{
 "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 distrax\n",
    "except:\n",
    "    %pip install --upgrade  distrax\n",
    "    import distrax\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",
    "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{\\hid}{\\hidden}\n",
    "\\newcommand{\\observed}{\\vy}\n",
    "\\newcommand{\\obs}{\\observed}\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 <ssm-ar>` \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",
    "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 <ssm-simplified>` \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",
    "```\n",
    "\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)=\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 \n",
    "{numref}`Figure %s <casino-fig>`\n",
    "%{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",
    "$p(\\obs_t|\\hidden_t=j)$ 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 in 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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING:absl:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<distrax._src.utils.hmm.HMM object at 0x7fb570b7e1f0>\n"
     ]
    }
   ],
   "source": [
    "import distrax\n",
    "from distrax import HMM\n",
    "\n",
    "\n",
    "hmm = HMM(trans_dist=distrax.Categorical(probs=A),\n",
    "            init_dist=distrax.Categorical(probs=pi),\n",
    "            obs_dist=distrax.Categorical(probs=B))\n",
    "\n",
    "print(hmm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "Let's sample from the model. We will generate a sequence of latent states, $\\hid_{1:T}$,\n",
    "which we then convert to a sequence of observations, $\\obs_{1:T}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Printing sample observed/latent...\n",
      "x: 633665342652353616444236412331351246651613325161656366246242\n",
      "z: 222222211111111111111111111111111111111222111111112222211111\n"
     ]
    }
   ],
   "source": [
    "\n",
    "\n",
    "\n",
    "seed = 314\n",
    "n_samples = 300\n",
    "z_hist, x_hist = hmm.sample(seed=PRNGKey(seed), seq_len=n_samples)\n",
    "\n",
    "z_hist_str = \"\".join((np.array(z_hist) + 1).astype(str))[:60]\n",
    "x_hist_str = \"\".join((np.array(x_hist) + 1).astype(str))[:60]\n",
    "\n",
    "print(\"Printing sample observed/latent...\")\n",
    "print(f\"x: {x_hist_str}\")\n",
    "print(f\"z: {z_hist_str}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<pre style=\"white-space:pre;overflow-x:auto;line-height:normal;font-family:Menlo,'DejaVu Sans Mono',consolas,'Courier New',monospace\">  def sample<span style=\"font-weight: bold\">(</span>self,\n",
       "             *,\n",
       "             seed: chex.PRNGKey,\n",
       "             seq_len: chex.Array<span style=\"font-weight: bold\">)</span> -&gt; Tuple:\n",
       "    <span style=\"color: #008000; text-decoration-color: #008000\">\"\"</span>\"Sample from this HMM.\n",
       "\n",
       "    Samples an observation of given length according to this\n",
       "    Hidden Markov Model and gives the sequence of the hidden states\n",
       "    as well as the observation.\n",
       "\n",
       "    Args:\n",
       "      seed: Random key of shape <span style=\"font-weight: bold\">(</span><span style=\"color: #000080; text-decoration-color: #000080; font-weight: bold\">2</span>,<span style=\"font-weight: bold\">)</span> and dtype uint32.\n",
       "      seq_len: The length of the observation sequence.\n",
       "\n",
       "    Returns:\n",
       "      Tuple of hidden state sequence, and observation sequence.\n",
       "    <span style=\"color: #008000; text-decoration-color: #008000\">\"\"</span>\"\n",
       "    rng_key, rng_init = jax.random.split<span style=\"font-weight: bold\">(</span>seed<span style=\"font-weight: bold\">)</span>\n",
       "    initial_state = self._init_dist.sample<span style=\"font-weight: bold\">(</span><span style=\"color: #808000; text-decoration-color: #808000\">seed</span>=<span style=\"color: #800080; text-decoration-color: #800080\">rng_init</span><span style=\"font-weight: bold\">)</span>\n",
       "\n",
       "    def draw_state<span style=\"font-weight: bold\">(</span>prev_state, key<span style=\"font-weight: bold\">)</span>:\n",
       "      state = self._trans_dist.sample<span style=\"font-weight: bold\">(</span><span style=\"color: #808000; text-decoration-color: #808000\">seed</span>=<span style=\"color: #800080; text-decoration-color: #800080\">key</span><span style=\"font-weight: bold\">)</span>\n",
       "      return state, state\n",
       "\n",
       "    rng_state, rng_obs = jax.random.split<span style=\"font-weight: bold\">(</span>rng_key<span style=\"font-weight: bold\">)</span>\n",
