Density of the ‘banana-shaped distribution’ evaluated analytically in the TensorFlow Distributions framework.

Building Probability Distributions with the TensorFlow Probability Bijector API

TensorFlow Distributions, now under the broader umbrella of TensorFlow Probability, is a fantastic TensorFlow library for efficient and composable manipulation of probability distributions1.

Among the many features it has to offer, one of the most powerful in my opinion is the Bijector API, which provide the modular building blocks necessary to construct a broad class of probability distributions. Instead of describing it any further in the abstract, let’s dive right in with a simple example.

Example: Banana-shaped distribution

Consider the banana-shaped distribution, a commonly-used testbed for adaptive MCMC methods2. Denote the density of this distribution as $p_{Y}(\mathbf{y})$. To illustrate, 1k samples randomly drawn from this distribution are shown below:

Banana distribution samples

The underlying process that generates samples $\tilde{\mathbf{y}} \sim p_{Y}(\mathbf{y})$ is simple to describe, and is of the general form,

$$ \tilde{\mathbf{y}} \sim p_{Y}(\mathbf{y}) \quad \Leftrightarrow \quad \tilde{\mathbf{y}} = G(\tilde{\mathbf{x}}), \quad \tilde{\mathbf{x}} \sim p_{X}(\mathbf{x}). $$

In other words, a sample $\tilde{\mathbf{y}}$ is the output of a transformation $G$, given a sample $\tilde{\mathbf{x}}$ drawn from some underlying base distribution $p_{X}(\mathbf{x})$.

However, it is not as straightforward to compute an analytical expression for density $p_{Y}(\mathbf{y})$. In fact, this is only possible if $G$ is a differentiable and invertible transformation (a diffeomorphism3), and if there is an analytical expression for $p_{X}(\mathbf{x})$.

Transformations that fail to satisfy these conditions (which includes something as simple as a multi-layer perceptron with non-linear activations) give rise to implicit distributions, and will be the subject of many posts to come. But for now, we will restrict our attention to diffeomorphisms.

Base distribution

Following on with our example, the base distribution $p_{X}(\mathbf{x})$ is given by a two-dimensional Gaussian with unit variances and covariance $\rho = 0.95$:

$$ p_{X}(\mathbf{x}) = \mathcal{N}(\mathbf{x} | \mathbf{0}, \mathbf{\Sigma}), \qquad \mathbf{\Sigma} = \begin{bmatrix} 1 & 0.95 \newline 0.95 & 1 \end{bmatrix} $$

This can be encapsulated by an instance of MultivariateNormalTriL, which is parameterized by a lower-triangular matrix. First let’s import TensorFlow Distributions:

import tensorflow.contrib.distributions as tfd

Then we create the lower-triangular matrix and the instantiate the distribution:

>>> rho = 0.95
>>> Sigma = np.float32(np.eye(N=2) + rho * np.eye(N=2)[::-1])
>>> Sigma
array([[1.  , 0.95],
       [0.95, 1.  ]], dtype=float32)
>>> p_x = tfd.MultivariateNormalTriL(scale_tril=tf.cholesky(Sigma))

As with all subclasses of tfd.Distribution, we can evaluated the probability density function of this distribution by calling the p_x.prob method. Evaluating this on an uniformly-spaced grid yields the equiprobability contour plot below:

Base density

Forward Transformation

The required transformation $G$ is defined as:

$$ G(\mathbf{x}) = \begin{bmatrix} x_1 \newline x_2 - x_1^2 - 1 \newline \end{bmatrix} $$

We implement this in the _forward function below4:

def _forward(x):

    y_0 = x[..., 0:1]
    y_1 = x[..., 1:2] - y_0**2 - 1
    y_tail = x[..., 2:-1]

    return tf.concat([y_0, y_1, y_tail], axis=-1)

We can now use this to generate samples from $p_{Y}(\mathbf{y})$. To do this we first sample from the base distribution $p_{X}(\mathbf{x})$ by calling p_x.sample. For this illustration, we generate 1k samples, which is specified through the sample_shape argument. We then transform these samples through $G$ by calling _forward on them.

>>> x_samples = p_x.sample(1000)
>>> y_samples = _forward(x_samples)

The figure below contains scatterplots of the 1k samples x_samples (left) and the transformed y_samples (right):

Banana and base samples

Instantiating a TransformedDistribution with a Bijector

Having specified the forward transformation and the underlying distribution, we have now fully described the sample generation process, which is the bare minimum necessary to define a probability distribution.

