In [1]:
%pylab inline
import numpy as np
import scipy.stats as stats
import pymc3 as pm
from theano import shared
import theano
import theano.tensor as tt
floatX = "float32"
%config InlineBackend.figure_format = 'retina'
%qtconsole --colors=linux
plt.style.use('ggplot')
A walkthrough of implementing a Conditional Autoregressive (CAR) model in PyMC3, with WinBugs/PyMC2 and STAN code as references.
PyMC3 issue#2022, issue#2066 and comments. I would like to thank @denadai2, @aseyboldt, and @twiecki for the helpful discussion.As a probabilistic language, there are some fundamental differences between PyMC3 and other alternatives such as WinBugs, JAGS, and STAN. In this notebook, I will summarise some heuristics and intuition I got over the past two years using PyMC3. I will outline some thinking process of how I approach a modelling problem using PyMC3, and how thinking in linear algebra solves most of the programming problem. I hope this notebook will shed some light into the design and feature of PyMC3, and similar language built on linear algebra package with a static world view (e.g., Edward, which is based on Tensorflow).
For more resources comparing between PyMC3 codes and other probabilistic languages:
Supposed we want to implement a Conditional Autoregressive (CAR) model with some reference codes in WinBugs/PyMC2 and STAN.
For the sake of brevity, I will not go into the details of the CAR model. The essential idea of this kind model is autocorrelation, which is informally speaking "correlation with itself". In a CAR model, the probability of values estimated at any given location $y_i$ are conditional on some neighboring values $y_j, _{j \neq i}$ (in another word, correlated/covariated with these values):
where $\sigma_i^{2}$ is a spatially varying covariance parameter, and $b_{ii} = 0$.
Here we will demonstrate the implementation of a CAR model using the canonical example: the lip cancer risk data in Scotland between 1975 and 1980. The original data is from (Kemp et al. 1985). This data set includes observed lip cancer case counts at 56 spatial units in Scotland, with the expected number of cases as intercept, and an area-specific continuous variable coded for the proportion of the population employed in agriculture, fishing, or forestry (AFF). We want to model how lip cancer rates (O below) relate to AFF (aff below), as exposure to sunlight is a risk factor.
Setting up the data:
In [2]:
county = np.array(["skye_lochalsh", "banff_buchan", "caithness,berwickshire", "ross_cromarty",
"orkney", "moray", "shetland", "lochaber", "gordon", "western_isles",
"sutherland", "nairn", "wigtown", "NE.fife", "kincardine", "badenoch",
"ettrick", "inverness", "roxburgh", "angus", "aberdeen", "argyll_bute",
"clydesdale", "kirkcaldy", "dunfermline", "nithsdale", "east_lothian",
"perth_kinross", "west_lothian", "cumnock_doon", "stewartry", "midlothian",
"stirling", "kyle_carrick", "inverclyde", "cunninghame", "monklands",
"dumbarton", "clydebank", "renfrew", "falkirk", "clackmannan", "motherwell",
"edinburgh", "kilmarnock", "east_kilbride", "hamilton", "glasgow", "dundee",
"cumbernauld", "bearsden", "eastwood", "strathkelvin", "tweeddale",
"annandale"])
# observed
O = np.array([9, 39, 11, 9, 15, 8, 26, 7, 6, 20, 13, 5, 3, 8, 17, 9, 2, 7, 9, 7, 16,
31, 11, 7, 19, 15, 7, 10, 16, 11, 5, 3, 7, 8, 11, 9, 11, 8, 6, 4, 10,
8, 2, 6, 19, 3, 2, 3, 28, 6, 1, 1, 1, 1, 0, 0])
N = len(O)
# expected (E) rates, based on the age of the local population
E = np.array([1.4, 8.7, 3.0, 2.5, 4.3, 2.4, 8.1, 2.3, 2.0, 6.6, 4.4, 1.8, 1.1, 3.3,
7.8, 4.6, 1.1, 4.2, 5.5, 4.4, 10.5, 22.7, 8.8, 5.6, 15.5, 12.5, 6.0,
9.0, 14.4, 10.2, 4.8, 2.9, 7.0, 8.5, 12.3, 10.1, 12.7, 9.4, 7.2, 5.3,
18.8, 15.8, 4.3, 14.6, 50.7, 8.2, 5.6, 9.3, 88.7, 19.6, 3.4, 3.6, 5.7,
7.0, 4.2, 1.8])
logE = np.log(E)
# proportion of the population engaged in agriculture, forestry, or fishing (AFF)
aff = np.array([16, 16, 10, 24, 10, 24, 10, 7, 7, 16, 7, 16, 10, 24, 7, 16, 10, 7,
7, 10, 7, 16, 10, 7, 1, 1, 7, 7, 10, 10, 7, 24, 10, 7, 7, 0, 10, 1,
16, 0, 1, 16, 16, 0, 1, 7, 1, 1, 0, 1, 1, 0, 1, 1, 16, 10])/10.
