{ "cells": [ { "cell_type": "markdown", "id": "42c1db60", "metadata": {}, "source": [ "# Dynamic Models \n", "\n", "- **[1]** (##) Given the Markov property\n", "\\begin{equation*}\n", "p(x_n|x_{n-1},x_{n-2},\\ldots,x_1) = p(x_n|x_{n-1}) \\tag{A1}\n", "\\end{equation*}\n", "proof that, for any $n$,\n", "\\begin{align*}\n", "p(x_n,x_{n-1},&\\ldots,x_{k+1},x_{k-1},\\ldots,x_1|x_k) = \\\\\n", "&p(x_n,x_{n-1},\\ldots,x_{k+1}|x_k) \\cdot p(x_{k-1},x_{k-2},\\ldots,x_1|x_k) \\tag{A2}\\,.\n", "\\end{align*}\n", "In other words, proof that, if the Markov property A1 holds, then, given the \"present\" ($x_k$), the \"future\" $(x_n,x_{n-1},\\ldots,x_{k+1})$ is _independent_ of the \"past\" $(x_{k-1},x_{k-2},\\ldots,x_1)$.\n", "\n", "\n", "- **[2]** (#) \n", " (a) What's the difference between a hidden Markov model and a linear Dynamical system? \n", " \n", " (b) For the same number of state variables, which of these two models has a larger memory capacity, and why? \n", " \n", "- **[3]** (#) \n", "(a) What is the 1st-order Markov assumption? \n", "(b) Derive the joint probability distribution $p(x_{1:T},z_{0:T})$ (where $x_t$ and $z_t$ are observed and latent variables respectively) for the state-space model with transition and observation models $p(z_t|z_{t-1})$ and $p(x_t|z_t)$. \n", "(c) What is a Hidden Markov Model (HMM)? \n", "(d) What is a Linear Dynamical System (LDS)? \n", "(e) What is a Kalman Filter? \n", "(f) How does the Kalman Filter relate to the LDS? \n", "(g) Explain the popularity of Kalman filtering and HMMs? \n", "(h) How relates a HMM to a GMM? \n", "\n" ] }, { "cell_type": "code", "execution_count": null, "id": "f2b614bd", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.5.2", "language": "julia", "name": "julia-1.5" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.5.2" } }, "nbformat": 4, "nbformat_minor": 5 }