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'
plt.style.use('ggplot')

Populating the interactive namespace from numpy and matplotlib


# PyMC3 Modeling tips and heuristic¶

A walkthrough of implementing a Conditional Autoregressive (CAR) model in PyMC3, with WinBugs/PyMC2 and STAN code as references.

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 in how I approach a modelling problem using PyMC3, and how thinking in linear algebra solves most of the programming problems. I hope this notebook will shed some light onto the design and features of PyMC3, and similar languages that are built on linear algebra packages with a static world view (e.g., Edward, which is based on Tensorflow).

For more resources comparing between PyMC3 codes and other probabilistic languages: * PyMC3 port of “Doing Bayesian Data Analysis” - PyMC3 vs WinBugs/JAGS/STAN * PyMC3 port of “Bayesian Cognitive Modeling” - PyMC3 vs WinBugs/JAGS/STAN * [WIP] PyMC3 port of “Statistical Rethinking” - PyMC3 vs STAN

## Background information¶

Suppose we want to implement a Conditional Autoregressive (CAR) model with examples in WinBugs/PyMC2 and STAN.
For the sake of brevity, I will not go into the details of the CAR model. The essential idea is autocorrelation, which is informally “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):
$y_i \mid y_j, j \neq i \sim \mathcal{N}(\alpha \sum_{j = 1}^n b_{ij} y_j, \sigma_i^{2})$

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 a canonical example: the lip cancer risk data in Scotland between 1975 and 1980. The original data is from (Kemp et al. 1985). This dataset 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.

$O_i \sim \mathcal{Poisson}(\text{exp}(\beta_0 + \beta_1*aff + \phi_i + \log(\text{E}_i)))$
$\phi_i \mid \phi_j, j \neq i \sim \mathcal{N}(\alpha \sum_{j = 1}^n b_{ij} \phi_j, \sigma_i^{2})$

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.

[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)

# 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])


## A WinBugs/PyMC2 implementation¶

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 porting this model to PyMC3 is the car.normal function in WinBugs. It is a likelihood function that conditions each realization on some neigbour realization (a smoothed property). In PyMC2, it could be implemented as a custom likelihood function (a @stochastic node) mu_phi <http://glau.ca/?p=340>__:

@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 just define mu_phi similarly and wrap it in a pymc3.DensityDist, however, doing so usually results in a very slow model (both in compling and sampling). In general, porting pymc2 code into pymc3 (or even generally porting WinBugs, JAGS, or STAN code into PyMC3) that use a for loops tend to perform poorly in theano, the backend of PyMC3.

The underlying mechanism in PyMC3 is very different compared to PyMC2, using for loops 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 slows down compilation appreciably.

The easiest way is to move the loop out 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 compute it before defining your Model.

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.

## PyMC3 implementation using theano.scan¶

So lets try to implement the CAR model using theano.scan. First we create a theano function with theano.scan and check if it really works by comparing its result to the for-loop.

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

# 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))

[-0.22175201 -0.14781145 -0.53384127  0.01320356 -0.16192564 -1.10971794
-0.07910109 -0.83692263 -0.47747822  0.59007413  0.33055391 -0.1670654
-0.86906285 -0.1248809  -0.73033361  0.10636681  0.02969443  0.29081879
0.24585857  0.72974921  0.25590465  0.55421142  0.12301765 -0.04841463
-0.53751571 -0.13495914  0.62703638  0.50535261 -0.29189189  0.89895098
-0.13630391 -0.4486971   0.34870552  0.28891306 -0.01254449 -0.04516954
0.1885583   0.7594217   0.42341675  0.67432005  0.35642887 -0.3177663
0.11514166  0.78772165 -0.10791141  0.10196948  0.97384957  0.73794546
0.69991521 -0.54984279  0.93403301  1.03208351  0.45207161  0.44287877
0.2902995   0.29128088]
[-0.22175201 -0.14781145 -0.53384127  0.01320356 -0.16192564 -1.10971794
-0.07910109 -0.83692263 -0.47747822  0.59007413  0.33055391 -0.1670654
-0.86906285 -0.1248809  -0.73033361  0.10636681  0.02969443  0.29081879
0.24585857  0.72974921  0.25590465  0.55421142  0.12301765 -0.04841463
-0.53751571 -0.13495914  0.62703638  0.50535261 -0.29189189  0.89895098
-0.13630391 -0.4486971   0.34870552  0.28891306 -0.01254449 -0.04516954
0.1885583   0.7594217   0.42341675  0.67432005  0.35642887 -0.3177663
0.11514166  0.78772165 -0.10791141  0.10196948  0.97384957  0.73794546
0.69991521 -0.54984279  0.93403301  1.03208351  0.45207161  0.44287877
0.2902995   0.29128088]