The forward transformation is also the first of three operations needed to fully specify a Bijector, which can be used to instantiate a TransformedDistribution that encapsulates the banana-shaped distribution.

Creating a Bijector

First, let’s subclass Bijector to define the Banana bijector and implement the forward transformation as an instance method:

class Banana(tfd.bijectors.Bijector):

    def __init__(self, name="banana"):
        super(Banana, self).__init__(inverse_min_event_ndims=1,
                                     name=name)

    def _forward(self, x):

        y_0 = x[..., 0:1]
        y_1 = x[..., 1:2] - y_0**2 - 1
        y_tail = x[..., 2:-1]

        return tf.concat([y_0, y_1, y_tail], axis=-1)

Note that we need to specify either forward_min_event_ndims or inverse_min_event_ndims, the number of dimensions the forward or inverse transformation operate on (which can sometimes differ). In our example, both the inverse and forward transformation operate on vectors (rank 1 tensors), so we set inverse_min_event_ndims=1.

With an instance of the Banana bijector, we can call the forward method on x_samples to produce y_samples as before:

>>> y_samples = Banana().forward(x_samples)

Instantiating a TransformedDistribution

More importantly, we can now create a TransformedDistribution with the base distribution p_x and an instance of the Banana bijector:

>>> p_y = tfd.TransformedDistribution(distribution=p_x, bijector=Banana())

This now allows us to directly sample from p_y just as we could with p_x, and any other TensorFlow Probability Distribution:

>>> y_samples = p_y.sample(1000)

Neat!

Probability Density Function

Although we can now sample from this distribution, we have yet to define the operations necessary to evaluate its probability density function—the remaining two of three operations needed to fully specify a Bijector

Indeed, calling p_y.prob at this stage would simply raise a NotImplementedError exception. So what else do we need to define?

Recall the probability density of $p_{Y}(\mathbf{y})$ is given by:

$$ p_{Y}(\mathbf{y}) = p_{X}(G^{-1}(\mathbf{y})) \mathrm{det} \left ( \frac{\partial}{\partial\mathbf{y}} G^{-1}(\mathbf{y}) \right ) $$

Hence we need to specify the inverse transformation $G^{-1}(\mathbf{y})$ and its Jacobian determinant $\mathrm{det} \left ( \frac{\partial}{\partial\mathbf{y}} G^{-1}(\mathbf{y}) \right )$.

For numerical stability, the Bijector API requires that this be defined in log-space. Hence, it is useful to recall that the forward and inverse log determinant Jacobians differ only in their signs5,

$$ \begin{align} \log \mathrm{det} \left ( \frac{\partial}{\partial\mathbf{y}} G^{-1}(\mathbf{y}) \right ) & = - \log \mathrm{det} \left ( \frac{\partial}{\partial\mathbf{x}} G(\mathbf{x}) \right ), \end{align} $$

which gives us the option of implementing either (or both). However, do note the following from the official tf.contrib.distributions.bijectors.Bijector API docs:

Generally its preferable to directly implement the inverse Jacobian determinant. This should have superior numerical stability and will often share subgraphs with the _inverse implementation.

Inverse Transformation

So let’s implement the inverse transform $G^{-1}$, which is given by:

$$ G^{-1}(\mathbf{y}) = \begin{bmatrix} y_1 \newline y_2 + y_1^2 + 1 \newline \end{bmatrix} $$

We define this in the _inverse function below:

def _inverse(y):

    x_0 = y[..., 0:1]
    x_1 = y[..., 1:2] + x_0**2 + 1
    x_tail = y[..., 2:-1]

    return tf.concat([x_0, x_1, x_tail], axis=-1)

Jacobian determinant

Now we compute the log determinant of the Jacobian of the inverse transformation. In this simple example, the transformation is volume-preserving, meaning its Jacobian determinant is equal to 1.

This is easy to verify:

$$ \begin{align} \mathrm{det} \left ( \frac{\partial}{\partial\mathbf{y}} G^{-1}(\mathbf{y}) \right ) & = \mathrm{det} \begin{pmatrix} \frac{\partial}{\partial y_1} y_1 & \frac{\partial}{\partial y_2} y_1 \newline \frac{\partial}{\partial y_1} y_2 + y_1^2 + 1 & \frac{\partial}{\partial y_2} y_2 + y_1^2 + 1 \newline \end{pmatrix} \newline & = \mathrm{det} \begin{pmatrix} 1 & 0 \newline 2 y_1 & 1 \newline \end{pmatrix} = 1 \end{align} $$

Hence, the log determinant Jacobian is given by zeros shaped like input y, up to the last inverse_min_event_ndims=1 dimensions:

def _inverse_log_det_jacobian(y):

    return tf.zeros(shape=y.shape[:-1])

Since the log determinant Jacobian is constant, i.e. independent of the input, we can just specify it for one input by setting the flag is_constant_jacobian=True6, and the Bijector class will handle the necessary shape inference for us.

