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# import some dependencies
import torch

import pyro
import pyro.distributions as dist


# Models in Pyro: From Primitive Distributions to Stochastic Functions¶

The basic unit of Pyro programs is the stochastic function. This is an arbitrary Python callable that combines two ingredients:

• deterministic Python code; and
• primitive stochastic functions

Concretely, a stochastic function can be any Python object with a __call__() method, like a function, a method, or a PyTorch nn.Module.

Throughout the tutorials and documentation, we will often call stochastic functions models, since stochastic functions can be used to represent simplified or abstract descriptions of a process by which data are generated. Expressing models as stochastic functions in Pyro means that models can be composed, reused, imported, and serialized just like regular Python callables.

Without further ado, let’s introduce one of our basic building blocks: primitive stochastic functions.

## Primitive Stochastic Functions¶

Primitive stochastic functions, or distributions, are an important class of stochastic functions for which we can explicitly compute the probability of the outputs given the inputs. As of PyTorch 0.4 and Pyro 0.2, Pyro uses PyTorch’s distribution library. You can also create custom distributions using transforms.

Using primitive stochastic functions is easy. For example, to draw a sample x from the unit normal distribution $$\mathcal{N}(0,1)$$ we do the following:

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loc = 0.   # mean zero
scale = 1. # unit variance
normal = dist.Normal(loc, scale) # create a normal distribution object
x = normal.sample() # draw a sample from N(0,1)
print("sample", x)
print("log prob", normal.log_prob(x)) # score the sample from N(0,1)


Here, dist.Normal is a callable instance of the Distribution class that takes parameters and provides sample and score methods. Note that the parameters passed to dist.Normal are torch.Tensors. This is necessary because we want to make use of PyTorch’s fast tensor math and autograd capabilities during inference.

## The pyro.sample Primitive¶

One of the core language primitives in Pyro is the pyro.sample statement. Using pyro.sample is as simple as calling a primitive stochastic function with one important difference:

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x = pyro.sample("my_sample", dist.Normal(loc, scale))
print(x)


Just like a direct call to dist.Normal().sample(), this returns a sample from the unit normal distribution. The crucial difference is that this sample is named. Pyro’s backend uses these names to uniquely identify sample statements and change their behavior at runtime depending on how the enclosing stochastic function is being used. As we will see, this is how Pyro can implement the various manipulations that underlie inference algorithms.

## A Simple Model¶

Now that we’ve introduced pyro.sample and pyro.distributions we can write a simple model. Since we’re ultimately interested in probabilistic programming because we want to model things in the real world, let’s choose something concrete.

Let’s suppose we have a bunch of data with daily mean temperatures and cloud cover. We want to reason about how temperature interacts with whether it was sunny or cloudy. A simple stochastic function that does that is given by:

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def weather():
cloudy = pyro.sample('cloudy', dist.Bernoulli(0.3))
cloudy = 'cloudy' if cloudy.item() == 1.0 else 'sunny'
mean_temp = {'cloudy': 55.0, 'sunny': 75.0}[cloudy]
scale_temp = {'cloudy': 10.0, 'sunny': 15.0}[cloudy]
temp = pyro.sample('temp', dist.Normal(mean_temp, scale_temp))
return cloudy, temp.item()

for _ in range(3):
print(weather())


Let’s go through this line-by-line. First, in lines 2-3 we use pyro.sample to define a binary random variable ‘cloudy’, which is given by a draw from the bernoulli distribution with a parameter of 0.3. Since the bernoulli distributions returns 0s or 1s, in line 4 we convert the value cloudy to a string so that return values of weather are easier to parse. So according to this model 30% of the time it’s cloudy and 70% of the time it’s sunny.

In lines 5-6 we define the parameters we’re going to use to sample the temperature in lines 7-9. These parameters depend on the particular value of cloudy we sampled in line 2. For example, the mean temperature is 55 degrees (Fahrenheit) on cloudy days and 75 degrees on sunny days. Finally we return the two values cloudy and temp in line 10.

Procedurally, weather() is a non-deterministic Python callable that returns two random samples. Because the randomness is invoked with pyro.sample, however, it is much more than that. In particular weather() specifies a joint probability distribution over two named random variables: cloudy and temp. As such, it defines a probabilistic model that we can reason about using the techniques of probability theory. For example we might ask: if I observe a temperature of 70 degrees, how likely is it to be cloudy? How to formulate and answer these kinds of questions will be the subject of the next tutorial.

We’ve now seen how to define a simple model. Building off of it is easy. For example:

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def ice_cream_sales():
cloudy, temp = weather()
expected_sales = 200. if cloudy == 'sunny' and temp > 80.0 else 50.
ice_cream = pyro.sample('ice_cream', dist.Normal(expected_sales, 10.0))
return ice_cream


This kind of modularity, familiar to any programmer, is obviously very powerful. But is it powerful enough to encompass all the different kinds of models we’d like to express?

## Universality: Stochastic Recursion, Higher-order Stochastic Functions, and Random Control Flow¶

Because Pyro is embedded in Python, stochastic functions can contain arbitrarily complex deterministic Python and randomness can freely affect control flow. For example, we can construct recursive functions that terminate their recursion nondeterministically, provided we take care to pass pyro.sample unique sample names whenever it’s called. For example we can define a geometric distribution like so:

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def geometric(p, t=None):
if t is None:
t = 0
x = pyro.sample("x_{}".format(t), dist.Bernoulli(p))
if x.item() == 0:
return x
else:
return x + geometric(p, t + 1)

print(geometric(0.5))


Note that the names x_0, x_1, etc., in geometric() are generated dynamically and that different executions can have different numbers of named random variables.

We are also free to define stochastic functions that accept as input or produce as output other stochastic functions:

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def normal_product(loc, scale):
z1 = pyro.sample("z1", dist.Normal(loc, scale))
z2 = pyro.sample("z2", dist.Normal(loc, scale))
y = z1 * z2
return y

def make_normal_normal():
mu_latent = pyro.sample("mu_latent", dist.Normal(0, 1))
fn = lambda scale: normal_product(mu_latent, scale)
return fn

print(make_normal_normal()(1.))


Here make_normal_normal() is a stochastic function that takes one argument and which, upon execution, generates three named random variables.

The fact that Pyro supports arbitrary Python code like this—iteration, recursion, higher-order functions, etc.—in conjuction with random control flow means that Pyro stochastic functions are universal, i.e. they can be used to represent any computable probability distribution. As we will see in subsequent tutorials, this is incredibly powerful.

It is worth emphasizing that this is one reason why Pyro is built on top of PyTorch: dynamic computational graphs are an important ingredient in allowing for universal models that can benefit from GPU-accelerated tensor math.

## Next Steps¶

We’ve shown how we can use stochastic functions and primitive distributions to represent models in Pyro. In order to learn models from data and reason about them we need to be able to do inference. This is the subject of the next tutorial.