Since it produces the same result as the orignial for-loop, we will wrap it as a new distribution with a log-likelihood 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 [7]:

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})

Auto-assigning NUTS sampler...
Average Loss = 202.97:   8%|▊         | 16437/200000 [00:20<03:40, 834.33it/s]
Convergence archived at 16500
Interrupted at 16,500 [8%]: Average Loss = 337
100%|██████████| 4000/4000.0 [28:03<00:00, 12.27it/s]

In [8]:

pm.traceplot(trace1, varnames=['alpha', 'sd_h', 'sd_c']);

In [9]:

pm.plot_posterior(trace1, varnames=['alpha']);


theano.scan is much faster than using a python for loop, but it is still quite slow. One approach for improving it is to use linear algebra. That is, we should try to find a way to use matrix multiplication instead of looping (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.

## PyMC3 implementation using matrix “trick”¶

Again, we try on some simulation data to make sure the implementation is correct.

In [10]:

maxwz = max([sum(w) for w in weights])
N = len(weights)
wmat2 = np.zeros((N, N))
amat2 = np.zeros((N, N), dtype='int32')
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))

[-0.79847225 -0.35566929  0.50726277 -0.1279199  -0.55775445  0.09553115
-0.05925132  0.34764192 -0.84118472 -0.20047967 -0.17107106 -0.58662655
0.13991989 -0.65204496 -1.12490181 -0.2620437  -0.33765774 -0.00992251
-0.08191449 -0.0993801  -0.83830122 -0.00728558 -0.44663178  0.0221561
-0.54458272 -0.88404449 -0.32445576  0.58966362 -0.01684126  0.2838882
0.53790262  0.02117891 -0.07079652 -0.88753984 -0.66015978 -0.32685328
0.17664941 -0.59170726 -0.34773551  0.64698776 -0.55797083 -0.5152577
-0.28250522  0.31772902  0.11282101 -0.53837003 -0.16239299  0.75354403
-0.21765241 -0.45637515 -0.23264213 -0.64534271  0.75149658  0.11944715
0.25436263  0.57070506]
[-0.79847225 -0.35566929  0.50726277 -0.1279199  -0.55775445  0.09553115
-0.05925132  0.34764192 -0.84118472 -0.20047967 -0.17107106 -0.58662655
0.13991989 -0.65204496 -1.12490181 -0.2620437  -0.33765774 -0.00992251
-0.08191449 -0.0993801  -0.83830122 -0.00728558 -0.44663178  0.0221561
-0.54458272 -0.88404449 -0.32445576  0.58966362 -0.01684126  0.2838882
0.53790262  0.02117891 -0.07079652 -0.88753984 -0.66015978 -0.32685328
0.17664941 -0.59170726 -0.34773551  0.64698776 -0.55797083 -0.5152577
-0.28250522  0.31772902  0.11282101 -0.53837003 -0.16239299  0.75354403
-0.21765241 -0.45637515 -0.23264213 -0.64534271  0.75149658  0.11944715
0.25436263  0.57070506]


Now create a new CAR distribution with the matrix multiplication instead of theano.scan to get the mu

In [11]:

class CAR2(distribution.Continuous):
"""
Conditional Autoregressive (CAR) distribution

Parameters
----------
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 [12]:

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})

Auto-assigning NUTS sampler...
Average Loss = 203.33:   8%|▊         | 16206/200000 [00:02<00:30, 6007.72it/s]
Convergence archived at 16500
Interrupted at 16,500 [8%]: Average Loss = 337
100%|██████████| 4000/4000.0 [02:36<00:00, 25.58it/s]


As you can see, it is appreciably faster using matrix multiplication.

