Documentation last updated on 13 November 2002

# How to use BNT for DBNs

- Model specification
  - HMMs
  - Kalman filters
  - Coupled HMMs
  - Water network
  - BAT network
- Inference
  - Discrete hidden nodes
  - Continuous hidden nodes
- Learning
  - Parameter learning
  - Structure learning

Note: you are recommended to read an introduction to DBNs first, such as this book chapter.

# Model specification

Dynamic Bayesian Networks (DBNs) are directed graphical models of stochastic processes. They generalise hidden Markov models (HMMs) and linear dynamical systems (LDSs) by representing the hidden (and observed) state in terms of state variables, which can have complex interdependencies. The graphical structure provides an easy way to specify these conditional independencies, and hence to provide a compact parameterization of the model.

Note that "temporal Bayesian network" would be a better name than "dynamic Bayesian network", since it is assumed that the model structure does not change, but the term DBN has become entrenched. We also normally assume that the parameters do not change, i.e., the model is time-invariant. However, we can always add extra hidden nodes to represent the current "regime", thereby creating mixtures of models to capture periodic non-stationarities.

There are some cases where the size of the state space can change over time, e.g., tracking a variable, but unknown, number of objects. In this case, we need to change the model structure over time. BNT does not support this.

## Hidden Markov Models (HMMs)

The simplest kind of DBN is a Hidden Markov Model (HMM), which has one discrete hidden node and one discrete or continuous observed node per slice. We illustrate this below. As before, circles denote continuous nodes, squares denote discrete nodes, clear means hidden, shaded means observed.

We have "unrolled" the model for three "time slices" -- the structure and parameters are assumed to repeat as the model is unrolled further. Hence to specify a DBN, we need to define the intra-slice topology (within a slice), the inter-slice topology (between two slices), as well as the parameters for the first two slices. (Such a two-slice temporal Bayes net is often called a 2TBN.)

We can specify the topology as follows.

    intra = zeros(2);
    intra(1,2) = 1; % node 1 in slice t connects to node 2 in slice t

    inter = zeros(2);
    inter(1,1) = 1; % node 1 in slice t-1 connects to node 1 in slice t

We can specify the parameters as follows, where for simplicity we assume the observed node is discrete.

    Q = 2; % num hidden states
    O = 2; % num observable symbols

    ns = [Q O];
    dnodes = 1:2;
    bnet = mk_dbn(intra, inter, ns, 'discrete', dnodes);
    for i=1:4
      bnet.CPD{i} = tabular_CPD(bnet, i);
    end

We assume the distributions P(X(t) | X(t-1)) and P(Y(t) | X(t)) are independent of t for t > 1. Hence the CPD for nodes 5, 7, ... is the same as for node 3, so we say they are in the same equivalence class, with node 3 being the "representative" for this class. In other words, we have tied the parameters for nodes 3, 5, 7, ... Similarly, nodes 4, 6, 8, ... are tied. Note, however, that (the parameters for) nodes 1 and 2 are not tied to subsequent slices.

Above we assumed the observation model P(Y(t) | X(t)) is independent of t for t>1, but it is conventional to assume this is true for all t. So we would like to put nodes 2, 4, 6, ... all in the same class. We can do this by explicitely defining the equivalence classes, as follows (see here for more details on parameter tying).

We define eclass1(i) to be the equivalence class that node i in slice 1 belongs to. Similarly, we define eclass2(i) to be the equivalence class that node i in slice 2, 3, ..., belongs to. For an HMM, we have

    eclass1 = [1 2];
    eclass2 = [3 2];
    eclass = [eclass1 eclass2];

This ties the observation model across slices, since e.g., eclass(4) = eclass(2) = 2.

By default, eclass1 = 1:ss, and eclass2 = (1:ss)+ss, where ss = slice size = the number of nodes per slice. But by using the above tieing pattern, we now only have 3 CPDs to specify, instead of 4:

    bnet = mk_dbn(intra, inter, ns, 'discrete', dnodes, 'eclass1', eclass1, 'eclass2', eclass2);
    prior0 = normalise(rand(Q,1));
    transmat0 = mk_stochastic(rand(Q,Q));
    obsmat0 = mk_stochastic(rand(Q,O));
    bnet.CPD{1} = tabular_CPD(bnet, 1, prior0);
    bnet.CPD{2} = tabular_CPD(bnet, 2, obsmat0);
    bnet.CPD{3} = tabular_CPD(bnet, 3, transmat0);

We discuss how to do inference and learning on this model below. (See also my HMM toolbox, which is included with BNT.)

