def _multi_gamma_sequence(self, a, p, name='multi_gamma_sequence'):
"""Creates sequence used in multivariate (di)gamma; shape = shape(a)+[p]."""
with tf.name_scope(name):
# Linspace only takes scalars, so we'll add in the offset afterwards.
seq = ps.linspace(tf.constant(0., dtype=self.dtype), 0.5 - 0.5 * p,
ps.cast(p, tf.int32))
return seq + a[..., tf.newaxis]
def resample_systematic(log_probs,
event_size,
sample_shape,
seed=None,
name=None):
"""A systematic resampler for sequential Monte Carlo.
The value returned from this function is similar to sampling with
```python
expanded_sample_shape = tf.concat([[event_size], sample_shape]), axis=-1)
tfd.Categorical(logits=log_probs).sample(expanded_sample_shape)`
```
but with values sorted along the first axis. It can be considered to be
sampling events made up of a length-`event_size` vector of draws from
the `Categorical` distribution. However, although the elements of
this event have the appropriate marginal distribution, they are not
independent of each other. Instead they are drawn using a stratified
sampling method that in some sense reduces variance and is suitable for
use with Sequential Monte Carlo algorithms as described in
[Doucet et al. (2011)][2].
The sortedness is an unintended side effect of the algorithm that is
harmless in the context of simple SMC algorithms.
This implementation is based on the algorithms in [Maskell et al. (2006)][1]
where it is called minimum variance resampling.
Args:
log_probs: A tensor-valued batch of discrete log probability distributions.
event_size: the dimension of the vector considered a single draw.
sample_shape: the `sample_shape` determining the number of draws.
seed: PRNG seed; see `tfp.random.sanitize_seed` for details.
Default value: None (i.e. no seed).
name: Python `str` name for ops created by this method.
Default value: `None` (i.e., `'resample_systematic'`).
Returns:
resampled_indices: a tensor of samples.
#### References
[1]: S. Maskell, B. Alun-Jones and M. Macleod. A Single Instruction Multiple
Data Particle Filter.
In 2006 IEEE Nonlinear Statistical Signal Processing Workshop.
http://people.ds.cam.ac.uk/fanf2/hermes/doc/antiforgery/stats.pdf
[2]: A. Doucet & A. M. Johansen. Tutorial on Particle Filtering and
Smoothing: Fifteen Years Later
In 2011 The Oxford Handbook of Nonlinear Filtering
https://www.stats.ox.ac.uk/~doucet/doucet_johansen_tutorialPF2011.pdf
"""
with tf.name_scope(name or 'resample_systematic') as name:
log_probs = tf.convert_to_tensor(log_probs, dtype_hint=tf.float32)
log_probs = dist_util.move_dimension(log_probs,
source_idx=0,
dest_idx=-1)
working_shape = ps.concat(
[sample_shape, ps.shape(log_probs)[:-1]], axis=0)
points_shape = ps.concat([working_shape, [event_size]], axis=0)
# Draw a single offset for each event.
interval_width = ps.cast(1. / event_size, dtype=log_probs.dtype)
offsets = uniform.Uniform(low=ps.cast(0., dtype=log_probs.dtype),
high=interval_width).sample(
working_shape, seed=seed)[...,
tf.newaxis]
even_spacing = ps.linspace(start=ps.cast(0., dtype=log_probs.dtype),
stop=1 - interval_width,
num=event_size) + offsets
log_points = tf.broadcast_to(tf.math.log(even_spacing), points_shape)
resampled = _resample_using_log_points(log_probs, sample_shape,
log_points)
return dist_util.move_dimension(resampled, source_idx=-1, dest_idx=0)