Constructing the model

In BayesPy, the model is constructed by creating nodes which form a directed network. There are two types of nodes: stochastic and deterministic. A stochastic node corresponds to a random variable (or a set of random variables) from a specific probability distribution. A deterministic node corresponds to a deterministic function of its parents. For a list of built-in nodes, see the User API.

Creating nodes

Creating a node is basically like writing the conditional prior distribution of the variable in Python. The node is constructed by giving the parent nodes, that is, the conditioning variables as arguments. The number of parents and their meaning depend on the node. For instance, a Gaussian node is created by giving the mean vector and the precision matrix. These parents can be constant numerical arrays if they are known:

>>> from bayespy.nodes import Gaussian
>>> X = Gaussian([2, 5], [[1.0, 0.3], [0.3, 1.0]])

or other nodes if they are unknown and given prior distributions:

>>> from bayespy.nodes import Gaussian, Wishart
>>> mu = Gaussian([0, 0], [[1e-6, 0],[0, 1e-6]])
>>> Lambda = Wishart(2, [[1, 0], [0, 1]])
>>> X = Gaussian(mu, Lambda)

Nodes can also be named by providing name keyword argument:

>>> X = Gaussian(mu, Lambda, name='x')

The name may be useful when referring to the node using an inference engine.

For the parent nodes, there are two main restrictions: non-constant parent nodes must be conjugate and the parent nodes must be mutually independent in the posterior approximation.

Conjugacy of the parents

In Bayesian framework in general, one can give quite arbitrary probability distributions for variables. However, one often uses distributions that are easy to handle in practice. Quite often this means that the parents are given conjugate priors. This is also one of the limitations in BayesPy: only conjugate family prior distributions are accepted currently. Thus, although in principle one could give, for instance, gamma prior for the mean parameter mu, only Gaussian-family distributions are accepted because of the conjugacy. If the parent is not of a proper type, an error is raised. This conjugacy is checked automatically by BayesPy and NoConverterError is raised if a parent cannot be interpreted as being from a conjugate distribution.

Independence of the parents

Another a bit rarely encountered limitation is that the parents must be mutually independent (in the posterior factorization). Thus, a node cannot have the same stochastic node as several parents without intermediate stochastic nodes. For instance, the following leads to an error:

>>> from bayespy.nodes import Dot
>>> Y = Dot(X, X)
Traceback (most recent call last):
ValueError: Parent nodes are not independent

The error is raised because X is given as two parents for Y, and obviously X is not independent of X in the posterior approximation. Even if X is not given several times directly but there are some intermediate deterministic nodes, an error is raised because the deterministic nodes depend on their parents and thus the parents of Y would not be independent. However, it is valid that a node is a parent of another node via several paths if all the paths or all except one path has intermediate stochastic nodes. This is valid because the intermediate stochastic nodes have independent posterior approximations. Thus, for instance, the following construction does not raise errors:

>>> from bayespy.nodes import Dot
>>> Z = Gaussian(X, [[1,0], [0,1]])
>>> Y = Dot(X, Z)

This works because there is now an intermediate stochastic node Z on the other path from X node to Y node.

Effects of the nodes on inference

When constructing the network with nodes, the stochastic nodes actually define three important aspects:

  1. the prior probability distribution for the variables,

  2. the factorization of the posterior approximation,

  3. the functional form of the posterior approximation for the variables.

Prior probability distribution

First, the most intuitive feature of the nodes is that they define the prior distribution. In the previous example, mu was a stochastic GaussianARD node corresponding to \mu from the normal distribution, tau was a stochastic Gamma node corresponding to \tau from the gamma distribution, and y was a stochastic GaussianARD node corresponding to y from the normal distribution with mean \mu and precision \tau. If we denote the set of all stochastic nodes by \Omega, and by \pi_X the set of parents of a node X, the model is defined as

p(\Omega) = \prod_{X \in \Omega} p(X|\pi_X),

where nodes correspond to the terms p(X|\pi_X).

Posterior factorization

Second, the nodes define the structure of the posterior approximation. The variational Bayesian approximation factorizes with respect to nodes, that is, each node corresponds to an independent probability distribution in the posterior approximation. In the previous example, mu and tau were separate nodes, thus the posterior approximation factorizes with respect to them: q(\mu)q(\tau). Thus, the posterior approximation can be written as:

p(\tilde{\Omega}|\hat{\Omega}) \approx \prod_{X \in \tilde{\Omega}} q(X),

where \tilde{\Omega} is the set of latent stochastic nodes and \hat{\Omega} is the set of observed stochastic nodes. Sometimes one may want to avoid the factorization between some variables. For this purpose, there are some nodes which model several variables jointly without factorization. For instance, GaussianGammaISO is a joint node for \mu and \tau variables from the normal-gamma distribution and the posterior approximation does not factorize between \mu and \tau, that is, the posterior approximation is q(\mu,\tau).

Functional form of the posterior

Last, the nodes define the functional form of the posterior approximation. Usually, the posterior approximation has the same or similar functional form as the prior. For instance, Gamma uses gamma distribution to also approximate the posterior distribution. Similarly, GaussianARD uses Gaussian distribution for the posterior. However, the posterior approximation of GaussianARD uses a full covariance matrix although the prior assumes a diagonal covariance matrix. Thus, there can be slight differences in the exact functional form of the posterior approximation but the rule of thumb is that the functional form of the posterior approximation is the same as or more general than the functional form of the prior.

Using plate notation

Defining plates

Stochastic nodes take the optional parameter plates, which can be used to define plates of the variable. A plate defines the number of repetitions of a set of variables. For instance, a set of random variables \mathbf{y}_{mn} could be defined as

\mathbf{y}_{mn} \sim \mathcal{N}(\boldsymbol{\mu}, \mathbf{\Lambda}),\qquad m=0,\ldots,9, \quad n=0,\ldots,29.

This can also be visualized as a graphical model: