PyMC3 and Theano

What is Theano

Theano is a package that allows us to define functions involving array operations and linear algebra. When we define a PyMC3 model, we implicitly build up a Theano function from the space of our parameters to their posterior probability density up to a constant factor. We then use symbolic manipulations of this function to also get access to its gradient.

For a thorough introduction to Theano see the theano docs, but for the most part you don’t need detailed knowledge about it as long as you are not trying to define new distributions or other extensions of PyMC3. But let’s look at a simple example to get a rough idea about how it works. Say, we’d like to define the (completely arbitrarily chosen) function

\[\begin{split}f\colon \mathbb{R} \times \mathbb{R}^n \times \mathbb{N}^n \to \mathbb{R}\\ (a, x, y) \mapsto \sum_{i=0}^{n} \exp(ax_i^3 + y_i^2).\end{split}\]

First, we need to define symbolic variables for our inputs (this is similar to eg SymPy’s Symbol):

import theano
import theano.tensor as tt
# We don't specify the dtype of our input variables, so it
# defaults to using float64 without any special config.
a = tt.scalar('a')
x = tt.vector('x')
# `tt.ivector` creates a symbolic vector of integers.
y = tt.ivector('y')

Next, we use those variables to build up a symbolic representation of the output of our function. Note, that no computation is actually being done at this point. We only record what operations we need to do to compute the output:

inner = a * x**3 + y**2
out = tt.exp(inner).sum()

Note

In this example we use tt.exp to create a symbolic representation of the exponential of inner. Somewhat surprisingly, it would also have worked if we used np.exp. This is because numpy gives objects it operates on a chance to define the results of operations themselves. Theano variables do this for a large number of operations. We usually still prefer the theano functions instead of the numpy versions, as that makes it clear that we are working with symbolic input instead of plain arrays.

Now we can tell Theano to build a function that does this computation. With a typical configuration Theano generates C code, compiles it, and creates a python function which wraps the C function:

func = theano.function([a, x, y], [out])

We can call this function with actual arrays as many times as we want:

a_val = 1.2
x_vals = np.random.randn(10)
y_vals = np.random.randn(10)

out = func(a_val, x_vals, y_vals)

For the most part the symbolic Theano variables can be operated on like NumPy arrays. Most NumPy functions are available in theano.tensor (which is typically imported as tt). A lot of linear algebra operations can be found in tt.nlinalg and tt.slinalg (the NumPy and SciPy operations respectively). Some support for sparse matrices is available in theano.sparse. For a detailed overview of available operations, see the theano api docs.

A notable exception where theano variables do not behave like NumPy arrays are operations involving conditional execution:

Code like this won’t work as expected:

a = tt.vector('a')
if (a > 0).all():
    b = tt.sqrt(a)
else:
    b = -a

(a > 0).all() isn’t actually a boolean as it would be in NumPy, but still a symbolic variable. Python will convert this object to a boolean and according to the rules for this conversion, things that aren’t empty containers or zero are converted to True. So the code is equivalent to this:

a = tt.vector('a')
b = tt.sqrt(a)

To get the desired behaviour, we can use tt.switch:

a = tt.vector('a')
b = tt.switch((a > 0).all(), tt.sqrt(a), -a)

Indexing also works similarly to NumPy:

a = tt.vector('a')
# Access the 10th element. This will fail when a function build
# from this expression is executed with an array that is too short.
b = a[10]

# Extract a subvector
b = a[[1, 2, 10]]

Changing elements of an array is possible using tt.set_subtensor:

a = tt.vector('a')
b = tt.set_subtensor(a[:10], 1)

# is roughly equivalent to this (although theano avoids
# the copy if `a` isn't used anymore)
a = np.random.randn(10)
b = a.copy()
b[:10] = 1

How PyMC3 uses Theano

Now that we have a basic understanding of Theano we can look at what happens if we define a PyMC3 model. Let’s look at a simple example:

true_mu = 0.1
data = true_mu + np.random.randn(50)

with pm.Model() as model:
    mu = pm.Normal('mu', mu=0, sd=1)
    y = pm.Normal('y', mu=mu, sd=1, observed=data)

In this model we define two variables: mu and y. The first is a free variable that we want to infer, the second is an observed variable. To sample from the posterior we need to build the function

\[\begin{split}\log P(μ|y) + C = \log P(y|μ) + \log P(μ) =: \text{logp}(μ)\\\end{split}\]

where with the normal likelihood \(N(x|μ,σ^2)\)

\[\begin{split}\text{logp}\colon \mathbb{R} \to \mathbb{R}\\ μ \mapsto \log N(μ|0, 1) + \log N(y|μ, 1),\end{split}\]

To build that function we need to keep track of two things: The parameter space (the free variables) and the logp function. For each free variable we generate a Theano variable. And for each variable (observed or otherwise) we add a term to the global logp. In the background something similar to this is happening:

# For illustration only, those functions don't actually exist
# in exactly this way!
model = pm.Model()

mu = tt.scalar('mu')
model.add_free_variable(mu)
model.add_logp_term(pm.Normal.dist(0, 1).logp(mu))

model.add_logp_term(pm.Normal.dist(mu, 1).logp(data))

So calling pm.Normal() modifies the model: It changes the logp function of the model. If the observed keyword isn’t set it also creates a new free variable. In contrast, pm.Normal.dist() doesn’t care about the model, it just creates an object that represents the normal distribution. Calling logp on this object creates a theano variable for the logp probability or log probability density of the distribution, but again without changing the model in any way.

Continuous variables with support only on a subset of the real numbers are treated a bit differently. We create a transformed variable that has support on the reals and then modify this variable. For example:

with pm.Model() as model:
    mu = pm.Normal('mu', 0, 1)
    sd = pm.HalfNormal('sd', 1)
    y = pm.Normal('y', mu=mu, sd=sd, observed=data)

is roughly equivalent to this:

# For illustration only, not real code!
model = pm.Model()
mu = tt.scalar('mu')
model.add_free_variable(mu)
model.add_logp_term(pm.Normal.dist(0, 1).logp(mu))

sd_log__ = tt.scalar('sd_log__')
model.add_free_variable(sd_log__)
model.add_logp_term(corrected_logp_half_normal(sd_log__))

sd = tt.exp(sd_log__)
model.add_deterministic_variable(sd)

model.add_logp_term(pm.Normal.dist(mu, sd).logp(data))

The return values of the variable constructors are subclasses of theano variables, so when we define a variable we can use any theano operation on them:

design_matrix = np.array([[...]])
with pm.Model() as model:
    # beta is a tt.dvector
    beta = pm.Normal('beta', 0, 1, shape=len(design_matrix))
    predict = tt.dot(design_matrix, beta)
    sd = pm.HalfCauchy('sd', beta=2.5)
    pm.Normal('y', mu=predict, sd=sd, observed=data)