bayespy.nodes.CategoricalMarkovChain

class bayespy.nodes.CategoricalMarkovChain(pi, A, states=None, **kwargs)[source]

Node for categorical Markov chain random variables.

The node models a Markov chain which has a discrete set of K possible states and the next state depends only on the previous state and the state transition probabilities. The graphical model is shown below:

% tikzlibrary.code.tex
%
% Copyright 2010-2011 by Laura Dietz
% Copyright 2012 by Jaakko Luttinen
%
% This file may be distributed and/or modified
%
% 1. under the LaTeX Project Public License and/or
% 2. under the GNU General Public License.
%
% See the files LICENSE_LPPL and LICENSE_GPL for more details.

% Load other libraries
\usetikzlibrary{shapes}
\usetikzlibrary{fit}
\usetikzlibrary{chains}
\usetikzlibrary{arrows}

% Latent node
\tikzstyle{latent} = [circle,fill=white,draw=black,inner sep=1pt,
minimum size=20pt, font=\fontsize{10}{10}\selectfont, node distance=1]
% Observed node
\tikzstyle{obs} = [latent,fill=gray!25]
% Constant node
\tikzstyle{const} = [rectangle, inner sep=0pt, node distance=1]
% Factor node
\tikzstyle{factor} = [rectangle, fill=black,minimum size=5pt, inner
sep=0pt, node distance=0.4]
% Deterministic node
\tikzstyle{det} = [latent, diamond]

% Plate node
\tikzstyle{plate} = [draw, rectangle, rounded corners, fit=#1]
% Invisible wrapper node
\tikzstyle{wrap} = [inner sep=0pt, fit=#1]
% Gate
\tikzstyle{gate} = [draw, rectangle, dashed, fit=#1]

% Caption node
\tikzstyle{caption} = [font=\footnotesize, node distance=0] %
\tikzstyle{plate caption} = [caption, node distance=0, inner sep=0pt,
below left=5pt and 0pt of #1.south east] %
\tikzstyle{factor caption} = [caption] %
\tikzstyle{every label} += [caption] %

\tikzset{>={triangle 45}}

%\pgfdeclarelayer{b}
%\pgfdeclarelayer{f}
%\pgfsetlayers{b,main,f}

% \factoredge [options] {inputs} {factors} {outputs}
\newcommand{\factoredge}[4][]{ %
  % Connect all nodes #2 to all nodes #4 via all factors #3.
  \foreach \f in {#3} { %
    \foreach \x in {#2} { %
      \draw[-,#1] (\x) edge[-] (\f) ; %
    } ;
    \foreach \y in {#4} { %
      \draw[->,#1] (\f) -- (\y) ; %
    } ;
  } ;
}

% \edge [options] {inputs} {outputs}
\newcommand{\edge}[3][]{ %
  % Connect all nodes #2 to all nodes #3.
  \foreach \x in {#2} { %
    \foreach \y in {#3} { %
      \draw[->,#1] (\x) -- (\y) ;%
    } ;
  } ;
}

% \factor [options] {name} {caption} {inputs} {outputs}
\newcommand{\factor}[5][]{ %
  % Draw the factor node. Use alias to allow empty names.
  \node[factor, label={[name=#2-caption]#3}, name=#2, #1,
  alias=#2-alias] {} ; %
  % Connect all inputs to outputs via this factor
  \factoredge {#4} {#2-alias} {#5} ; %
}

% \plate [options] {name} {fitlist} {caption}
\newcommand{\plate}[4][]{ %
  \node[wrap=#3] (#2-wrap) {}; %
  \node[plate caption=#2-wrap] (#2-caption) {#4}; %
  \node[plate=(#2-wrap)(#2-caption), #1] (#2) {}; %
}

% \gate [options] {name} {fitlist} {inputs}
\newcommand{\gate}[4][]{ %
  \node[gate=#3, name=#2, #1, alias=#2-alias] {}; %
  \foreach \x in {#4} { %
    \draw [-*,thick] (\x) -- (#2-alias); %
  } ;%
}

% \vgate {name} {fitlist-left} {caption-left} {fitlist-right}
% {caption-right} {inputs}
\newcommand{\vgate}[6]{ %
  % Wrap the left and right parts
  \node[wrap=#2] (#1-left) {}; %
  \node[wrap=#4] (#1-right) {}; %
  % Draw the gate
  \node[gate=(#1-left)(#1-right)] (#1) {}; %
  % Add captions
  \node[caption, below left=of #1.north ] (#1-left-caption)
  {#3}; %
  \node[caption, below right=of #1.north ] (#1-right-caption)
  {#5}; %
  % Draw middle separation
  \draw [-, dashed] (#1.north) -- (#1.south); %
  % Draw inputs
  \foreach \x in {#6} { %
    \draw [-*,thick] (\x) -- (#1); %
  } ;%
}

