# Advanced usage of Theano in PyMC3¶

## Writing custom Theano Ops¶

While Theano includes a wide range of operations, there are cases where
it makes sense to write your own. But before doing this it is a good
idea to think hard if it is actually necessary. Especially if you want
to use algorithms that need gradient information — this includes NUTS and
all variational methods, and you probably *should* want to use those —
this is often quite a bit of work and also requires some math and
debugging skills for the gradients.

Good reasons for defining a custom Op might be the following:

- You require an operation that is not available in Theano and can’t be build up out of existing Theano operations. This could for example include models where you need to solve differential equations or integrals, or find a root or minimum of a function that depends on your parameters.
- You want to connect your PyMC3 model to some existing external code.
- After carefully considering different parametrizations and a lot of profiling your model is still too slow, but you know of a faster way to compute the gradient than what theano is doing. This faster way might be anything from clever maths to using more hardware. There is nothing stopping anyone from using a cluster via MPI in a custom node, if a part of the gradient computation is slow enough and sufficiently parallelizable to make the cost worth it. We would definitely like to hear about any such examples.

Theano has extensive documentation, about how to write new Ops.

### Finding the root of a function¶

We’ll use finding the root of a function as a simple example. Let’s say we want to define a model where a parameter is defined implicitly as the root of a function, that depends on another parameter:

First, we observe that this problem is well-defined, because \(R(\cdot, \theta)\) is monotone and has the image \((-1, \infty)\) for \(\mu, \theta \in \mathbb{R}^+\). To avoid overflows in \(\exp(\mu \theta)\) for large values of \(\mu\theta\) we instead find the root of

Also, we have

We can now use scipy.optimize.newton to find the root:

```
from scipy import optimize, special
import numpy as np
def func(mu, theta):
thetamu = theta * mu
value = np.log(mu) + np.logaddexp(0, thetamu)
return value
def jac(mu, theta):
thetamu = theta * mu
jac = theta * special.expit(thetamu) + 1 / mu
return jac
def mu_from_theta(theta):
return optimize.newton(func, 1, fprime=jac, args=(0.4,))
```

We could wrap mu_from_theta with tt.as_op and use gradient-free methods like Metropolis, but to get NUTS and ADVI working, we also need to define the derivative of mu_from_theta. We can find this derivative using the implicit function theorem, or equivalently we take the derivative with respect of \(\theta\) for both sides of \(R(\mu(\theta), \theta) = 0\) and solve for \(\frac{d\mu}{d\theta}\). This isn’t hard to do by hand, but for the fun of it, let’s do it using sympy:

```
import sympy
mu = sympy.Function('mu')
theta = sympy.Symbol('theta')
R = mu(theta) + mu(theta) * sympy.exp(theta * mu(theta)) - 1
solution = sympy.solve(R.diff(theta), mu(theta).diff(theta))[0]
```

We get

Now, we use this to define a theano op, that also computes the gradient:

```
import theano
import theano.tensor as tt
import theano.tests.unittest_tools
class MuFromTheta(tt.Op):
itypes = [tt.dscalar]
otypes = [tt.dscalar]
def perform(self, node, inputs, outputs):
theta, = inputs
mu = mu_from_theta(theta)
outputs[0][0] = np.array(mu)
def grad(self, inputs, g):
theta, = inputs
mu = self(theta)
thetamu = theta * mu
return [- g[0] * mu ** 2 / (1 + thetamu + tt.exp(-thetamu))]
```

If you value your sanity, always check that the gradient is ok:

```
theano.tests.unittest_tools.verify_grad(MuFromTheta(), [np.array(0.2)])
theano.tests.unittest_tools.verify_grad(MuFromTheta(), [np.array(1e-5)])
theano.tests.unittest_tools.verify_grad(MuFromTheta(), [np.array(1e5)])
```

We can now define our model using this new op:

```
import pymc3 as pm
tt_mu_from_theta = MuFromTheta()
with pm.Model() as model:
theta = pm.HalfNormal('theta', sd=1)
mu = pm.Deterministic('mu', tt_mu_from_theta(theta))
pm.Normal('y', mu=mu, sd=0.1, observed=[0.2, 0.21, 0.3])
trace = pm.sample()
```