{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "# meta-data does not work yet in VScode\n", "# https://github.com/microsoft/vscode-jupyter/issues/1121\n", "\n", "{\n", " \"tags\": [\n", " \"hide-cell\"\n", " ]\n", "}\n", "\n", "\n", "### Install necessary libraries\n", "\n", "try:\n", " import jax\n", "except:\n", " # For cuda version, see https://github.com/google/jax#installation\n", " %pip install --upgrade \"jax[cpu]\" \n", " import jax\n", "\n", "try:\n", " import jsl\n", "except:\n", " %pip install git+https://github.com/probml/jsl\n", " import jsl\n", "\n", "try:\n", " import rich\n", "except:\n", " %pip install rich\n", " import rich\n", "\n", "\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "{\n", " \"tags\": [\n", " \"hide-cell\"\n", " ]\n", "}\n", "\n", "\n", "### Import standard libraries\n", "\n", "import abc\n", "from dataclasses import dataclass\n", "import functools\n", "import itertools\n", "\n", "from typing import Any, Callable, NamedTuple, Optional, Union, Tuple\n", "\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "\n", "\n", "import jax\n", "import jax.numpy as jnp\n", "from jax import lax, vmap, jit, grad\n", "from jax.scipy.special import logit\n", "from jax.nn import softmax\n", "from functools import partial\n", "from jax.random import PRNGKey, split\n", "\n", "import inspect\n", "import inspect as py_inspect\n", "import rich\n", "\n", "from rich import inspect as r_inspect\n", "from rich import print as r_print\n", "\n", "def print_source(fname):\n", " r_print(py_inspect.getsource(fname))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "```{math}\n", "\n", "\\newcommand\\floor[1]{\\lfloor#1\\rfloor}\n", "\n", "\\newcommand{\\real}{\\mathbb{R}}\n", "\n", "% Numbers\n", "\\newcommand{\\vzero}{\\boldsymbol{0}}\n", "\\newcommand{\\vone}{\\boldsymbol{1}}\n", "\n", "% Greek https://www.latex-tutorial.com/symbols/greek-alphabet/\n", "\\newcommand{\\valpha}{\\boldsymbol{\\alpha}}\n", "\\newcommand{\\vbeta}{\\boldsymbol{\\beta}}\n", "\\newcommand{\\vchi}{\\boldsymbol{\\chi}}\n", "\\newcommand{\\vdelta}{\\boldsymbol{\\delta}}\n", "\\newcommand{\\vDelta}{\\boldsymbol{\\Delta}}\n", "\\newcommand{\\vepsilon}{\\boldsymbol{\\epsilon}}\n", "\\newcommand{\\vzeta}{\\boldsymbol{\\zeta}}\n", "\\newcommand{\\vXi}{\\boldsymbol{\\Xi}}\n", "\\newcommand{\\vell}{\\boldsymbol{\\ell}}\n", "\\newcommand{\\veta}{\\boldsymbol{\\eta}}\n", "%\\newcommand{\\vEta}{\\boldsymbol{\\Eta}}\n", "\\newcommand{\\vgamma}{\\boldsymbol{\\gamma}}\n", "\\newcommand{\\vGamma}{\\boldsymbol{\\Gamma}}\n", "\\newcommand{\\vmu}{\\boldsymbol{\\mu}}\n", "\\newcommand{\\vmut}{\\boldsymbol{\\tilde{\\mu}}}\n", "\\newcommand{\\vnu}{\\boldsymbol{\\nu}}\n", "\\newcommand{\\vkappa}{\\boldsymbol{\\kappa}}\n", "\\newcommand{\\vlambda}{\\boldsymbol{\\lambda}}\n", "\\newcommand{\\vLambda}{\\boldsymbol{\\Lambda}}\n", "\\newcommand{\\vLambdaBar}{\\overline{\\vLambda}}\n", "%\\newcommand{\\vnu}{\\boldsymbol{\\nu}}\n", "\\newcommand{\\vomega}{\\boldsymbol{\\omega}}\n", "\\newcommand{\\vOmega}{\\boldsymbol{\\Omega}}\n", "\\newcommand{\\vphi}{\\boldsymbol{\\phi}}\n", "\\newcommand{\\vvarphi}{\\boldsymbol{\\varphi}}\n", "\\newcommand{\\vPhi}