In [13]:

pm.traceplot(trace2, varnames=['alpha', 'sd_h', 'sd_c']);

In [14]:

pm.plot_posterior(trace2, varnames=['alpha']);


## PyMC3 implementation using Matrix multiplication¶

There are almost always multiple ways to formulate a particular model. Some approaches work better than the others under different contexts (size of your dataset, properties of the sampler, etc).

In this case, we can expressed the CAR prior as:

$\phi \sim \mathcal{N}(0, [D_\tau (I - \alpha B)]^{-1}).$

You can find more details in the original Stan case study. You might come across similar constructs 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 [15]:

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 [16]:

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=(1, N))

# Mean model
mu = pm.Deterministic('mu', tt.exp(tt.dot(X, beta) + phi.T + log_offset))

# Likelihood
Yi = pm.Poisson('Yi', mu=mu.ravel(), observed=O)

trace3 = pm.sample(3e3, njobs=2, tune=1000)

Auto-assigning NUTS sampler...
Average Loss = 174.97:   8%|▊         | 16657/200000 [00:14<02:58, 1029.80it/s]
Convergence archived at 16700
Interrupted at 16,700 [8%]: Average Loss = 265.66
100%|██████████| 4000/4000.0 [01:37<00:00, 40.96it/s]

In [17]:

pm.traceplot(trace3, varnames=['alpha', 'beta', 'tau']);

In [18]:

pm.plot_posterior(trace3, varnames=['alpha']);


Notice that since the model parameterization is different than in the WinBugs model, the alpha can’t be interpreted in the same way.

## PyMC3 implementation using Sparse Matrix¶

Note that in the node $$\phi \sim \mathcal{N}(0, [D_\tau (I - \alpha B)]^{-1})$$, we are computing the log-likelihood for a multivariate Gaussian distribution, which might not scale well in high-dimensions. We can take advantage of the fact that the covariance matrix here $$[D_\tau (I - \alpha B)]^{-1}$$ is sparse, and there are faster ways to compute its log-likelihood.

For example, a more efficient sparse representation of the CAR in STAN:

functions {
/**
* Return the log probability of a proper conditional autoregressive (CAR) prior
* with a sparse representation for the adjacency matrix
*
* @param phi Vector containing the parameters with a CAR prior
* @param tau Precision parameter for the CAR prior (real)
* @param alpha Dependence (usually spatial) parameter for the CAR prior (real)
* @param W_sparse Sparse representation of adjacency matrix (int array)
* @param n Length of phi (int)
* @param W_n Number of adjacent pairs (int)
* @param D_sparse Number of neighbors for each location (vector)
* @param lambda Eigenvalues of D^{-1/2}*W*D^{-1/2} (vector)
*
* @return Log probability density of CAR prior up to additive constant
*/
real sparse_car_lpdf(vector phi, real tau, real alpha,
int[,] W_sparse, vector D_sparse, vector lambda, int n, int W_n) {
row_vector[n] phit_D; // phi' * D
row_vector[n] phit_W; // phi' * W
vector[n] ldet_terms;

phit_D = (phi .* D_sparse)';
phit_W = rep_row_vector(0, n);
for (i in 1:W_n) {
phit_W[W_sparse[i, 1]] = phit_W[W_sparse[i, 1]] + phi[W_sparse[i, 2]];
phit_W[W_sparse[i, 2]] = phit_W[W_sparse[i, 2]] + phi[W_sparse[i, 1]];
}

for (i in 1:n) ldet_terms[i] = log1m(alpha * lambda[i]);
return 0.5 * (n * log(tau)
+ sum(ldet_terms)
- tau * (phit_D * phi - alpha * (phit_W * phi)));
}
}


with the data transformed in the model:

transformed data {
int W_sparse[W_n, 2];   // adjacency pairs
vector[n] D_sparse;     // diagonal of D (number of neigbors for each site)
vector[n] lambda;       // eigenvalues of invsqrtD * W * invsqrtD