Some common variants on HMMs are shown below. BNT can handle all of these.

## Linear Dynamical Systems (LDSs) and Kalman filters

A Linear Dynamical System (LDS) has the same topology as an HMM, but all the nodes are assumed to have linear-Gaussian distributions, i.e.,

       x(t+1) = A*x(t) + w(t),  w ~ N(0, Q),  x(0) ~ N(init_x, init_V)
       y(t)   = C*x(t) + v(t),  v ~ N(0, R)

Some simple variants are shown below.

We can create a regular LDS in BNT as follows.

    intra = zeros(2);
    intra(1,2) = 1;
    inter = zeros(2);
    inter(1,1) = 1;
    n = 2;

    X = 2; % size of hidden state
    Y = 2; % size of observable state

    ns = [X Y];
    dnodes = [];
    onodes = [2];
    eclass1 = [1 2];
    eclass2 = [3 2];
    bnet = mk_dbn(intra, inter, ns, 'discrete', dnodes, 'eclass1', eclass1, 'eclass2', eclass2);

    x0 = rand(X,1);
    V0 = eye(X); % must be positive semi definite!
    C0 = rand(Y,X);
    R0 = eye(Y);
    A0 = rand(X,X);
    Q0 = eye(X);

    bnet.CPD{1} = gaussian_CPD(bnet, 1, 'mean', x0, 'cov', V0, 'cov_prior_weight', 0);
    bnet.CPD{2} = gaussian_CPD(bnet, 2, 'mean', zeros(Y,1), 'cov', R0, 'weights', C0, ...
    			   'clamp_mean', 1, 'cov_prior_weight', 0);
    bnet.CPD{3} = gaussian_CPD(bnet, 3, 'mean', zeros(X,1), 'cov', Q0, 'weights', A0, ...
    			   'clamp_mean', 1, 'cov_prior_weight', 0);

We discuss how to do inference and learning on this model below. (See also my Kalman filter toolbox, which is included with BNT.)

## Coupled HMMs

Here is an example of a coupled HMM with N=5 chains, unrolled for T=3 slices. Each hidden discrete node has a private observed Gaussian child.

We can make this using the function

    Q = 2; % binary hidden nodes
    discrete_obs = 0; % cts observed nodes
    Y = 1; % scalar observed nodes
    bnet = mk_chmm(N, Q, Y, discrete_obs);

## Water network

Consider the following model of a water purification plant, developed by Finn V. Jensen, Uffe Kjærulff, Kristian G. Olesen, and Jan Pedersen.

We now show how to specify this model in BNT.

    ss = 12; % slice size
    intra = zeros(ss);
    intra(1,9) = 1;
    intra(3,10) = 1;
    intra(4,11) = 1;
    intra(8,12) = 1;

    inter = zeros(ss);
    inter(1, [1 3]) = 1; % node 1 in slice 1 connects to nodes 1 and 3 in slice 2
    inter(2, [2 3 7]) = 1;
    inter(3, [3 4 5]) = 1;
    inter(4, [3 4 6]) = 1;
    inter(5, [3 5 6]) = 1;
    inter(6, [4 5 6]) = 1;
    inter(7, [7 8]) = 1;
    inter(8, [6 7 8]) = 1;

    onodes = 9:12; % observed
    dnodes = 1:ss; % discrete
    ns = 2*ones(1,ss); % binary nodes
    eclass1 = 1:12;
    eclass2 = [13:20 9:12];
    eclass = [eclass1 eclass2];
    bnet = mk_dbn(intra, inter, ns, 'discrete', dnodes, 'eclass1', eclass1, 'eclass2', eclass2);
    for e=1:max(eclass)
      bnet.CPD{e} = tabular_CPD(bnet, e);
    end

We have tied the observation parameters across all slices. Click here for a more complex example of parameter tieing.

## BATnet

As an example of a more complicated DBN, consider the following example, which is a model of a car's high level state, as might be used by an automated car. (The model is from Forbes, Huang, Kanazawa and Russell, "The BATmobile: Towards a Bayesian Automated Taxi", IJCAI 95. The figure is from Boyen and Koller, "Tractable Inference for Complex Stochastic Processes", UAI98. For simplicity, we only show the observed nodes for slice 2.)

