Author: Austin Rochford
Outside of the beta-binomial model, the multivariate normal model is likely the most studied Bayesian model in history. PyMC3 supports sampling from the LKJ distribution. The LKJ distribution represents the distribution on correlation matrices and is conjugate to the multivariate normal distribution. This post will show how to fit a simple multivariate normal model using pymc3 with a normal-LKJ prior.
First, we generate some two-dimensional sample data.
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%matplotlib inline
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from matplotlib.patches import Ellipse
from matplotlib import pyplot as plt
import numpy as np
import pymc3 as pm
import scipy as sp
import seaborn as sns
from theano import tensor as tt
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np.random.seed(3264602) # from random.org
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N = 100
mu_actual = sp.stats.uniform.rvs(-5, 10, size=2)
cov_actual_sqrt = sp.stats.uniform.rvs(0, 2, size=(2, 2))
cov_actual = np.dot(cov_actual_sqrt.T, cov_actual_sqrt)
x = sp.stats.multivariate_normal.rvs(mu_actual, cov_actual, size=N)
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var, U = np.linalg.eig(cov_actual)
angle = 180. / np.pi * np.arccos(np.abs(U[0, 0]))
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fig, ax = plt.subplots(figsize=(8, 6))
blue = sns.color_palette()[0]
e = Ellipse(mu_actual, 2 * np.sqrt(5.991 * var[0]), 2 * np.sqrt(5.991 * var[1]), angle=-angle)
e.set_alpha(0.5)
e.set_facecolor('gray')
e.set_zorder(10);
ax.add_artist(e);
ax.scatter(x[:, 0], x[:, 1], c='k', alpha=0.5, zorder=11);
rect = plt.Rectangle((0, 0), 1, 1, fc='gray', alpha=0.5)
ax.legend([rect], ['95% true credible region'], loc=2);
The sampling distribution for our model is $x_i \sim N(\mu, \Lambda)$, where $\Lambda$ is the precision matrix) of the distribution. The precision matrix is the inverse of the covariance matrix. The support of the LKJ distribution is the set of correlation matrices, not covariance matrices. We will use the separation strategy from Barnard et al. to combine an LKJ prior on the correlation matrix with a prior on the standard deviations of each dimension to produce a prior on the covariance matrix.
Let $\sigma$ be the vector of standard deviations of each component of our normal distribution, and $\mathbf{C}$ be the correlation matrix. The relationship
$$\Sigma = \operatorname{diag}(\sigma)\ \mathbf{C} \operatorname{diag}(\sigma)$$shows that priors on $\sigma$ and $\mathbf{C}$ will induce a prior on $\Sigma$. Following Barnard et al., we place a standard lognormal prior each the elements $\sigma$, and an LKJ prior on the correlation matric $\mathbf{C}$. The LKJ distribution requires a shape parameter $\nu > 0$. If $\mathbf{C} \sim LKJ(\nu)$, then $f(\mathbf{C}) \propto |\mathbf{C}|^{\nu - 1}$ (here $|\cdot|$ is the determinant).
We can now begin to build this model in pymc3. As you can see, we are passing summary statistics to testval which will be used as the starting point when we do inference further below. When summary statistics are available, it is always a good idea to use them in this manner.
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init_sigma = np.std(x, axis=0)
init_corr = np.corrcoef(x, rowvar=0)[0, 1]
with pm.Model() as model:
sigma = pm.Lognormal('sigma', np.zeros(2), np.ones(2), shape=2, testval=init_sigma)
nu = pm.Uniform('nu', 0, 5)
C_triu = pm.LKJCorr('C_triu', nu, 2, testval=init_corr)
There is a slight complication in pymc3's handling of the LKJCorr distribution; pymc3 represents the support of this distribution as a one-dimensional vector of the upper triangular elements of the full covariance matrix.
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C_triu.tag.test_value.shape
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In order to build a the full correlation matric $\mathbf{C}$, we first build a $2 \times 2$ tensor whose values are all C_triu and then set the diagonal entries to one. (Recall that a correlation matrix must be symmetric and positive definite with all diagonal entries equal to one.) We can then proceed to build the covariance matrix $\Sigma$ and the precision matrix $\Lambda$.
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with model:
C = pm.Deterministic('C', tt.fill_diagonal(C_triu[np.zeros((2, 2), dtype=np.int64)], 1.))
sigma_diag = pm.Deterministic('sigma_mat', tt.nlinalg.diag(sigma))
cov = pm.Deterministic('cov', tt.nlinalg.matrix_dot(sigma_diag, C, sigma_diag))
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cov.tag.test_value
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While defining C in terms of C_triu was simple in this case because our sampling distribution is two-dimensional, the example from this StackOverflow question shows how to generalize this transformation to arbitrarily many dimensions.
Finally, we define the prior on $\mu$ and the sampling distribution.
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with model:
mu = pm.MvNormal('mu', 0, cov, shape=2, testval=np.mean(x, axis=0))
x_ = pm.MvNormal('x', mu, cov, observed=x)
We are now ready to fit this model using pymc3.
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n_samples = 4000
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with model:
trace_ = pm.sample(n_samples, pm.Metropolis())
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trace = trace_[50:]
We see that the posterior estimate of $\mu$ is reasonably accurate.
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pm.traceplot(trace, varnames=['mu']);
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mu_actual
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The estimates of the standard deviations are certainly biased.
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pm.traceplot(trace, varnames=['sigma']);
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trace['sigma'].mean(axis=0)
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np.sqrt(var)
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However, the 95% posterior credible region is visually quite close to the true credible region, so we can be fairly satisfied with our model.
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post_cov = trace['cov'].mean(axis=0)
post_sigma, post_U = np.linalg.eig(post_cov)
post_angle = 180. / np.pi * np.arccos(np.abs(post_U[0, 0]))
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fig, ax = plt.subplots(figsize=(8, 6))
blue = sns.color_palette()[0]
e = Ellipse(mu_actual, 2 * np.sqrt(5.991 * post_sigma[0]), 2 * np.sqrt(5.991 * post_sigma[1]), angle=-post_angle)
e.set_alpha(0.5)
e.set_facecolor(blue)
e.set_zorder(9);
ax.add_artist(e);
e = Ellipse(mu_actual, 2 * np.sqrt(5.991 * var[0]), 2 * np.sqrt(5.991 * var[1]), angle=-angle)
e.set_alpha(0.5)
e.set_facecolor('gray')
e.set_zorder(10);
ax.add_artist(e);
ax.scatter(x[:, 0], x[:, 1], c='k', alpha=0.5, zorder=11);
rect = plt.Rectangle((0, 0), 1, 1, fc='gray', alpha=0.5)
post_rect = plt.Rectangle((0, 0), 1, 1, fc=blue, alpha=0.5)
ax.legend([rect, post_rect],
['95% true credible region',
'95% posterior credible region'],
loc=2);