..
   Copyright (C) 2014 Jaakko Luttinen

   This file is licensed under the MIT License. See LICENSE for a text of the
   license.


.. testsetup::

    from matplotlib import pyplot
    pyplot.ion()
    import numpy
    numpy.random.seed(1)


Quick start guide
=================

This short guide shows the key steps in using BayesPy for variational
Bayesian inference by applying BayesPy to a simple problem. The key
steps in using BayesPy are the following:

-  Construct the model

-  Observe some of the variables by providing the data in a proper
   format

-  Run variational Bayesian inference

-  Examine the resulting posterior approximation

To demonstrate BayesPy, we'll consider a very simple problem: we have a
set of observations from a Gaussian distribution with unknown mean and
variance, and we want to learn these parameters. In this case, we do not
use any real-world data but generate some artificial data. The dataset
consists of ten samples from a Gaussian distribution with mean 5 and
standard deviation 10. This dataset can be generated with NumPy as
follows:

>>> import numpy as np
>>> data = np.random.normal(5, 10, size=(10,))

Constructing the model
----------------------

Now, given this data we would like to estimate the mean and the standard
deviation as if we didn't know their values. The model can be defined as
follows:

.. math::


   \begin{split}
   p(\mathbf{y}|\mu,\tau) &= \prod^{9}_{n=0} \mathcal{N}(y_n|\mu,\tau) \\
   p(\mu) &= \mathcal{N}(\mu|0,10^{-6}) \\
   p(\tau) &= \mathcal{G}(\tau|10^{-6},10^{-6})
   \end{split}

where :math:`\mathcal{N}` is the Gaussian distribution parameterized by
its mean and precision (i.e., inverse variance), and :math:`\mathcal{G}`
is the gamma distribution parameterized by its shape and rate
parameters. Note that we have given quite uninformative priors for the
variables :math:`\mu` and :math:`\tau`\ . This simple model can also be
shown as a directed factor graph:

.. bayesnet:: Directed factor graph of the example model.

   \node[obs]                                  (y)     {$y_n$} ;
   \node[latent, above left=1.5 and 0.5 of y]  (mu)    {$\mu$} ;
   \node[latent, above right=1.5 and 0.5 of y] (tau)   {$\tau$} ;
   \factor[above=of mu] {mu-f} {left:$\mathcal{N}$} {} {mu} ;
   \factor[above=of tau] {tau-f} {left:$\mathcal{G}$} {} {tau} ;

   \factor[above=of y] {y-f} {left:$\mathcal{N}$} {mu,tau}     {y};

   \plate {} {(y)(y-f)(y-f-caption)} {$n=0,\ldots,9$} ;
                
This model can be constructed in BayesPy as follows:

>>> from bayespy.nodes import GaussianARD, Gamma
>>> mu = GaussianARD(0, 1e-6)
>>> tau = Gamma(1e-6, 1e-6)
>>> y = GaussianARD(mu, tau, plates=(10,))
                
.. currentmodule:: bayespy.nodes

This is quite self-explanatory given the model definitions above. We have used
two types of nodes :class:`GaussianARD` and :class:`Gamma` to represent Gaussian
and gamma distributions, respectively. There are much more distributions in
:mod:`bayespy.nodes` so you can construct quite complex conjugate exponential
family models. The node :code:`y` uses keyword argument :code:`plates` to define
the plates :math:`n=0,\ldots,9`.
                
Performing inference
--------------------

Now that we have created the model, we can provide our data by setting
``y`` as observed:

>>> y.observe(data)

Next we want to estimate the posterior distribution. In principle, we could use
different inference engines (e.g., MCMC or EP) but currently only variational
Bayesian (VB) engine is implemented. The engine is initialized by giving all the
nodes of the model:

>>> from bayespy.inference import VB
>>> Q = VB(mu, tau, y)

The inference algorithm can be run as long as wanted (max. 20 iterations
in this case):

>>> Q.update(repeat=20)
Iteration 1: loglike=-6.020956e+01 (... seconds)
Iteration 2: loglike=-5.820527e+01 (... seconds)
Iteration 3: loglike=-5.820290e+01 (... seconds)
Iteration 4: loglike=-5.820288e+01 (... seconds)
Converged at iteration 4.

Now the algorithm converged after four iterations, before the requested 20
iterations. VB approximates the true posterior :math:`p(\mu,\tau|\mathbf{y})`
with a distribution which factorizes with respect to the nodes:
:math:`q(\mu)q(\tau)`\ .

Examining posterior approximation
---------------------------------

The resulting approximate posterior distributions :math:`q(\mu)` and
:math:`q(\tau)` can be examined, for instance, by plotting the marginal
probability density functions:

>>> import bayespy.plot as bpplt
>>> bpplt.pyplot.subplot(2, 1, 1)
<matplotlib.axes...AxesSubplot object at 0x...>
>>> bpplt.pdf(mu, np.linspace(-10, 20, num=100), color='k', name=r'\mu')
[<matplotlib.lines.Line2D object at 0x...>]
>>> bpplt.pyplot.subplot(2, 1, 2)
<matplotlib.axes...AxesSubplot object at 0x...>
>>> bpplt.pdf(tau, np.linspace(1e-6, 0.08, num=100), color='k', name=r'\tau')
[<matplotlib.lines.Line2D object at 0x...>]
>>> bpplt.pyplot.tight_layout()
>>> bpplt.pyplot.show()

.. plot::

    import numpy as np
    np.random.seed(1)
    data = np.random.normal(5, 10, size=(10,))
    from bayespy.nodes import GaussianARD, Gamma
    mu = GaussianARD(0, 1e-6)
    tau = Gamma(1e-6, 1e-6)
    y = GaussianARD(mu, tau, plates=(10,))
    y.observe(data)
    from bayespy.inference import VB
    Q = VB(mu, tau, y)
    Q.update(repeat=20)
    import bayespy.plot as bpplt
    bpplt.pyplot.subplot(2, 1, 1)
    bpplt.pdf(mu, np.linspace(-10, 20, num=100), color='k', name=r'\mu')
    bpplt.pyplot.subplot(2, 1, 2)
    bpplt.pdf(tau, np.linspace(1e-6, 0.08, num=100), color='k', name=r'\tau')
    bpplt.pyplot.tight_layout()
    bpplt.pyplot.show()

This example was a very simple introduction to using BayesPy. The model
can be much more complex and each phase contains more options to give
the user more control over the inference. The following sections give
more details about the phases.