Since this topology is so complicated, it is useful to be able to refer to the nodes by name, instead of number.

    names = {'LeftClr', 'RightClr', 'LatAct', ... 'Bclr', 'BYdotDiff'};
    ss = length(names);

We can specify the intra-slice topology using a cell array as follows, where each row specifies a connection between two named nodes:

    intrac = {...
       'LeftClr', 'LeftClrSens';
      'RightClr', 'RightClrSens';
      ...
      'BYdotDiff', 'BcloseFast'};

Finally, we can convert this cell array to an adjacency matrix using the following function:

    [intra, names] = mk_adj_mat(intrac, names, 1);

This function also permutes the names so that they are in topological order. Given this ordering of the names, we can make the inter-slice connectivity matrix as follows:

    interc = {...
       'LeftClr', 'LeftClr';
       'LeftClr', 'LatAct';
       ...
       'FBStatus', 'LatAct'};

    inter = mk_adj_mat(interc, names, 0);

To refer to a node, we must know its number, which can be computed as in the following example:

    obs = {'LeftClrSens', 'RightClrSens', 'TurnSignalSens', 'XdotSens', 'YdotSens', 'FYdotDiffSens', ...
          'FclrSens', 'BXdotSens', 'BclrSens', 'BYdotDiffSens'};
    for i=1:length(obs)
      onodes(i) = stringmatch(obs{i}, names);
    end
    onodes = sort(onodes);

(We sort the onodes since most BNT routines assume that set-valued arguments are in sorted order.) We can now make the DBN:

    dnodes = 1:ss;
    ns = 2*ones(1,ss); % binary nodes
    bnet = mk_dbn(intra, inter, ns, 'iscrete', dnodes);

To specify the parameters, we must know the order of the parents. See the function BNT/general/mk_named_CPT for a way to do this in the case of tabular nodes. For simplicity, we just generate random parameters:

    for i=1:2*ss
      bnet.CPD{i} = tabular_CPD(bnet, i);
    end

A complete version of this example is available in BNT/examples/dynamic/bat1.m.

# Inference

The general inference problem for DBNs is to compute P(X(i,t0) | Y(:, t1:t2)), where X(i,t) represents the i'th hidden variable at time t and Y(:,t1:t2) represents all the evidence between times t1 and t2. There are several special cases of interest, illustrated below. The arrow indicates t0: it is X(t0) that we are trying to estimate. The shaded region denotes t1:t2, the available data.

BNT can currently only handle offline smoothing. (The HMM engine handles filtering and, to a limited extent, prediction.) The usage is similar to static inference engines, except now the evidence is a 2D cell array of size ss*T, where ss is the number of nodes per slice (ss = slice sizee) and T is the number of slices. Also, 'marginal_nodes' takes two arguments, the nodes and the time-slice. For example, to compute P(X(i,t) | y(:,1:T)), we proceed as follows (where onodes are the indices of the observedd nodes in each slice, which correspond to y):

    ev = sample_dbn(bnet, T);
    evidence = cell(ss,T);
    evidence(onodes,:) = ev(onodes, :); % all cells besides onodes are empty
    [engine, ll] = enter_evidence(engine, evidence);
    marg = marginal_nodes(engine, i, t);

## Discrete hidden nodes

If all the hidden nodes are discrete, we can use the junction tree algorithm to perform inference. The simplest approach, `jtree_unrolled_dbn_inf_engine`, unrolls the DBN into a static network and applies jtree; however, for long sequences, this can be very slow and can result in numerical underflow. A better approach is to apply the jtree algorithm to pairs of neighboring slices at a time; this is implemented in `jtree_dbn_inf_engine`.

A DBN can be converted to an HMM if all the hidden nodes are discrete. In this case, you can use `hmm_inf_engine`. This is faster than jtree for small models because the constant factors of the algorithm are lower, but can be exponentially slower for models with many variables (e.g., > 6 binary hidden nodes).

The use of both `jtree_dbn_inf_engine` and `hmm_inf_engine` is deprecated. A better approach is to construct a smoother engine out of lower-level engines, which implement forward/backward operators. You can create these engines as follows.

    engine = smoother_engine(hmm_2TBN_inf_engine(bnet));
    or
    engine = smoother_engine(jtree_2TBN_inf_engine(bnet));

You then call them in the usual way:

    engine = enter_evidence(engine, evidence);
    m = marginal_nodes(engine, nodes, t);

Note: you must declare the observed nodes in the bnet before using hmm_2TBN_inf_engine.

