Hidden Markov Models
Contents
# meta-data does not work yet in VScode
# https://github.com/microsoft/vscode-jupyter/issues/1121
{
"tags": [
"hide-cell"
]
}
### Install necessary libraries
try:
import jax
except:
# For cuda version, see https://github.com/google/jax#installation
%pip install --upgrade "jax[cpu]"
import jax
try:
import distrax
except:
%pip install --upgrade distrax
import distrax
try:
import jsl
except:
%pip install git+https://github.com/probml/jsl
import jsl
try:
import rich
except:
%pip install rich
import rich
{
"tags": [
"hide-cell"
]
}
### Import standard libraries
import abc
from dataclasses import dataclass
import functools
import itertools
from typing import Any, Callable, NamedTuple, Optional, Union, Tuple
import matplotlib.pyplot as plt
import numpy as np
import jax
import jax.numpy as jnp
from jax import lax, vmap, jit, grad
from jax.scipy.special import logit
from jax.nn import softmax
from functools import partial
from jax.random import PRNGKey, split
import inspect
import inspect as py_inspect
import rich
from rich import inspect as r_inspect
from rich import print as r_print
def print_source(fname):
r_print(py_inspect.getsource(fname))
\[ \begin{align}\begin{aligned}\newcommand\floor[1]{\lfloor#1\rfloor}\\\newcommand{\real}{\mathbb{R}}\\% Numbers
\newcommand{\vzero}{\boldsymbol{0}}
\newcommand{\vone}{\boldsymbol{1}}\\% Greek https://www.latex-tutorial.com/symbols/greek-alphabet/
\newcommand{\valpha}{\boldsymbol{\alpha}}
\newcommand{\vbeta}{\boldsymbol{\beta}}
\newcommand{\vchi}{\boldsymbol{\chi}}
\newcommand{\vdelta}{\boldsymbol{\delta}}
\newcommand{\vDelta}{\boldsymbol{\Delta}}
\newcommand{\vepsilon}{\boldsymbol{\epsilon}}
\newcommand{\vzeta}{\boldsymbol{\zeta}}
\newcommand{\vXi}{\boldsymbol{\Xi}}
\newcommand{\vell}{\boldsymbol{\ell}}
\newcommand{\veta}{\boldsymbol{\eta}}
%\newcommand{\vEta}{\boldsymbol{\Eta}}
\newcommand{\vgamma}{\boldsymbol{\gamma}}
\newcommand{\vGamma}{\boldsymbol{\Gamma}}
\newcommand{\vmu}{\boldsymbol{\mu}}
\newcommand{\vmut}{\boldsymbol{\tilde{\mu}}}
\newcommand{\vnu}{\boldsymbol{\nu}}
\newcommand{\vkappa}{\boldsymbol{\kappa}}
\newcommand{\vlambda}{\boldsymbol{\lambda}}
\newcommand{\vLambda}{\boldsymbol{\Lambda}}
\newcommand{\vLambdaBar}{\overline{\vLambda}}
%\newcommand{\vnu}{\boldsymbol{\nu}}
\newcommand{\vomega}{\boldsymbol{\omega}}
\newcommand{\vOmega}{\boldsymbol{\Omega}}
\newcommand{\vphi}{\boldsymbol{\phi}}
\newcommand{\vvarphi}{\boldsymbol{\varphi}}
\newcommand{\vPhi}{\boldsymbol{\Phi}}
\newcommand{\vpi}{\boldsymbol{\pi}}
\newcommand{\vPi}{\boldsymbol{\Pi}}
\newcommand{\vpsi}{\boldsymbol{\psi}}
\newcommand{\vPsi}{\boldsymbol{\Psi}}
\newcommand{\vrho}{\boldsymbol{\rho}}
\newcommand{\vtheta}{\boldsymbol{\theta}}
\newcommand{\vthetat}{\boldsymbol{\tilde{\theta}}}
\newcommand{\vTheta}{\boldsymbol{\Theta}}
\newcommand{\vsigma}{\boldsymbol{\sigma}}
\newcommand{\vSigma}{\boldsymbol{\Sigma}}
\newcommand{\vSigmat}{\boldsymbol{\tilde{\Sigma}}}
\newcommand{\vsigmoid}{\vsigma}
\newcommand{\vtau}{\boldsymbol{\tau}}
\newcommand{\vxi}{\boldsymbol{\xi}}\\
% Lower Roman (Vectors)
\newcommand{\va}{\mathbf{a}}
\newcommand{\vb}{\mathbf{b}}
\newcommand{\vBt}{\mathbf{\tilde{B}}}
\newcommand{\vc}{\mathbf{c}}
\newcommand{\vct}{\mathbf{\tilde{c}}}
\newcommand{\vd}{\mathbf{d}}
\newcommand{\ve}{\mathbf{e}}
\newcommand{\vf}{\mathbf{f}}
\newcommand{\vg}{\mathbf{g}}
\newcommand{\vh}{\mathbf{h}}
%\newcommand{\myvh}{\mathbf{h}}
\newcommand{\vi}{\mathbf{i}}
\newcommand{\vj}{\mathbf{j}}
\newcommand{\vk}{\mathbf{k}}
\newcommand{\vl}{\mathbf{l}}
\newcommand{\vm}{\mathbf{m}}
\newcommand{\vn}{\mathbf{n}}
\newcommand{\vo}{\mathbf{o}}
\newcommand{\vp}{\mathbf{p}}
\newcommand{\vq}{\mathbf{q}}
\newcommand{\vr}{\mathbf{r}}
\newcommand{\vs}{\mathbf{s}}
\newcommand{\vt}{\mathbf{t}}
\newcommand{\vu}{\mathbf{u}}
\newcommand{\vv}{\mathbf{v}}
\newcommand{\vw}{\mathbf{w}}
\newcommand{\vws}{\vw_s}
\newcommand{\vwt}{\mathbf{\tilde{w}}}
\newcommand{\vWt}{\mathbf{\tilde{W}}}
\newcommand{\vwh}{\hat{\vw}}
\newcommand{\vx}{\mathbf{x}}
%\newcommand{\vx}{\mathbf{x}}
\newcommand{\vxt}{\mathbf{\tilde{x}}}
\newcommand{\vy}{\mathbf{y}}
\newcommand{\vyt}{\mathbf{\tilde{y}}}
\newcommand{\vz}{\mathbf{z}}
%\newcommand{\vzt}{\mathbf{\tilde{z}}}\\
% Upper Roman (Matrices)
\newcommand{\vA}{\mathbf{A}}
\newcommand{\vB}{\mathbf{B}}
\newcommand{\vC}{\mathbf{C}}
\newcommand{\vD}{\mathbf{D}}
\newcommand{\vE}{\mathbf{E}}
\newcommand{\vF}{\mathbf{F}}
\newcommand{\vG}{\mathbf{G}}
\newcommand{\vH}{\mathbf{H}}
\newcommand{\vI}{\mathbf{I}}
\newcommand{\vJ}{\mathbf{J}}
\newcommand{\vK}{\mathbf{K}}
\newcommand{\vL}{\mathbf{L}}
\newcommand{\vM}{\mathbf{M}}
\newcommand{\vMt}{\mathbf{\tilde{M}}}
\newcommand{\vN}{\mathbf{N}}
\newcommand{\vO}{\mathbf{O}}
\newcommand{\vP}{\mathbf{P}}
\newcommand{\vQ}{\mathbf{Q}}
\newcommand{\vR}{\mathbf{R}}
\newcommand{\vS}{\mathbf{S}}
\newcommand{\vT}{\mathbf{T}}
\newcommand{\vU}{\mathbf{U}}
\newcommand{\vV}{\mathbf{V}}
\newcommand{\vW}{\mathbf{W}}
\newcommand{\vX}{\mathbf{X}}
%\newcommand{\vXs}{\vX_{\vs}}
\newcommand{\vXs}{\vX_{s}}
\newcommand{\vXt}{\mathbf{\tilde{X}}}
\newcommand{\vY}{\mathbf{Y}}
\newcommand{\vZ}{\mathbf{Z}}
\newcommand{\vZt}{\mathbf{\tilde{Z}}}
\newcommand{\vzt}{\mathbf{\tilde{z}}}\\
%%%%
\newcommand{\hidden}{\vz}
\newcommand{\hid}{\hidden}
\newcommand{\observed}{\vy}
\newcommand{\obs}{\observed}
\newcommand{\inputs}{\vu}
\newcommand{\input}{\inputs}\\\newcommand{\hmmTrans}{\vA}
\newcommand{\hmmObs}{\vB}
\newcommand{\hmmInit}{\vpi}
\newcommand{\hmmhid}{\hidden}
\newcommand{\hmmobs}{\obs}\\\newcommand{\ldsDyn}{\vA}
\newcommand{\ldsObs}{\vC}
\newcommand{\ldsDynIn}{\vB}
\newcommand{\ldsObsIn}{\vD}
\newcommand{\ldsDynNoise}{\vQ}
\newcommand{\ldsObsNoise}{\vR}\\\newcommand{\ssmDynFn}{f}
\newcommand{\ssmObsFn}{h}\\
%%%
\newcommand{\gauss}{\mathcal{N}}\\\newcommand{\diag}{\mathrm{diag}}\end{aligned}\end{align} \]