       "    keys = jax.random.split<span style=\"font-weight: bold\">(</span>rng_state, seq_len - <span style=\"color: #000080; text-decoration-color: #000080; font-weight: bold\">1</span><span style=\"font-weight: bold\">)</span>\n",
       "    _, states = jax.lax.scan<span style=\"font-weight: bold\">(</span>draw_state, initial_state, keys<span style=\"font-weight: bold\">)</span>\n",
       "    states = jnp.append<span style=\"font-weight: bold\">(</span>initial_state, states<span style=\"font-weight: bold\">)</span>\n",
       "\n",
       "    def draw_obs<span style=\"font-weight: bold\">(</span>state, key<span style=\"font-weight: bold\">)</span>:\n",
       "      return self._obs_dist.sample<span style=\"font-weight: bold\">(</span><span style=\"color: #808000; text-decoration-color: #808000\">seed</span>=<span style=\"color: #800080; text-decoration-color: #800080\">key</span><span style=\"font-weight: bold\">)</span>\n",
       "\n",
       "    keys = jax.random.split<span style=\"font-weight: bold\">(</span>rng_obs, seq_len<span style=\"font-weight: bold\">)</span>\n",
       "    obs_seq = jax.vmap<span style=\"font-weight: bold\">(</span>draw_obs, <span style=\"color: #808000; text-decoration-color: #808000\">in_axes</span>=<span style=\"font-weight: bold\">(</span><span style=\"color: #000080; text-decoration-color: #000080; font-weight: bold\">0</span>, <span style=\"color: #000080; text-decoration-color: #000080; font-weight: bold\">0</span><span style=\"font-weight: bold\">))(</span>states, keys<span style=\"font-weight: bold\">)</span>\n",
       "\n",
       "    return states, obs_seq\n",
       "\n",
       "</pre>\n"
      ],
      "text/plain": [
       "<rich.jupyter.JupyterRenderable at 0x7fb54800ebe0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Here is the source code for the sampling algorithm.\n",
    "\n",
    "print_source(hmm.sample)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our primary goal will be to infer the latent state from the observations,\n",
    "so we can detect if the casino is being dishonest or not. This will\n",
    "affect how we choose to gamble our money.\n",
    "We discuss various ways to perform this inference below."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear Gaussian SSMs\n",
    "\n",
    "Blah blah"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(sec:tracking-lds)=\n",
    "## Example: model for 2d tracking\n",
    "\n",
    "Blah blah"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(sec:inference)=\n",
    "# Inferential goals\n",
    "\n",
    "```{figure} /figures/dbn-inference-problems.png\n",
    ":scale: 100%\n",
    ":name: dbn-inference\n",
    "\n",
    "Illustration of the different kinds of inference in an SSM.\n",
    " The main kinds of inference for state-space models.\n",
    "    The shaded region is the interval for which we have data.\n",
    "    The arrow represents the time step at which we want to perform inference.\n",
    "    $t$ is the current time,  $T$ is the sequence length,\n",
    "$\\ell$ is the lag and $h$ is the prediction horizon.\n",
    "```\n",
    "\n",
    "\n",
    "\n",
    "Given the sequence of observations, and a known model,\n",
    "one of the main tasks with SSMs\n",
    "to perform posterior inference,\n",
    "about the hidden states; this is also called\n",
    "state estimation.\n",
    "At each time step $t$,\n",
    "there are multiple forms of posterior we may be interested in computing,\n",
    "including the following:\n",
    "- the filtering distribution\n",
    "$p(\\hmmhid_t|\\hmmobs_{1:t})$\n",
    "- the smoothing distribution\n",
    "$p(\\hmmhid_t|\\hmmobs_{1:T})$ (note that this conditions on future data $T>t$)\n",
    "- the fixed-lag smoothing distribution\n",
    "$p(\\hmmhid_{t-\\ell}|\\hmmobs_{1:t})$ (note that this\n",
    "infers $\\ell$ steps in the past given data up to the present).\n",
    "\n",
    "We may also want to compute the\n",
    "predictive distribution $h$ steps into the future:\n",
    "\\begin{align}\n",
    "p(\\hmmobs_{t+h}|\\hmmobs_{1:t})\n",
    "&= \\sum_{\\hmmhid_{t+h}} p(\\hmmobs_{t+h}|\\hmmhid_{t+h}) p(\\hmmhid_{t+h}|\\hmmobs_{1:t})\n",
    "\\end{align}\n",
    "where the hidden state predictive distribution is\n",
    "\\begin{align}\n",
    "p(\\hmmhid_{t+h}|\\hmmobs_{1:t})\n",
    "&= \\sum_{\\hmmhid_{t:t+h-1}}\n",
    " p(\\hmmhid_t|\\hmmobs_{1:t}) \n",
    " p(\\hmmhid_{t+1}|\\hmmhid_{t})\n",
    " p(\\hmmhid_{t+2}|\\hmmhid_{t+1})\n",
    "\\cdots\n",
    " p(\\hmmhid_{t+h}|\\hmmhid_{t+h-1})\n",
    "\\end{align}\n",
    "See \\cref{fig:dbn_inf_problems} for a summary of these distributions.\n",
    "\n",
    "In addition  to comuting posterior marginals,\n",
    "we may want to compute the most probable hidden sequence,\n",
    "i.e., the joint MAP estimate\n",
    "```{math}\n",
    "\\arg \\max_{\\hmmhid_{1:T}} p(\\hmmhid_{1:T}|\\hmmobs_{1:T})\n",