Putting it all together in the Banana bijector subclass, we have:

class Banana(tfd.bijectors.Bijector):

    def __init__(self, name="banana"):
        super(Banana, self).__init__(inverse_min_event_ndims=1,
                                     is_constant_jacobian=True,
                                     name=name)

    def _forward(self, x):

        y_0 = x[..., 0:1]
        y_1 = x[..., 1:2] - y_0**2 - 1
        y_tail = x[..., 2:-1]

        return tf.concat([y_0, y_1, y_tail], axis=-1)

    def _inverse(self, y):

        x_0 = y[..., 0:1]
        x_1 = y[..., 1:2] + x_0**2 + 1
        x_tail = y[..., 2:-1]

        return tf.concat([x_0, x_1, x_tail], axis=-1)

    def _inverse_log_det_jacobian(self, y):

        return tf.zeros(shape=())

Finally, we can instantiate distribution p_y by calling tfd.TransformedDistribution as we did before et voilĂ , we can now simply call p_y.prob to evaluate the probability density function.

Evaluating this on the same uniformly-spaced grid as before yields the following equiprobability contour plot:

Banana density

Inline Bijector

Before we conclude, we note that instead of creating a subclass, one can also opt for a more lightweight and functional approach by creating an Inline bijector:

banana = tfd.bijectors.Inline(
    forward_fn=_forward, 
    inverse_fn=_inverse,
    inverse_log_det_jacobian_fn=_inverse_log_det_jacobian,
    inverse_min_event_ndims=1,
    is_constant_jacobian=True,
)
p_y = tfd.TransformedDistribution(distribution=p_x, bijector=banana)

Summary

In this post, we showed that using diffeomorphisms—mappings that are differentiable and invertible, it is possible transform standard distributions into interesting and complicated distributions, while still being able to compute their densities analytically.

The Bijector API provides an interface that encapsulates the basic properties of a diffeomorphism needed to transform a distribution. These are: the forward transform itself, its inverse and the determinant of their Jacobians.

Using this, TransformedDistribution automatically implements perhaps the two most important methods of a probability distribution: sampling (sample), and density evaluation (prob).

Needless to say, this is a very powerful combination. Through the Bijector API, the number of possible distributions that can be implemented and used directly with other functionalities in the TensorFlow Probability ecosystem effectively becomes endless.


  1. Dillon, J.V., Langmore, I., Tran, D., Brevdo, E., Vasudevan, S., Moore, D., Patton, B., Alemi, A., Hoffman, M. and Saurous, R.A., 2017. TensorFlow Distributions. arXiv preprint arXiv:1711.10604. ^
  2. Haario, H., Saksman, E., & Tamminen, J. (1999). Adaptive proposal distribution for random walk Metropolis algorithm. Computational Statistics, 14(3), 375-396. ^
  3. for the transformation to be a diffeomorphism, it also needs to be smooth. ^
  4. we implement this for the general case of $K \geq 2$ dimensional inputs since this actually turns out to be easier and cleaner (a phenomenon known as Inventor’s paradox). ^
  5. this is a straightforward consequence of the inverse function theorem which says the matrix inverse of the Jacobian of $G$ is the Jacobian of its inverse $G^{-1}$, $$ \frac{\partial}{\partial\mathbf{y}} G^{-1}(\mathbf{y}) = \left ( \frac{\partial}{\partial\mathbf{x}} G(\mathbf{x}) \right )^{-1} $$ Taking the determinant of both sides, we get: $$ \begin{align} \mathrm{det} \left ( \frac{\partial}{\partial\mathbf{y}} G^{-1}(\mathbf{y}) \right ) & = \mathrm{det} \left ( \left ( \frac{\partial}{\partial\mathbf{x}} G(\mathbf{x}) \right )^{-1} \right ) \newline & = \mathrm{det} \left ( \frac{\partial}{\partial\mathbf{x}} G(\mathbf{x}) \right )^{-1} \end{align} $$ as required.

    ^
  6. See description of is_constant_jacobian argument for further details. ^

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