# Spatial adjacency information
adj = np.array([[5, 9, 11,19],
[7, 10],
[6, 12],
[18,20,28],
[1, 11,12,13,19],
[3, 8],
[2, 10,13,16,17],
[6],
[1, 11,17,19,23,29],
[2, 7, 16,22],
[1, 5, 9, 12],
[3, 5, 11],
[5, 7, 17,19],
[31,32,35],
[25,29,50],
[7, 10,17,21,22,29],
[7, 9, 13,16,19,29],
[4,20, 28,33,55,56],
[1, 5, 9, 13,17],
[4, 18,55],
[16,29,50],
[10,16],
[9, 29,34,36,37,39],
[27,30,31,44,47,48,55,56],
[15,26,29],
[25,29,42,43],
[24,31,32,55],
[4, 18,33,45],
[9, 15,16,17,21,23,25,26,34,43,50],
[24,38,42,44,45,56],
[14,24,27,32,35,46,47],
[14,27,31,35],
[18,28,45,56],
[23,29,39,40,42,43,51,52,54],
[14,31,32,37,46],
[23,37,39,41],
[23,35,36,41,46],
[30,42,44,49,51,54],
[23,34,36,40,41],
[34,39,41,49,52],
[36,37,39,40,46,49,53],
[26,30,34,38,43,51],
[26,29,34,42],
[24,30,38,48,49],
[28,30,33,56],
[31,35,37,41,47,53],
[24,31,46,48,49,53],
[24,44,47,49],
[38,40,41,44,47,48,52,53,54],
[15,21,29],
[34,38,42,54],
[34,40,49,54],
[41,46,47,49],
[34,38,49,51,52],
[18,20,24,27,56],
[18,24,30,33,45,55]])
# Change to Python indexing (i.e. -1)
for i in range(len(adj)):
for j in range(len(adj[i])):
adj[i][j] = adj[i][j]-1
# spatial weight
weights = np.array([[1,1,1,1],
[1,1],
[1,1],
[1,1,1],
[1,1,1,1,1],
[1,1],
[1,1,1,1,1],
[1],
[1,1,1,1,1,1],
[1,1,1,1],
[1,1,1,1],
[1,1,1],
[1,1,1,1],
[1,1,1],
[1,1,1],
[1,1,1,1,1,1],
[1,1,1,1,1,1],
[1,1,1,1,1,1],
[1,1,1,1,1],
[1,1,1],
[1,1,1],
[1,1],
[1,1,1,1,1,1],
[1,1,1,1,1,1,1,1],
[1,1,1],
[1,1,1,1],
[1,1,1,1],
[1,1,1,1],
[1,1,1,1,1,1,1,1,1,1,1],
[1,1,1,1,1,1],
[1,1,1,1,1,1,1],
[1,1,1,1],
[1,1,1,1],
[1,1,1,1,1,1,1,1,1],
[1,1,1,1,1],
[1,1,1,1],
[1,1,1,1,1],
[1,1,1,1,1,1],
[1,1,1,1,1],
[1,1,1,1,1],
[1,1,1,1,1,1,1],
[1,1,1,1,1,1],
[1,1,1,1],
[1,1,1,1,1],
[1,1,1,1],
[1,1,1,1,1,1],
[1,1,1,1,1,1],
[1,1,1,1],
[1,1,1,1,1,1,1,1,1],
[1,1,1],
[1,1,1,1],
[1,1,1,1],
[1,1,1,1],
[1,1,1,1,1],
[1,1,1,1,1],
[1,1,1,1,1,1]])
Wplus = np.asarray([sum(w) for w in weights])
The classical WinBugs implementation (more information here):
model
{
for (i in 1 : regions) {
O[i] ~ dpois(mu[i])
log(mu[i]) <- log(E[i]) + beta0 + beta1*aff[i]/10 + phi[i] + theta[i]
theta[i] ~ dnorm(0.0,tau.h)
}
phi[1:regions] ~ car.normal(adj[], weights[], Wplus[], tau.c)
beta0 ~ dnorm(0.0, 1.0E-5) # vague prior on grand intercept
beta1 ~ dnorm(0.0, 1.0E-5) # vague prior on covariate effect
tau.h ~ dgamma(3.2761, 1.81)
tau.c ~ dgamma(1.0, 1.0)
sd.h <- sd(theta[]) # marginal SD of heterogeneity effects
sd.c <- sd(phi[]) # marginal SD of clustering (spatial) effects
alpha <- sd.c / (sd.h + sd.c)