Unfortunately, when all the hiddden nodes are discrete, exact inference takes O(2^n) time, where n is the number of hidden nodes per slice, even if the model is sparse. The basic reason for this is that two nodes become correlated, even if there is no direct connection between them in the 2TBN, by virtue of sharing common ancestors in the past. Hence we need to use approximations.

A popular approximate inference algorithm for discrete DBNs, known as BK, is described in

- Tractable inference for complex stochastic processes , Boyen and Koller, UAI 1998
- Approximate learning of dynamic models, Boyen and Koller, NIPS 1998.

This approximates the belief state with a product of marginals on a specified set of clusters. For example, in the water network, we might use the following clusters:

    engine = bk_inf_engine(bnet, { [1 2], [3 4 5 6], [7 8] });

This engine can now be used just like the jtree engine. Two special cases of the BK algorithm are supported: 'ff' (fully factored) means each node has its own cluster, and 'exact' means there is 1 cluster that contains the whole slice. These can be created as follows:

    engine = bk_inf_engine(bnet, 'ff');
    engine = bk_inf_engine(bnet, 'exact');

For pedagogical purposes, an implementation of BK-FF that uses an HMM instead of junction tree is available at `bk_ff_hmm_inf_engine`.

## Continuous hidden nodes

If all the hidden nodes are linear-Gaussian, and the observed nodes are linear-Gaussian, the model is a linear dynamical system (LDS). A DBN can be converted to an LDS if all the hidden nodes are linear-Gaussian and if they are all persistent. In this case, you can use `kalman_inf_engine`. For more general linear-gaussian models, you can use `jtree_dbn_inf_engine` or `jtree_unrolled_dbn_inf_engine`.

For nonlinear systems with Gaussian noise, the unscented Kalman filter (UKF), due to Julier and Uhlmann, is far superior to the well-known extended Kalman filter (EKF), both in theory and practice. The key idea of the UKF is that it is easier to estimate a Gaussian distribution from a set of points than to approximate an arbitrary non-linear function. We start with points that are plus/minus sigma away from the mean along each dimension, and then pipe them through the nonlinearity, and then fit a Gaussian to the transformed points. (No need to compute Jacobians, unlike the EKF!)

For systems with non-Gaussian noise, I recommend Particle filtering (PF), which is a popular sequential Monte Carlo technique.

The EKF can be used as a proposal distribution for a PF. This method is better than either one alone. See The Unscented Particle Filter, by R van der Merwe, A Doucet, JFG de Freitas and E Wan, May 2000. Matlab software for the UPF is also available.

Note: none of this software is part of BNT.

# Learning

Learning in DBNs can be done online or offline. Currently only offline learning is implemented in BNT.

## Parameter learning

Offline parameter learning is very similar to learning in static networks, except now the training data is a cell-array of 2D cell-arrays. For example, cases{l}{i,t} is the value of node i in slice t in sequence l, or [] if unobserved. Each sequence can be a different length, and may have missing values in arbitrary locations. Here is a typical code fragment for using EM.

    ncases = 2;
    cases = cell(1, ncases);
    for i=1:ncases
      ev = sample_dbn(bnet, T);
      cases{i} = cell(ss,T);
      cases{i}(onodes,:) = ev(onodes, :);
    end
    [bnet2, LLtrace] = learn_params_dbn_em(engine, cases, 'max_iter', 10);

If the observed node is vector-valued and stored in an OxT array, you need to assign each vector to a single cell, as in the following example.

    data = [xpos(:)'; ypos(:)'];
    ncases = 1;
    cases = cell(1, ncases);
    onodes = bnet.observed;
    for i=1:ncases
      cases{i} = cell(ss,T);
      cases{i}(onodes,:) = num2cell(data(:,1:T), 1);
    end

For a complete code listing of how to do EM in a simple DBN, click here.

## Structure learning

There is currently only one structure learning algorithm for DBNs. This assumes all nodes are tabular and observed, and that there are no intra-slice connections. Hence we can find the optimal set of parents for each node separately, without worrying about directed cycles or node orderings. The function is called as follows

    inter = learn_struct_dbn_reveal(cases, ns, max_fan_in, penalty)

A full example is given in BNT/examples/dynamic/reveal1.m. Setting the penalty term to 0 gives the maximum likelihood model; this is equivalent to maximizing the mutual information between parents and child (in the bioinformatics community, this is known as the REVEAL algorithm). A non-zero penalty invokes the BIC criterion, which lessens the chance of overfitting.

Dirk Husmeier has extended MCMC model selection to DBNs.