Student-t Process

PyMC3 also includes T-process priors. They are a generalization of a Gaussian process prior to the multivariate Student’s T distribution. The usage is identical to that of gp.Latent, except they require a degrees of freedom parameter when they are specified in the model. For more information, see chapter 9 of Rasmussen+Williams, and Shah et al..

Note that T processes aren’t additive in the same way as GPs, so addition of TP objects are not supported.

Samples from a TP prior

The following code draws samples from a T process prior with 3 degrees of freedom and a Gaussian process, both with the same covariance matrix.

In [1]:
import pymc3 as pm
import theano.tensor as tt
import numpy as np

import matplotlib.pyplot as plt
%matplotlib inline
In [2]:
# set the seed
np.random.seed(1)

n = 100 # The number of data points
X = np.linspace(0, 10, n)[:, None] # The inputs to the GP, they must be arranged as a column vector

# Define the true covariance function and its parameters
_true = 1.0
η_true = 3.0
cov_func = η_true**2 * pm.gp.cov.Matern52(1, _true)

# A mean function that is zero everywhere
mean_func = pm.gp.mean.Zero()

# The latent function values are one sample from a multivariate normal
# Note that we have to call `eval()` because PyMC3 built on top of Theano
tp_samples = pm.MvStudentT.dist(mu=mean_func(X).eval(), cov=cov_func(X).eval(), nu=3).random(size=8)

## Plot samples from TP prior
fig = plt.figure(figsize=(12,5)); ax = fig.gca()
ax.plot(X.flatten(), tp_samples.T, lw=3, alpha=0.6);
ax.set_xlabel("X"); ax.set_ylabel("y"); ax.set_title("Samples from TP with DoF=3");


gp_samples = pm.MvNormal.dist(mu=mean_func(X).eval(), cov=cov_func(X).eval()).random(size=8)
fig = plt.figure(figsize=(12,5)); ax = fig.gca()
ax.plot(X.flatten(), gp_samples.T, lw=3, alpha=0.6);
ax.set_xlabel("X"); ax.set_ylabel("y"); ax.set_title("Samples from GP");
../_images/notebooks_GP-TProcess_3_0.png
../_images/notebooks_GP-TProcess_3_1.png

Poisson data generated by a T process

For the Poisson rate, we take the square of the function represented by the T process prior.

In [11]:
np.random.seed(7)

n = 150 # The number of data points
X = np.linspace(0, 10, n)[:, None] # The inputs to the GP, they must be arranged as a column vector

# Define the true covariance function and its parameters
_true = 1.0
η_true = 3.0
cov_func = η_true**2 * pm.gp.cov.ExpQuad(1, _true)

# A mean function that is zero everywhere
mean_func = pm.gp.mean.Zero()

# The latent function values are one sample from a multivariate normal
# Note that we have to call `eval()` because PyMC3 built on top of Theano
f_true = pm.MvStudentT.dist(mu=mean_func(X).eval(), cov=cov_func(X).eval(), nu=3).random(size=1)
y = np.random.poisson(f_true**2)

fig = plt.figure(figsize=(12,5)); ax = fig.gca()
ax.plot(X, f_true**2, "dodgerblue", lw=3, label="True f");
ax.plot(X, y, 'ok', ms=3, label="Data");
ax.set_xlabel("X"); ax.set_ylabel("y"); plt.legend();
../_images/notebooks_GP-TProcess_5_0.png
In [12]:
with pm.Model() as model:
     = pm.Gamma("ℓ", alpha=2, beta=2)
    η = pm.HalfCauchy("η", beta=3)
    cov = η**2 * pm.gp.cov.ExpQuad(1, )

    # informative prior on degrees of freedom < 5
    ν = pm.Gamma("ν", alpha=2, beta=1)
    tp = pm.gp.TP(cov_func=cov, nu=ν)
    f = tp.prior("f", X=X)

    # adding a small constant seems to help with numerical stability here
    y_ = pm.Poisson("y", mu=tt.square(f) + 1e-6, observed=y)

    tr = pm.sample(1000)
Auto-assigning NUTS sampler...
Initializing NUTS using advi+adapt_diag...
Average Loss = 1,379.5:   6%|▌         | 11278/200000 [01:05<15:56, 197.36it/s]
Convergence archived at 11300
Interrupted at 11,300 [5%]: Average Loss = 2,068
100%|██████████| 1500/1500 [31:34<00:00,  1.46s/it]/home/bill/pymc3/pymc3/step_methods/hmc/nuts.py:467: UserWarning: Chain 0 contains 11 diverging samples after tuning. If increasing `target_accept` does not help try to reparameterize.
  % (self._chain_id, n_diverging))

In [13]:
pm.traceplot(tr, varnames=["ℓ", "ν", "η"], lines={"ℓ": _true, "η": η_true, "ν": 3});
../_images/notebooks_GP-TProcess_7_0.png
In [14]:
n_new = 200
X_new = np.linspace(0, 15, n_new)[:,None]

# add the GP conditional to the model, given the new X values
with model:
    f_pred = tp.conditional("f_pred", X_new)

# Sample from the GP conditional distribution
with model:
    pred_samples = pm.sample_ppc(tr, vars=[f_pred], samples=1000)
100%|██████████| 1000/1000 [01:44<00:00,  7.08it/s]
In [18]:
fig = plt.figure(figsize=(12,5)); ax = fig.gca()
from pymc3.gp.util import plot_gp_dist
plot_gp_dist(ax, np.square(pred_samples["f_pred"]), X_new);
plt.plot(X, np.square(f_true), "dodgerblue", lw=3, label="True f");
plt.plot(X, y, 'ok', ms=3, alpha=0.5, label="Observed data");
plt.xlabel("X"); plt.ylabel("True f(x)"); plt.ylim([-2, 20])
plt.title("Conditional distribution of f_*, given f"); plt.legend();
../_images/notebooks_GP-TProcess_9_0.png