}The main challenge to port this model to PyMC3 is the car.normal in WinBugs. It is a likelihood function that each realization is conditioned on some neigbour realization (a smoothed property). In PyMC2, it could be implemented as a custom likelihood function (a @stochastic node) mu_phi:
@stochastic
def mu_phi(tau=tau_c, value=np.zeros(N)):
# Calculate mu based on average of neighbours
mu = np.array([ sum(weights[i]*value[adj[i]])/Wplus[i] for i in xrange(N)])
# Scale precision to the number of neighbours
taux = tau*Wplus
return normal_like(value,mu,taux)
We can of course just define mu_phi similarly and wrap it in a pymc3.DensityDist, however, doing so is usually resulting a very slow model (both in compling and sampling). It is a general challenge of porting pymc2 code into pymc3 (or even generally porting WinBugs, JAGS, or STAN code into PyMC3), as using a for-loop under pm.Model perform poorly in theano, the backend of PyMC3.
The underlying mechanism in PyMC3 is very different compared to PyMC2, using for-loop to generate RV or stacking multiple RV with arguments such as [pm.Binomial('obs%'%i, p[i], n) for i in range(K)] generate unnecessary large number of nodes in theano graph, which then slow down the compiling to an unbearable amount.
The easiest way is to move the for-loop outside of pm.Model. And usually is not difficult to do. For example, in STAN you can have a transformed data{} block, in PyMC3 you just need to computed it before defining your Model.
And if it is absolutely necessary to use a for-loop, you can use a theano loop (i.e., theano.scan), which you can find some introduction on the theano website and see a usecase in PyMC3 timeseries distribution.
In [3]:
value = np.asarray(np.random.randn(N,), dtype=theano.config.floatX)
maxwz = max([sum(w) for w in weights])
N = len(weights)
wmat = np.zeros((N, maxwz))
amat = np.zeros((N, maxwz), dtype='int32')
for i, w in enumerate(weights):
wmat[i, np.arange(len(w))] = w
amat[i, np.arange(len(w))] = adj[i]
# defining the tensor variables
x = tt.vector("x")
x.tag.test_value = value
w = tt.matrix("w")
# provide Theano with a default test-value
w.tag.test_value = wmat
a = tt.matrix("a", dtype='int32')
a.tag.test_value = amat
def get_mu(w, a):
a1 = tt.cast(a, 'int32')
return tt.sum(w*x[a1])/tt.sum(w)
results, _ = theano.scan(fn=get_mu, sequences=[w, a])
compute_elementwise = theano.function(inputs=[x, w, a], outputs=results)
print(compute_elementwise(value, wmat, amat))
def mu_phi(value):
N = len(weights)
# Calculate mu based on average of neighbours
mu = np.array([np.sum(weights[i]*value[adj[i]])/Wplus[i] for i in range(N)])
return mu
print(mu_phi(value))
Since it produce the same result as the orignial for-loop, we now wrap it as a new distribution with a loglikelihood function in PyMC3.