    "```\n",
    "or sample sequences from the posterior\n",
    "```{math}\n",
    "\\hmmhid_{1:T} \\sim p(\\hmmhid_{1:T}|\\hmmobs_{1:T})\n",
    "```\n",
    "\n",
    "Algorithms for all these task are discussed in the following chapters,\n",
    "since the details depend on the form of the SSM.\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: inference in the casino HMM\n",
    "\n",
    "We now illustrate filtering, smoothing and MAP decoding applied\n",
    "to the casino HMM from {ref}`sec:casino`. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/opt/anaconda3/lib/python3.8/site-packages/jax/_src/numpy/lax_numpy.py:5154: UserWarning: Explicitly requested dtype <class 'jax._src.numpy.lax_numpy.int64'> requested in astype is not available, and will be truncated to dtype int32. To enable more dtypes, set the jax_enable_x64 configuration option or the JAX_ENABLE_X64 shell environment variable. See https://github.com/google/jax#current-gotchas for more.\n",
      "  lax_internal._check_user_dtype_supported(dtype, \"astype\")\n"
     ]
    }
   ],
   "source": [
    "# Call inference engine\n",
    "\n",
    "filtered_dist, _, smoothed_dist, loglik = hmm.forward_backward(x_hist)\n",
    "map_path = hmm.viterbi(x_hist)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Find the span of timesteps that the    simulated systems turns to be in state 1\n",
    "def find_dishonest_intervals(z_hist):\n",
    "    spans = []\n",
    "    x_init = 0\n",
    "    for t, _ in enumerate(z_hist[:-1]):\n",
    "        if z_hist[t + 1] == 0 and z_hist[t] == 1:\n",
    "            x_end = t\n",
    "            spans.append((x_init, x_end))\n",
    "        elif z_hist[t + 1] == 1 and z_hist[t] == 0:\n",
    "            x_init = t + 1\n",
    "    return spans"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Plot posterior\n",
    "def plot_inference(inference_values, z_hist, ax, state=1, map_estimate=False):\n",
    "    n_samples = len(inference_values)\n",
    "    xspan = np.arange(1, n_samples + 1)\n",
    "    spans = find_dishonest_intervals(z_hist)\n",
    "    if map_estimate:\n",
    "        ax.step(xspan, inference_values, where=\"post\")\n",
    "    else:\n",
    "        ax.plot(xspan, inference_values[:, state])\n",
    "\n",
    "    for span in spans:\n",
    "        ax.axvspan(*span, alpha=0.5, facecolor=\"tab:gray\", edgecolor=\"none\")\n",
    "    ax.set_xlim(1, n_samples)\n",
    "    # ax.set_ylim(0, 1)\n",
    "    ax.set_ylim(-0.1, 1.1)\n",
    "    ax.set_xlabel(\"Observation number\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Filtered')"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "[encoded data, 40804 characters]\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "filenames": {
       "image/png": "/Users/kpmurphy/github/ssm-book/_build/jupyter_execute/chapters/ssm/ssm_19_1.png"
      },
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    " # Filtering\n",
    "fig, ax = plt.subplots()\n",
    "plot_inference(filtered_dist, z_hist, ax)\n",
    "ax.set_ylabel(\"p(loaded)\")\n",
    "ax.set_title(\"Filtered\")\n",
    " \n",
    "\n",
    "  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Smoothed')"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "[encoded data, 33688 characters]\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
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     "metadata": {
      "filenames": {
       "image/png": "/Users/kpmurphy/github/ssm-book/_build/jupyter_execute/chapters/ssm/ssm_20_1.png"
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      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Smoothing\n",
    "fig, ax = plt.subplots()\n",
    "plot_inference(smoothed_dist, z_hist, ax)\n",
    "ax.set_ylabel(\"p(loaded)\")\n",
    "ax.set_title(\"Smoothed\")\n",
    "\n",
    "\n",
    " "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Viterbi')"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "[encoded data, 8672 characters]\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "filenames": {
       "image/png": "/Users/kpmurphy/github/ssm-book/_build/jupyter_execute/chapters/ssm/ssm_21_1.png"
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      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# MAP estimation\n",
    "fig, ax = plt.subplots()\n",
    "plot_inference(map_path, z_hist, ax, map_estimate=True)\n",
    "ax.set_ylabel(\"MAP state\")\n",
    "ax.set_title(\"Viterbi\")\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# TODO: posterior samples\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Example: inference in the tracking SSM\n",
    "\n",
    "We now illustrate filtering, smoothing and MAP decoding applied\n",
    "to the 2d tracking HMM from {ref}`sec:tracking-lds`. "
   ]
  }
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