{\\boldsymbol{\\Phi}}\n", "\\newcommand{\\vpi}{\\boldsymbol{\\pi}}\n", "\\newcommand{\\vPi}{\\boldsymbol{\\Pi}}\n", "\\newcommand{\\vpsi}{\\boldsymbol{\\psi}}\n", "\\newcommand{\\vPsi}{\\boldsymbol{\\Psi}}\n", "\\newcommand{\\vrho}{\\boldsymbol{\\rho}}\n", "\\newcommand{\\vtheta}{\\boldsymbol{\\theta}}\n", "\\newcommand{\\vthetat}{\\boldsymbol{\\tilde{\\theta}}}\n", "\\newcommand{\\vTheta}{\\boldsymbol{\\Theta}}\n", "\\newcommand{\\vsigma}{\\boldsymbol{\\sigma}}\n", "\\newcommand{\\vSigma}{\\boldsymbol{\\Sigma}}\n", "\\newcommand{\\vSigmat}{\\boldsymbol{\\tilde{\\Sigma}}}\n", "\\newcommand{\\vsigmoid}{\\vsigma}\n", "\\newcommand{\\vtau}{\\boldsymbol{\\tau}}\n", "\\newcommand{\\vxi}{\\boldsymbol{\\xi}}\n", "\n", "\n", "% Lower Roman (Vectors)\n", "\\newcommand{\\va}{\\mathbf{a}}\n", "\\newcommand{\\vb}{\\mathbf{b}}\n", "\\newcommand{\\vBt}{\\mathbf{\\tilde{B}}}\n", "\\newcommand{\\vc}{\\mathbf{c}}\n", "\\newcommand{\\vct}{\\mathbf{\\tilde{c}}}\n", "\\newcommand{\\vd}{\\mathbf{d}}\n", "\\newcommand{\\ve}{\\mathbf{e}}\n", "\\newcommand{\\vf}{\\mathbf{f}}\n", "\\newcommand{\\vg}{\\mathbf{g}}\n", "\\newcommand{\\vh}{\\mathbf{h}}\n", "%\\newcommand{\\myvh}{\\mathbf{h}}\n", "\\newcommand{\\vi}{\\mathbf{i}}\n", "\\newcommand{\\vj}{\\mathbf{j}}\n", "\\newcommand{\\vk}{\\mathbf{k}}\n", "\\newcommand{\\vl}{\\mathbf{l}}\n", "\\newcommand{\\vm}{\\mathbf{m}}\n", "\\newcommand{\\vn}{\\mathbf{n}}\n", "\\newcommand{\\vo}{\\mathbf{o}}\n", "\\newcommand{\\vp}{\\mathbf{p}}\n", "\\newcommand{\\vq}{\\mathbf{q}}\n", "\\newcommand{\\vr}{\\mathbf{r}}\n", "\\newcommand{\\vs}{\\mathbf{s}}\n", "\\newcommand{\\vt}{\\mathbf{t}}\n", "\\newcommand{\\vu}{\\mathbf{u}}\n", "\\newcommand{\\vv}{\\mathbf{v}}\n", "\\newcommand{\\vw}{\\mathbf{w}}\n", "\\newcommand{\\vws}{\\vw_s}\n", "\\newcommand{\\vwt}{\\mathbf{\\tilde{w}}}\n", "\\newcommand{\\vWt}{\\mathbf{\\tilde{W}}}\n", "\\newcommand{\\vwh}{\\hat{\\vw}}\n", "\\newcommand{\\vx}{\\mathbf{x}}\n", "%\\newcommand{\\vx}{\\mathbf{x}}\n", "\\newcommand{\\vxt}{\\mathbf{\\tilde{x}}}\n", "\\newcommand{\\vy}{\\mathbf{y}}\n", "\\newcommand{\\vyt}{\\mathbf{\\tilde{y}}}\n", "\\newcommand{\\vz}{\\mathbf{z}}\n", "%\\newcommand{\\vzt}{\\mathbf{\\tilde{z}}}\n", "\n", "\n", "% Upper Roman (Matrices)\n", "\\newcommand{\\vA}{\\mathbf{A}}\n", "\\newcommand{\\vB}{\\mathbf{B}}\n", "\\newcommand{\\vC}{\\mathbf{C}}\n", "\\newcommand{\\vD}{\\mathbf{D}}\n", "\\newcommand{\\vE}{\\mathbf{E}}\n", "\\newcommand{\\vF}{\\mathbf{F}}\n", "\\newcommand{\\vG}{\\mathbf{G}}\n", "\\newcommand{\\vH}{\\mathbf{H}}\n", "\\newcommand{\\vI}{\\mathbf{I}}\n", "\\newcommand{\\vJ}{\\mathbf{J}}\n", "\\newcommand{\\vK}{\\mathbf{K}}\n", "\\newcommand{\\vL}{\\mathbf{L}}\n", "\\newcommand{\\vM}{\\mathbf{M}}\n", "\\newcommand{\\vMt}{\\mathbf{\\tilde{M}}}\n", "\\newcommand{\\vN}{\\mathbf{N}}\n", "\\newcommand{\\vO}{\\mathbf{O}}\n", "\\newcommand{\\vP}{\\mathbf{P}}\n", "\\newcommand{\\vQ}{\\mathbf{Q}}\n", "\\newcommand{\\vR}{\\mathbf{R}}\n", "\\newcommand{\\vS}{\\mathbf{S}}\n", "\\newcommand{\\vT}{\\mathbf{T}}\n", "\\newcommand{\\vU}{\\mathbf{U}}\n", "\\newcommand{\\vV}{\\mathbf{V}}\n", "\\newcommand{\\vW}{\\mathbf{W}}\n", "\\newcommand{\\vX}{\\mathbf{X}}\n", "