2

I'm trying to understand how to implement the following model in Tensorflow probability.

  1. An angle, theta, has uniform prior probability in the range [-pi / 2, +pi / 2];
  2. A direction-flip probability, beta, has uniform prior probability in range [0, 1];
  3. theta' is set to either:
    • theta' = theta + pi with probability beta; or
    • theta' = theta with probability (1 - beta);
  4. Concentration c has a HalfCauchy prior probability; and
  5. An observation, alpha is drawn from the von Mises distribution, centred on theta' with concentration c.

So far what I've tried is

import tensorflow_probability as tfp
import numpy as np
tfd = tfp.distributions

model = tfd.JointDistributionSequential(
    [
        tfd.Uniform(-np.pi / 2, +np.pi / 2, name='theta'), # theta
        tfd.Uniform(0.0, 1.0, name='beta'), # beta
        tfd.HalfCauchy(loc=0, scale=1), # c
        lambda c, beta, theta: tfd.VonMises(
            loc=theta + np.pi * tfd.Binomial(probs=beta),
            concentration=c,
            name='observed'
        ), # Observation, alpha
    ]
)

Calling this gives an error on the binomial part: TypeError: __init__() missing 1 required positional argument: 'total_count'. What am I doing wrong?

Updated 2020-03-17

Latest code is as follows. I'm still trying to find out how to implement part (3) of my model, i.e. flip the direction of my angle, theta, by adding pi with probability beta. Any help on this would be appreciated! What I have so far doesn't work because I can't multiply the Bernoulli object by a float.

model = tfd.JointDistributionSequential(
    [
        tfd.Uniform(-np.pi / 2, +np.pi / 2, name='theta'), # theta
        tfd.Uniform(0.0, 1.0, name='beta'), # beta
        tfd.HalfCauchy(loc=0, scale=1), # c
        lambda c, beta, theta: tfd.VonMises(
            loc=theta + np.pi * tfd.Bernoulli(probs=beta, dtype=tf.float32),
            concentration=c,
            name='observed'
        ), # Observation, alpha
    ]
)
tobycoleman
  • 1,664
  • 1
  • 18
  • 34

2 Answers2

3

The problem of multiplying a number by a distribution in the loc parameter calculation can be fixed by sampling from the Bernoulli distribution i.e.

...
loc=theta + np.pi*tfd.Bernoulli(probs=beta, dtype=tf.float32).sample(),
...

This allows sampling from the joint distribution but I'm not sure that it's correct.

Another approach is to pull out the flip random variable and scale using a bijector i.e.

tpb = tfp.bijectors
model = tfd.JointDistributionSequential(
[
    tfd.Uniform(-np.pi / 2, +np.pi / 2, name='theta'), # theta
    tfd.Uniform(0.0, 1.0, name='beta'), # beta
    lambda beta, theta: tfb.Scale(np.pi)(tfd.Bernoulli(probs=beta, dtype=tf.float32)), #flip
    tfd.HalfCauchy(loc=0, scale=1), # c
    lambda c, flip, beta, theta: tfd.VonMises(
        loc=theta + flip ,
        concentration=c,
        name='observed'
    ), # Observation, alpha
]

)

This also allows sampling from the joint dictribution and has the advantage of being able to see when flips happen.

Matt McDonnell
  • 111
  • 2
  • I think the second approach is probably better, though now note you have a discrete value, which will make sampling hard. You could try RelaxedBernoulli to get a continuous 0-1 value with concentrations around 0 and 1. – Brian Patton Apr 29 '20 at 15:02
1

Swap out Bernoulli for Binomial. Binomial is the sum of a total_count number of bernoulli draws.

Brian Patton
  • 1,046
  • 5
  • 4
  • BTW if you are using `tfp-nightly` I'd suggest looking at some of the `JointDistribution*AutoBatched` classes. Docs here https://www.tensorflow.org/probability/api_docs/python/tfp/distributions?version=nightly. – Brian Patton Mar 10 '20 at 16:22
  • Thanks - understand I need to use Bernoulli instead of Binomial. How do I implement step (3) in my model (setting theta' to theta + pi depending on the Bernoulli draw). What I've done isn't right because it raises `unsupported operand type(s) for *: 'float' and 'Bernoulli'` – tobycoleman Mar 11 '20 at 15:23
  • Pass dtype=tf.float32 to the constructor. https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/Bernoulli – Brian Patton Mar 11 '20 at 19:23
  • Sorry I still don't quite understand. I'm now using `theta + np.pi * tfd.Bernoulli(probs=beta, dtype=tf.float32)` to set the `loc` of the VonMises, but still get the same error. – tobycoleman Mar 12 '20 at 13:37
  • [See Matt McDonnell's answer.] – Brian Patton Apr 29 '20 at 14:59