In [4]:
from theano import scan
floatX = "float32"
from pymc3.distributions import continuous
from pymc3.distributions import distribution
In [5]:
class CAR(distribution.Continuous):
"""
Conditional Autoregressive (CAR) distribution
Parameters
----------
a : list of adjacency information
w : list of weight information
tau : precision at each location
"""
def __init__(self, w, a, tau, *args, **kwargs):
super(CAR, self).__init__(*args, **kwargs)
self.a = a = tt.as_tensor_variable(a)
self.w = w = tt.as_tensor_variable(w)
self.tau = tau*tt.sum(w, axis=1)
self.mode = 0.
def get_mu(self, x):
def weigth_mu(w, a):
a1 = tt.cast(a, 'int32')
return tt.sum(w*x[a1])/tt.sum(w)
mu_w, _ = scan(fn=weigth_mu,
sequences=[self.w, self.a])
return mu_w
def logp(self, x):
mu_w = self.get_mu(x)
tau = self.tau
return tt.sum(continuous.Normal.dist(mu=mu_w, tau=tau).logp(x))
We then use it in our PyMC3 version of the CAR model:
In [6]:
with pm.Model() as model1:
# Vague prior on intercept
beta0 = pm.Normal('beta0', mu=0.0, tau=1.0e-5)
# Vague prior on covariate effect
beta1 = pm.Normal('beta1', mu=0.0, tau=1.0e-5)
# Random effects (hierarchial) prior
tau_h = pm.Gamma('tau_h', alpha=3.2761, beta=1.81)
# Spatial clustering prior
tau_c = pm.Gamma('tau_c', alpha=1.0, beta=1.0)
# Regional random effects
theta = pm.Normal('theta', mu=0.0, tau=tau_h, shape=N)
mu_phi = CAR('mu_phi', w=wmat, a=amat, tau=tau_c, shape=N)
# Zero-centre phi
phi = pm.Deterministic('phi', mu_phi-tt.mean(mu_phi))
# Mean model
mu = pm.Deterministic('mu', tt.exp(logE + beta0 + beta1*aff + theta + phi))
# Likelihood
Yi = pm.Poisson('Yi', mu=mu, observed=O)
# Marginal SD of heterogeniety effects
sd_h = pm.Deterministic('sd_h', tt.std(theta))
# Marginal SD of clustering (spatial) effects
sd_c = pm.Deterministic('sd_c', tt.std(phi))
# Proportion sptial variance
alpha = pm.Deterministic('alpha', sd_c/(sd_h+sd_c))
trace1 = pm.sample(3e3, njobs=2, tune=1000, nuts_kwargs={'max_treedepth': 15})
In [7]:
pm.traceplot(trace1, varnames=['alpha', 'sd_h', 'sd_c']);
In [8]:
pm.plot_posterior(trace1, varnames=['alpha']);
theano.scan is much faster than using a python for loop, but it is still quite slow. One of the way to improve is to use linear algebra and matrix multiplication. In another word, we should try to find a way to use matrix multiplication instead of for-loop (if you have experience in using MATLAB, it is the same philosophy). In our case, we can totally do that.
For a similar problem, you can also have a look of my port of Lee and Wagenmakers' book. For example, in Chapter 19, the STAN code use a for loop to generate the likelihood function, and I generate the matrix outside and use matrix multiplication etc to archive the same purpose.
In [9]:
maxwz = max([sum(w) for w in weights])
N = len(weights)
wmat2 = np.zeros((N, N))
amat2 = np.zeros((N, N), dtype='int32')
for i, a in enumerate(adj):
amat2[i, a] = 1
wmat2[i, a] = weights[i]
value = np.asarray(np.random.randn(N,), dtype=theano.config.floatX)
print(np.sum(value*amat2, axis=1)/np.sum(wmat2, axis=1))
def mu_phi(value):
N = len(weights)
# Calculate mu based on average of neighbours
mu = np.array([np.sum(weights[i]*value[adj[i]])/Wplus[i] for i in range(N)])
return mu
print(mu_phi(value))
Now create a new CAR distribution with the matrix multiplication instead of theano.scan to get the mu
In [10]:
class CAR2(distribution.Continuous):
"""
Conditional Autoregressive (CAR) distribution
Parameters
----------
a : adjacency matrix
w : weight matrix
tau : precision at each location
"""
def __init__(self, w, a, tau, *args, **kwargs):
super(CAR2, self).__init__(*args, **kwargs)
self.a = a = tt.as_tensor_variable(a)
self.w = w = tt.as_tensor_variable(w)
self.tau = tau*tt.sum(w, axis=1)
self.mode = 0.