%\\newcommand{\\vXs}{\\vX_{\\vs}}\n", "\\newcommand{\\vXs}{\\vX_{s}}\n", "\\newcommand{\\vXt}{\\mathbf{\\tilde{X}}}\n", "\\newcommand{\\vY}{\\mathbf{Y}}\n", "\\newcommand{\\vZ}{\\mathbf{Z}}\n", "\\newcommand{\\vZt}{\\mathbf{\\tilde{Z}}}\n", "\\newcommand{\\vzt}{\\mathbf{\\tilde{z}}}\n", "\n", "\n", "%%%%\n", "\\newcommand{\\hidden}{\\vz}\n", "\\newcommand{\\obs}{\\vy}\n", "\\newcommand{\\inputs}{\\vu}\n", "\\newcommand{\\input}{\\inputs}\n", "\n", "\\newcommand{\\hmmTrans}{\\vA}\n", "\\newcommand{\\hmmObs}{\\vB}\n", "\\newcommand{\\hmmInit}{\\vpi}\n", "\\newcommand{\\hmmhid}{\\hidden}\n", "\\newcommand{\\hmmobs}{\\obs}\n", "\n", "\\newcommand{\\ldsDyn}{\\vA}\n", "\\newcommand{\\ldsObs}{\\vC}\n", "\\newcommand{\\ldsDynIn}{\\vB}\n", "\\newcommand{\\ldsObsIn}{\\vD}\n", "\\newcommand{\\ldsDynNoise}{\\vQ}\n", "\\newcommand{\\ldsObsNoise}{\\vR}\n", "\n", "\\newcommand{\\ssmDyn}{f}\n", "\\newcommand{\\ssmObs}{h}\n", "```\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "(sec:ssm-intro)=\n", "# What are State Space Models?\n", "\n", "\n", "A state space model or SSM\n", "is a partially observed Markov model,\n", "in which the hidden state, $\\hidden_t$,\n", "evolves over time according to a Markov process,\n", "possibly conditional on external inputs or controls $\\input_t$,\n", "and each hidden state generates some\n", "observations $\\obs_t$ at each time step.\n", "(In this book, we mostly focus on discrete time systems,\n", "although we consider the continuous-time case in XXX.)\n", "We get to see the observations, but not the hidden state.\n", "Our main goal is to infer the hidden state given the observations.\n", "However, we can also use the model to predict future observations,\n", "by first predicting future hidden states, and then predicting\n", "what observations they might generate.\n", "By using a hidden state $\\hidden_t$\n", "to represent the past observations, $\\obs_{1:t-1}$,\n", "the model can have ``infinite'' memory,\n", "unlike a standard Markov model.\n", "\n", "Formally we can define an SSM \n", "as the following joint distribution:\n", "```{math}\n", ":label: SSMfull\n", "p(\\hmmobs_{1:T},\\hmmhid_{1:T}|\\inputs_{1:T})\n", " = \\left[ p(\\hmmhid_1|\\inputs_1) \\prod_{t=2}^{T}\n", " p(\\hmmhid_t|\\hmmhid_{t-1},\\inputs_t) \\right]\n", " \\left[ \\prod_{t=1}^T p(\\hmmobs_t|\\hmmhid_t, \\inputs_t, \\hmmobs_{t-1}) \\right]\n", "```\n", "where $p(\\hmmhid_t|\\hmmhid_{t-1},\\inputs_t)$ is the\n", "transition model,\n", "$p(\\hmmobs_t|\\hmmhid_t, \\inputs_t, \\hmmobs_{t-1})$ is the\n", "observation model,\n", "and $\\inputs_{t}$ is an optional input or action.\n", "See {numref}`Figure %s ` \n", "for an illustration of the corresponding graphical model.\n", "\n", "\n", "```{figure} /figures/SSM-AR-inputs.png\n", ":scale: 100%\n", ":name: ssm-ar\n", "\n", "Illustration of an SSM as a graphical model.\n", "```\n", "\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We often consider a simpler setting in which there\n", "are no external inputs,\n", "and the observations are conditionally independent of each other\n", "(rather than having Markovian dependencies) given the hidden state.