{ // generate sparse representation for W
int counter;
counter = 1;
// loop over upper triangular part of W to identify neighbor pairs
for (i in 1:(n - 1)) {
for (j in (i + 1):n) {
if (W[i, j] == 1) {
W_sparse[counter, 1] = i;
W_sparse[counter, 2] = j;
counter = counter + 1;
}
}
}
}
for (i in 1:n) D_sparse[i] = sum(W[i]);
{
vector[n] invsqrtD;
for (i in 1:n) {
invsqrtD[i] = 1 / sqrt(D_sparse[i]);
}
}
}


and the likelihood:

model {
phi ~ sparse_car(tau, alpha, W_sparse, D_sparse, lambda, n, W_n);
}


This is quite a lot of code to digest, so my general approach is to compare the intermediate steps (whenever possible) with Stan. In this case, I will try to compute tau, alpha, W_sparse, D_sparse, lambda, n, W_n outside of the Stan model in R and compare with my own implementation.

Below is a Sparse CAR implementation in PyMC3 (see also here). Again, we try to avoid using any looping, as in STAN.

In [19]:

import scipy

class Sparse_CAR(distribution.Continuous):
"""
Sparse Conditional Autoregressive (CAR) distribution

Parameters
----------
alpha : spatial smoothing term
tau : precision at each location
"""

def __init__(self, alpha, W, tau, *args, **kwargs):
self.alpha = alpha = tt.as_tensor_variable(alpha)
self.tau = tau = tt.as_tensor_variable(tau)
D = W.sum(axis=0)
n, m = W.shape
self.n = n
self.median = self.mode = self.mean = 0
super(Sparse_CAR, self).__init__(*args, **kwargs)

# eigenvalues of D^−1/2 * W * D^−1/2
Dinv_sqrt = np.diag(1 / np.sqrt(D))
DWD = np.matmul(np.matmul(Dinv_sqrt, W), Dinv_sqrt)
self.lam = scipy.linalg.eigvalsh(DWD)

# sparse representation of W
w_sparse = scipy.sparse.csr_matrix(W)
self.W = theano.sparse.as_sparse_variable(w_sparse)
self.D = tt.as_tensor_variable(D)

# Presicion Matrix (inverse of Covariance matrix)
# d_sparse = scipy.sparse.csr_matrix(np.diag(D))
# self.D = theano.sparse.as_sparse_variable(d_sparse)
# self.Phi = self.tau * (self.D - self.alpha*self.W)

def logp(self, x):
logtau = self.n * tt.log(tau)
logdet = tt.log(1 - self.alpha * self.lam).sum()

# tau * ((phi .* D_sparse)' * phi - alpha * (phit_W * phi))
Wx = theano.sparse.dot(self.W, x)
tau_dot_x = self.D * x.T - self.alpha * Wx.ravel()
logquad = self.tau * tt.dot(x.ravel(), tau_dot_x.ravel())

# logquad = tt.dot(x.T, theano.sparse.dot(self.Phi, x)).sum()
return 0.5*(logtau + logdet - logquad)

In [20]:

with pm.Model() as model4:
# 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 = Sparse_CAR('phi', alpha, W, tau, 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)

trace4 = pm.sample(3e3, njobs=2, tune=1000)

Auto-assigning NUTS sampler...
Average Loss = 164.8:   8%|▊         | 16322/200000 [00:03<00:37, 4859.44it/s]
Convergence archived at 16700
Interrupted at 16,700 [8%]: Average Loss = 255.36
100%|██████████| 4000/4000.0 [00:16<00:00, 239.07it/s]

In [21]:

pm.traceplot(trace4, varnames=['alpha', 'beta', 'tau']);

In [22]:

pm.plot_posterior(trace4, varnames=['alpha']);


As you can see above, the sparse representation returns the same estimates, while being much faster than any other implementation.