def logp(self, x):
tau = self.tau
w = self.w
a = self.a
mu_w = tt.sum(x*a, axis=1)/tt.sum(w, axis=1)
return tt.sum(continuous.Normal.dist(mu=mu_w, tau=tau).logp(x))
In [11]:
with pm.Model() as model2:
# Vague prior on intercept
beta0 = pm.Normal('beta0', mu=0.0, tau=1.0e-5)
# Vague prior on covariate effect
beta1 = pm.Normal('beta1', mu=0.0, tau=1.0e-5)
# Random effects (hierarchial) prior
tau_h = pm.Gamma('tau_h', alpha=3.2761, beta=1.81)
# Spatial clustering prior
tau_c = pm.Gamma('tau_c', alpha=1.0, beta=1.0)
# Regional random effects
theta = pm.Normal('theta', mu=0.0, tau=tau_h, shape=N)
mu_phi = CAR2('mu_phi', w=wmat2, a=amat2, tau=tau_c, shape=N)
# Zero-centre phi
phi = pm.Deterministic('phi', mu_phi-tt.mean(mu_phi))
# Mean model
mu = pm.Deterministic('mu', tt.exp(logE + beta0 + beta1*aff + theta + phi))
# Likelihood
Yi = pm.Poisson('Yi', mu=mu, observed=O)
# Marginal SD of heterogeniety effects
sd_h = pm.Deterministic('sd_h', tt.std(theta))
# Marginal SD of clustering (spatial) effects
sd_c = pm.Deterministic('sd_c', tt.std(phi))
# Proportion sptial variance
alpha = pm.Deterministic('alpha', sd_c/(sd_h+sd_c))
trace2 = pm.sample(3e3, njobs=2, tune=1000, nuts_kwargs={'max_treedepth': 15})
As you can see, its 6x faster using matrix multiplication.
In [12]:
pm.traceplot(trace2, varnames=['alpha', 'sd_h', 'sd_c']);
In [13]:
pm.plot_posterior(trace2, varnames=['alpha']);
There are (almost) always many ways to formulate your model. And some works better than the others under different context (size of your dataset, properties of the sampler, etc). In this case, we can expressed the CAR prior as:
$$\phi \sim \mathcal{N}(0, [D_\tau (I - \alpha B)]^{-1}).$$For the sake of brevity, you can find more details in the original Stan case study. You might come across similar deviation in Gaussian Process, which result in a zero-mean Gaussian distribution conditioned on a covariance function.
In the Stan Code, matrix D is generated in the model using a transformed data{} block:
transformed data{
vector[n] zeros;
matrix<lower = 0>[n, n] D;
{
vector[n] W_rowsums;
for (i in 1:n) {
W_rowsums[i] = sum(W[i, ]);
}
D = diag_matrix(W_rowsums);
}
zeros = rep_vector(0, n);
}
We can generate the same matrix quite easily:
In [14]:
X = np.hstack((np.ones((N, 1)), stats.zscore(aff, ddof=1)[:, None]))
W = wmat2
D = np.diag(W.sum(axis=1))
log_offset = logE[:, None]
Then in the STAN model:
model {
phi ~ multi_normal_prec(zeros, tau * (D - alpha * W));
...
}
since the precision matrix just generated by some matrix multiplication, we can do just that in PyMC3:
In [15]:
with pm.Model() as model3:
# Vague prior on intercept and effect
beta = pm.Normal('beta', mu=0.0, tau=1.0, shape=(2, 1))
# Priors for spatial random effects
tau = pm.Gamma('tau', alpha=2., beta=2.)
alpha = pm.Uniform('alpha', lower=0, upper=1)
phi = pm.MvNormal('phi', mu=0, tau=tau*(D - alpha*W), shape=(N, 1))
# Mean model
mu = pm.Deterministic('mu', tt.exp(tt.dot(X, beta) + phi + log_offset))
# Likelihood
Yi = pm.Poisson('Yi', mu=mu.ravel(), observed=O)
trace3 = pm.sample(3e3, njobs=2, tune=1000)
In [16]:
pm.traceplot(trace3, varnames=['alpha', 'beta', 'tau']);