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# Some setting up stuff. This cell is hidden from the Sphinx-rendered documentation.
%matplotlib inline
%config InlineBackend.figure_format = 'png'
np.random.seed(42)
NOTE: Maximum likelihood estimation is very experimental in BayesPy. The whole messaging system is being rewritten in order to support much better MLE along with non-conjugate distributions and a bunch of other useful features. This example just shows how to do MLE currently if you need to but it doesn't work very well (the optimizer is bad).
Create some 3-dimensional inputs:
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import numpy as np
N = 200
X = np.random.randn(N, 1)
Pre-compute a squared distance matrix for the inputs:
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from scipy.spatial import distance
D = distance.squareform(distance.pdist(X, 'sqeuclidean'))
Define a covariance function (exponentiated square or squared exponential):
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def exponentiated_square(parameters):
lengthscale = np.exp(parameters[0])
magnitude = np.exp(parameters[1])
return magnitude**2 * np.exp(-D/lengthscale) + 1e-6 * np.identity(N)
Implement the backward gradient pass for the exponentiated square:
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def d_parameters(d_exponentiated_square, parameters):
""" Backward gradient of exponentiated square w.r.t. the parameters """
lengthscale = np.exp(parameters[0])
magnitude = np.exp(parameters[1])
K = magnitude**2 * np.exp(-D/lengthscale)
return [
np.sum(d_exponentiated_square * K * D / lengthscale),
np.sum(d_exponentiated_square * K * 2)
]
BayesPy uses precision matrix instead of covariance matrix for the Gaussian variables, thus we need to implement the matrix inversion and its gradient:
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def inverse(K):
return np.linalg.inv(K)
def d_covariance(d_inverse, K):
""" Backward gradient of inverse w.r.t. the covariance matrix """
invK = np.linalg.inv(K)
return -np.dot(invK, np.dot(d_inverse, invK))
Create a maximum likelihood node for the covariance hyperparameters. Because the maximum likelihood estimation assumes unbounded variables, the node represents the parameters in log-scale:
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import bayespy as bp
parameters = bp.nodes.MaximumLikelihood(np.log([1, 1]))
Create nodes that use our defined functions to compute the precision matrix from the parameters. Function takes the actual function as the first argument and then a 2-tuple for each input argument of the function. The first tuple elements are the input nodes and the second tuple elements are the corresponding gradient pass functions:
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Covariance = bp.nodes.Function(
exponentiated_square,
(parameters, d_parameters)
)
Lambda = bp.nodes.Function(
inverse,
(Covariance, d_covariance)
)
Create a noiseless latent Gaussian process node::
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latent = bp.nodes.Gaussian(np.zeros(N), Lambda)
Observation noise precision:
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tau = bp.nodes.Gamma(1e-3, 1e-3)
Node for the observations:
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Y = bp.nodes.GaussianARD(latent, tau)
Draw a sample from our model and use it as a data:
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K = exponentiated_square(np.log([0.3, 5]))
data = bp.nodes.Gaussian(np.zeros(N), np.linalg.inv(K + 0.1**2*np.identity(N))).random()
Y.observe(data)
Construct inference engine:
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Q = bp.inference.VB(Y, latent, tau, parameters)
Use gradient based optimization to learn the parameters. We collapse latent and tau so that the learning is more efficient because the coupling between latent and parameters is quite strong.
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Q.optimize(parameters, collapsed=[latent, tau], maxiter=100, verbose=False)
Show the learned parameters:
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print("Learned GP parameters:", np.exp(parameters.get_moments()[0]))
print("Learned noise std:", tau.get_moments()[0] ** (-0.5))
Simple plot of a posterior sample of the latent function values:
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%matplotlib inline
plt.plot(X[:,0], latent.random(), 'k.');