## A few other warnings¶

In Stan, there is an option to write a generated quantities block for sample generation. Doing the similar in pymc3, however, is not recommended.

Consider the following simple sample:

# Data
x = np.array([1.1, 1.9, 2.3, 1.8])
n = len(x)

with pm.Model() as model1:
# prior
mu = pm.Normal('mu', mu=0, tau=.001)
sigma = pm.Uniform('sigma', lower=0, upper=10)
# observed
xi = pm.Normal('xi', mu=mu, tau=1/(sigma**2), observed=x)
# generation
p = pm.Deterministic('p', pm.math.sigmoid(mu))
count = pm.Binomial('count', n=10, p=p, shape=10)


where we intended to use python count = pm.Binomial('count', n=10, p=p, shape=10) to generate posterior prediction. However, if the new RV added to the model is a discrete variable it can cause weird turbulence to the trace. You can see issue #1990 for related discussion.

## Final remarks¶

In this notebook, most of the parameter conventions (e.g., using tau when defining a Normal distribution) and choice of priors are strictly matched with the original code in Winbugs or Stan. However, it is important to note that merely porting the code from one probabilistic programming language to the another is not necessarily the best practice. The aim is not just to run the code in PyMC3, but to make sure the model is appropriate so that it returns correct estimates, and runs efficiently (fast sampling).

For example, as [@aseyboldt](https://github.com/aseyboldt) pointed out here and here, non-centered parametrizations are often a better choice than the centered parametrizations. In our case here, phi is following a zero-mean Normal distribution, thus it can be left out in the beginning and used to scale the values afterwards. Often, doing this can avoid correlations in the posterior (it will be slower in some cases, however).

Another thing to keep in mind is that models can be sensitive to choices of prior distributions; for example, you can have a hard time using Normal variables with a large sd as prior. Gelman often recommends Cauchy or StudentT (i.e., weakly-informative priors). More information on prior choice can be found on the Stan wiki.

There are always ways to improve code. Since our computational graph with pm.Model() consist of theano objects, we can always do print(VAR_TO_CHECK.tag.test_value) right after the declaration or computation to check the shape. For example, in our last example, as suggested by [@aseyboldt](https://github.com/pymc-devs/pymc3/pull/2080#issuecomment-297456574) there seem to be a lot of correlation in the posterior. That probably slows down NUTS quite a bit. As a debugging tool and guide for reparametrization you can look at the singular value decomposition of the standardized samples from the trace – basically the eigenvalues of the correlation matrix. If the problem is high dimensional you can use stuff from scipy.sparse.linalg to only compute the largest singular value:

from scipy import linalg, sparse

vals = np.array([model.dict_to_array(v) for v in trace[1000:]]).T
vals[:] -= vals.mean(axis=1)[:, None]
vals[:] /= vals.std(axis=1)[:, None]

U, S, Vh = sparse.linalg.svds(vals, k=20)


Then look at plt.plot(S) to see if any principal components are obvious, and check which variables are contributing by looking at the singular vectors: plt.plot(U[:, -1] ** 2). You can get the indices by looking at model.bijection.ordering.vmap.

Another great way to check the correlations in the posterior is to do a pairplot of the posterior (if your model doesn’t contain too many parameters). You can see quite clearly if and where the the posterior parameters are correlated.

In [23]:

import seaborn as sns
tracedf1 = pm.trace_to_dataframe(trace1, varnames=['beta0', 'beta1', 'tau_h', 'tau_c'])
sns.pairplot(tracedf1);

In [24]:

tracedf2 = pm.trace_to_dataframe(trace2, varnames=['beta0', 'beta1', 'tau_h', 'tau_c'])
sns.pairplot(tracedf2);

In [25]:

tracedf3 = pm.trace_to_dataframe(trace3, varnames=['beta', 'tau', 'alpha'])
sns.pairplot(tracedf3);

In [26]:

tracedf4 = pm.trace_to_dataframe(trace4, varnames=['beta', 'tau', 'alpha'])
sns.pairplot(tracedf4);