how to calculate log posterior of a GP over a trace in pymc3

问题

Use case

Suppose I have an observation y_0 at X_0 which I'd like to model with a Gaussian process with hyper params theta. Suppose I then determine a distribution in the hyper params theta by hierarchically sampling the posterior.

Now, I'd like to evaluate the log posterior probability of another observation say y_1 at X_1, averaged over the hyper param distribution, E_theta [ log P(y_1 | y_0, X_0, X_1, theta) ] Ideally, I'd draw from the posterior in theta and calculate log P(y_1 | y_0, X_0, X_1, theta) and then take the geometric mean.

Question 1

Suppose I have a have the result of a sampling, i.e. a trace for example:

with pm.Model() as model:
    ...
    trace = pm.sample(1000)

How do I evaluate another tensor over these samples (or a subset of them)? I can only find a partial solution using pm.Deterministic defined as part of the model,

with model:
    h = pm.Deterministic('h',thing_to_calculate_over_the_posterior)
    ### Make sure nothing then depends on h so it doesn't affect the sampling
    trace = pm.sample(1000)
h_sampled = trace['h']

This doesn't feel right exactly. I feel like you should be able to evaluate anything over a subset of the trace after they have been sampled.

Question 2

In pymc3 is there a method part of pm.gp that will create the tensor representing log P(y_1 | y_0 X_0 X_1 theta) OR must I create this myself which entails writing out the posterior mean and covariance (something already done inside pm.gp) and then calling cholesky decomp etc.

回答1:

I'll answer with an example for this use case. Essentially, separate sampling from the conditional definition, then use method 1 of method 2 below.

import numpy as np
import pymc3 as pm

# Data generation
X0 = np.linspace(0, 10, 100)[:,None]
y0 = X0**(0.5) + np.exp(-X0/5)*np.sin(10*X0)
y0 += 0.1*np.random.normal(size=y0.shape)
y0 = np.squeeze(y0)

# y1
X1 = np.linspace(0, 15, 200)[:,None]
y1 = X1**(0.5) + np.exp(-X1/6)*np.sin(8*X1)
y1 = np.squeeze(y1)

# 'Solve' the inference problem
with pm.Model() as model:
    l = pm.HalfNormal('l',5.)
    cov_func = pm.gp.cov.ExpQuad(1, ls=l)
    gp = pm.gp.Marginal(cov_func=cov_func)
    y0_ = gp.marginal_likelihood('y0',X0,y0,0.1)
    trace = pm.sample(100)

# Define the object P(y1 | X1 y0 X0)
with model:
    y1_ = gp.conditional('y1',X1,given={'X':X0,'y':y0,'noise':0.1})
    # Note the given=... is not strictly required as it's cached from above

###
# Method 1

logp = y1_.logp
logp_vals1 = []
for point in trace:
    point['y1'] = y1
    logp_vals1.append(logp(point))
    # note this is approximately 100x faster than logp_vals1.append(y1_.logp(point))
    # because logp is a property with a lot of overhead

###
# Method 2

import theano
y1_shr = theano.shared(y1)
with model:
    logp = pm.Deterministic('logp', y1_.distribution.logp(y1_shr))

logp_val2 = [pm.distributions.draw_values([logp], point) for point in trace]

Method 1 appears to be 2-3 times faster on my machine.

来源：https://stackoverflow.com/questions/48565385/how-to-calculate-log-posterior-of-a-gp-over-a-trace-in-pymc3

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python

theano

pymc3