A} - \tens{C}\tens{B}^{-1}\tens{C}^\trp)\\
+                    \bm{y} | \bm{x} &\sim \gaussian(\bm{b} + \tens{C}^\trp\tens{A}^{-1}(\bm{x} - \bm{a}),
+                    \tens{B} - \tens{C}^\trp\tens{A}^{-1}\tens{C})
+            \end{align*}
+        \column{0.5\textwidth}
+            \begin{align*}
+                \bm{x} &\sim \gaussian(\bm{m}, \tens{P})\\
+                \bm{y} | \bm{x} &\sim \gaussian(\tens{H}\bm{x} + \bm{u}, \tens{R})\\
+            \end{align*}
+            \begin{align*}
+                    \begin{bmatrix} \bm{x} \\ \bm{y}\end{bmatrix} &
+                    \sim \gaussian\left(\begin{bmatrix}\bm{m} \\ \tens{H}\bm{x} + \bm{u}\end{bmatrix},
+                    \begin{bmatrix}\tens{P} & \tens{P}\tens{H}^\trp \\ \tens{H}\tens{P} & \tens{H}\tens{P}\tens{H}^\trp + \tens{R}\end{bmatrix}\right)\\
+                    \bm{y} | \bm{x} &\sim \gaussian(\tens{H}\bm{x} + \bm{u}, \tens{H}\tens{P}\tens{H}^\trp + \tens{R})
+            \end{align*}
+    \end{columns}
+    \note<1>{
+    - Schur complement
+
+    - An interpretation of the conditional mean?
+
+    - Do exercise 3.5 from the book
+    }
+
+\end{frame}
\ No newline at end of file

lectures_beamer/include/update_step.tex

+\begin{frame}{Update step}
+    \begin{align*}
+        \mathcal{p}(\bm{x}_k|\bm{y}_{1:k}) &\propto \mathcal{p}(\bm{y}_k|\bm{x}_{k}) \mathcal{p}(\bm{x}_k|\bm{y}_{1:{k-1}})\\
+        \mathcal{p}(\bm{x}_k|\bm{y}_{1:{k-1}}) &= \bm{\int}{\mathcal{p}(\bm{x}_k|\bm{x}_{k-1}) \mathcal{p}(\bm{x}_{k-1}|\bm{y}_{1:{k-1}})\ud \bm{x}_{k-1}}
+    \end{align*}
+    \begin{align*}
+        \tens{S}_k &= \tens{H}_k \hat{\tens{P}}_k \tens{H}_k^\trp + \tens{R}_k\\
+        \tens{K}_k &= \hat{\tens{P}}_k \tens{H}^{\trp}\tens{S}_k^{-1}\\
+        \bm{m}_k &= \hat{\bm{m}}_k + \tens{K}_k\left(\bm{y}_k - \tens{H}_k \hat{\bm{m}}_k\right)\\
+        \tens{P}_k &= \hat{\tens{P}}_k - \tens{K}_k \tens{S}_k \tens{K}_k^{\trp}
+    \end{align*}
+    \note<1>{
+    - propagation of uncertainty to output using best estimate
+
+    - kalman gain
+
+    - update mean by projecting error on kalman gain
+
+    - update cov  by propagating uncertainty on measurement backward
+    }
+\end{frame}
+
+\begin{frame}{Kalman filter}
+    Predict (while no measurement):
+    \begin{align*}
+        \hat{\bm{m}}_k = \tens{A}_k\bm{m}_{k-1}\\
+        \tens{P}_k = \tens{A}_k\tens{P}_{k-1}\tens{A}_k^\trp + \tens{Q}_{k-1}
+    \end{align*}
+    Update (condition on measurement):
+    \begin{align*}
+        \tens{S}_k &= \tens{H}_k \hat{\tens{P}}_k \tens{H}_k^\trp + \tens{R}_k\\
+        \tens{K}_k &= \hat{\tens{P}}_k \tens{H}^{\trp}\tens{S}_k^{-1}\\
+        \bm{m}_k &= \hat{\bm{m}}_k + \tens{K}_k\left(\bm{y}_k - \tens{H}_k \hat{\bm{m}}_k\right)\\
+        \tens{P}_k &= \hat{\tens{P}}_k - \tens{K}_k \tens{S}_k \tens{K}_k^{\trp}
+    \end{align*}
+\end{frame}
\ No newline at end of file