\n", "In this case the joint simplifies to \n", "```{math}\n", ":label: SSMsimplified\n", "p(\\hmmobs_{1:T},\\hmmhid_{1:T})\n", " = \\left[ p(\\hmmhid_1) \\prod_{t=2}^{T}\n", " p(\\hmmhid_t|\\hmmhid_{t-1}) \\right]\n", " \\left[ \\prod_{t=1}^T p(\\hmmobs_t|\\hmmhid_t \\right]\n", "```\n", "See {numref}`Figure %s ` \n", "for an illustration of the corresponding graphical model.\n", "Compare {eq}`SSMfull` and {eq}`SSMsimplified`.\n", "\n", "\n", "```{figure} /figures/SSM-simplified.png\n", ":scale: 100%\n", ":name: ssm-simplified\n", "\n", "Illustration of a simplified SSM.\n", "```" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "(sec:hmm-intro)=\n", "# Hidden Markov Models\n", "\n", "In this section, we discuss the\n", "hidden Markov model or HMM,\n", "which is a state space model in which the hidden states\n", "are discrete, so $\\hmmhid_t \\in \\{1,\\ldots, K\\}$.\n", "The observations may be discrete,\n", "$\\hmmobs_t \\in \\{1,\\ldots, C\\}$,\n", "or continuous,\n", "$\\hmmobs_t \\in \\real^D$,\n", "or some combination,\n", "as we illustrate below.\n", "More details can be found in e.g., \n", "{cite}`Rabiner89,Fraser08,Cappe05`.\n", "For an interactive introduction,\n", "see https://nipunbatra.github.io/hmm/." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "(sec:casino-ex)=\n", "### Example: Casino HMM\n", "\n", "To illustrate HMMs with categorical observation model,\n", "we consider the \"Ocassionally dishonest casino\" model from {cite}`Durbin98`.\n", "There are 2 hidden states, representing whether the dice being used in the casino is fair or loaded.\n", "Each state defines a distribution over the 6 possible observations.\n", "\n", "The transition model is denoted by\n", "```{math}\n", "p(z_t=j|z_{t-1}=i) = \\hmmTrans_{ij}\n", "```\n", "Here the $i$'th row of $\\vA$ corresponds to the outgoing distribution from state $i$.\n", "This is a row stochastic matrix,\n", "meaning each row sums to one.\n", "We can visualize\n", "the non-zero entries in the transition matrix by creating a state transition diagram,\n", "as shown in {ref}`casino-fig`.\n", "\n", "```{figure} /figures/casino.png\n", ":scale: 50%\n", ":name: casino-fig\n", "\n", "Illustration of the casino HMM.\n", "```\n", "\n", "The observation model\n", "$p(\\obs_t|\\hiddden_t=j)$ has the form\n", "```{math}\n", "p(\\obs_t=k|\\hidden_t=j) = \\hmmObs_{jk} \n", "```\n", "This is represented by the histograms associated with each\n", "state in {ref}`casino-fig`.\n", "\n", "Finally,\n", "the initial state distribution is denoted by\n", "```{math}\n", "p(z_1=j) = \\hmmInit_j\n", "```\n", "\n", "Collectively we denote all the parameters by $\\vtheta=(\\hmmTrans, \\hmmObs, \\hmmInit)$.\n", "\n", "Now let us implement this model code." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "# state transition matrix\n", "A = np.array([\n", " [0.95, 0.05],\n", " [0.10, 0.90]\n", "])\n", "\n", "# observation matrix\n", "B = np.array([\n", " [1/6, 1/6, 1/6, 1/6, 1/6, 1/6], # fair die\n", " [1/10, 1/10, 1/10, 1/10, 1/10, 5/10] # loaded die\n", "])\n", "\n", "pi = np.array([0.5, 0.5])\n", "\n", "(nstates, nobs) = np.shape(B)\n" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 4 }