% \hgate {name} {fitlist-top} {caption-top} {fitlist-bottom}
% {caption-bottom} {inputs}
\newcommand{\hgate}[6]{ %
  % Wrap the left and right parts
  \node[wrap=#2] (#1-top) {}; %
  \node[wrap=#4] (#1-bottom) {}; %
  % Draw the gate
  \node[gate=(#1-top)(#1-bottom)] (#1) {}; %
  % Add captions
  \node[caption, above right=of #1.west ] (#1-top-caption)
  {#3}; %
  \node[caption, below right=of #1.west ] (#1-bottom-caption)
  {#5}; %
  % Draw middle separation
  \draw [-, dashed] (#1.west) -- (#1.east); %
  % Draw inputs
  \foreach \x in {#6} { %
    \draw [-*,thick] (\x) -- (#1); %
  } ;%
}

\tikzstyle{latent} += [minimum size=30pt];

\node[latent] (x0) {$x_0$};
\node[latent, right=of x0] (x1) {$x_1$};
\node[right=of x1] (dots) {$\cdots$};
\node[latent, right=of dots] (xn) {$x_{N-1}$};
\edge {x0}{x1};
\edge {x1}{dots};
\edge {dots}{xn};

\node[latent, above=of x0] (pi) {$\boldsymbol{\pi}$};
\node[latent, above=of dots] (A) {$\mathbf{A}$};
\edge {pi} {x0};
\edge {A} {x1,dots,xn};

where \boldsymbol{\pi} contains the probabilities for the initial state and \mathbf{A} is the state transition probability matrix. It is possible to have \mathbf{A} varying in time.

p(x_0, \ldots, x_{N-1}) &= p(x_0) \prod^{N-1}_{n=1} p(x_n|x_{n-1}),

where

p(x_0=k) &= \pi_k, \quad \text{for } k \in \{0,\ldots,K-1\},
\\
p(x_n=j|x_{n-1}=i) &= a_{ij}^{(n-1)} \quad \text{for } n=1,\ldots,N-1,\,
i\in\{1,\ldots,K-1\},\, j\in\{1,\ldots,K-1\}
\\
a_{ij}^{(n)} &= [\mathbf{A}_n]_{ij}

This node can be used to construct hidden Markov models by using Mixture for the emission distribution.

Parameters:

pi : Dirichlet-like node or (...,K)-array

\boldsymbol{\pi}, probabilities for the first state. K-dimensional Dirichlet.

A : Dirichlet-like node or (K,K)-array or (...,1,K,K)-array or (...,N-1,K,K)-array

\mathbf{A}, probabilities for state transitions. K-dimensional Dirichlet with plates (K,) or (...,1,K) or (...,N-1,K).

states : int, optional

N, the length of the chain.

__init__(pi, A, states=None, **kwargs)[source]

Create categorical Markov chain

Methods

__init__(pi, A[, states]) Create categorical Markov chain
add_plate_axis(to_plate)
broadcasting_multiplier(plates, *args)
delete() Delete this node and the children
get_gradient(rg) Computes gradient with respect to the natural parameters.
get_mask()
get_moments()
get_parameters() Return parameters of the VB distribution.
get_pdf_nodes()
get_riemannian_gradient() Computes the Riemannian/natural gradient.
get_shape(ind)
has_plotter() Return True if the node has a plotter
initialize_from_parameters(*args)
initialize_from_prior()
initialize_from_random() Set the variable to a random sample from the current distribution.
initialize_from_value(x, *args)
load(filename)
logpdf(X[, mask]) Compute the log probability density function Q(X) of this node.
lower_bound_contribution([gradient, ...]) Compute E[ log p(X|parents) - log q(X) ]
lowerbound()
move_plates(from_plate, to_plate)
observe(x, *args[, mask]) Fix moments, compute f and propagate mask.
pdf(X[, mask]) Compute the probability density function of this node.
plot([fig]) Plot the node distribution using the plotter of the node
random() Draw a random sample from the distribution.
save(filename)
set_parameters(x) Set the parameters of the VB distribution.
set_plotter(plotter)
show() Print the distribution using standard parameterization.
unobserve()
update([annealing])

Attributes

dims
plates
plates_multiplier Plate multiplier is applied to messages to parents