Tensor shapes in Pyro¶
You may also find it useful to read Eric J. Ma’s post Reasoning about Shapes and Probability Distributions. While this post is specifically about TensorFlow Probability, many of the same concepts apply.
Summary:¶
Tensors broadcast by aligning on the right:
torch.ones(3,4,5) + torch.ones(5).Distribution
.sample().shape == batch_shape + event_shape.Distribution
.log_prob(x).shape == batch_shape(but notevent_shape!).Use
.expand()to draw a batch of samples, or rely onplateto expand automatically.Use
my_dist.to_event(1)to declare a dimension as dependent.Use
with pyro.plate('name', size):to declare a dimension as conditionally independent.All dimensions must be declared either dependent or conditionally independent.
Try to support batching on the left. This lets Pyro auto-parallelize.
use negative indices like
x.sum(-1)rather thanx.sum(2)use ellipsis notation like
pixel = image[..., i, j]use Vindex if
i,jare enumerated,pixel = Vindex(image)[..., i, j]
When using
pyro.plate’s automatic subsampling, be sure to subsample your data:Either manually subample by capturing the index
with pyro.plate(...) as i: ...or automatically subsample via
batch = pyro.subsample(data, event_dim=...).
When debugging, examine all shapes in a trace using Trace.format_shapes().
Table of Contents¶
[1]:
import os
import torch
import pyro
from torch.distributions import constraints
from pyro.distributions import Bernoulli, Categorical, MultivariateNormal, Normal
from pyro.distributions.util import broadcast_shape
from pyro.infer import Trace_ELBO, TraceEnum_ELBO, config_enumerate
import pyro.poutine as poutine
from pyro.optim import Adam
smoke_test = ('CI' in os.environ)
assert pyro.__version__.startswith('1.9.0')
# We'll ue this helper to check our models are correct.
def test_model(model, guide, loss):
pyro.clear_param_store()
loss.loss(model, guide)
Distributions shapes: batch_shape and event_shape¶
PyTorch Tensors have a single .shape attribute, but Distributions have two shape attributions with special meaning: .batch_shape and .event_shape. These two combine to define the total shape of a sample
x = d.sample()
assert x.shape == d.batch_shape + d.event_shape
Indices over .batch_shape denote conditionally independent random variables, whereas indices over .event_shape denote dependent random variables (ie one draw from a distribution). Because the dependent random variables define probability together, the .log_prob() method only produces a single number for each event of shape .event_shape. Thus the total shape of .log_prob() is .batch_shape:
assert d.log_prob(x).shape == d.batch_shape
Note that the Distribution.sample() method also takes a sample_shape parameter that indexes over independent identically distributed (iid) random varables, so that
x2 = d.sample(sample_shape)
assert x2.shape == sample_shape + batch_shape + event_shape
In summary
| iid | independent | dependent
------+--------------+-------------+------------
shape = sample_shape + batch_shape + event_shape
For example univariate distributions have empty event shape (because each number is an independent event). Distributions over vectors like MultivariateNormal have len(event_shape) == 1. Distributions over matrices like InverseWishart have len(event_shape) == 2.
Examples¶
The simplest distribution shape is a single univariate distribution.
[2]:
d = Bernoulli(0.5)
assert d.batch_shape == ()
assert d.event_shape == ()
x = d.sample()
assert x.shape == ()
assert d.log_prob(x).shape == ()
Distributions can be batched by passing in batched parameters.
[3]:
d = Bernoulli(0.5 * torch.ones(3,4))
assert d.batch_shape == (3, 4)
assert d.event_shape == ()
x = d.sample()
assert x.shape == (3, 4)
assert d.log_prob(x).shape == (3, 4)
Another way to batch distributions is via the .expand() method. This only works if parameters are identical along the leftmost dimensions.
[4]:
d = Bernoulli(torch.tensor([0.1, 0.2, 0.3, 0.4])).expand([3, 4])
assert d.batch_shape == (3, 4)
assert d.event_shape == ()
x = d.sample()
assert x.shape == (3, 4)
assert d.log_prob(x).shape == (3, 4)
Multivariate distributions have nonempty .event_shape. For these distributions, the shapes of .sample() and .log_prob(x) differ:
[5]:
d = MultivariateNormal(torch.zeros(3), torch.eye(3, 3))
assert d.batch_shape == ()
assert d.event_shape == (3,)
x = d.sample()
assert x.shape == (3,) # == batch_shape + event_shape
assert d.log_prob(x).shape == () # == batch_shape
Reshaping distributions¶
In Pyro you can treat a univariate distribution as multivariate by calling the .to_event(n) property where n is the number of batch dimensions (from the right) to declare as dependent.
[6]:
d = Bernoulli(0.5 * torch.ones(3,4)).to_event(1)
assert d.batch_shape == (3,)
assert d.event_shape == (4,)
x = d.sample()
assert x.shape == (3, 4)
assert d.log_prob(x).shape == (3,)
While you work with Pyro programs, keep in mind that samples have shape batch_shape + event_shape, whereas .log_prob(x) values have shape batch_shape. You’ll need to ensure that batch_shape is carefully controlled by either trimming it down with .to_event(n) or by declaring dimensions as independent via pyro.plate.
It is always safe to assume dependence¶
Often in Pyro we’ll declare some dimensions as dependent even though they are in fact independent, e.g.
x = pyro.sample("x", Normal(0, 1).expand([10]).to_event(1))
assert x.shape == (10,)
This is useful for two reasons: First it allows us to easily swap in a MultivariateNormal distribution later. Second it simplifies the code a bit since we don’t need a plate (see below) as in
with pyro.plate("x_plate", 10):
x = pyro.sample("x", Normal(0, 1)) # .expand([10]) is automatic
assert x.shape == (10,)
The difference between these two versions is that the second version with plate informs Pyro that it can make use of conditional independence information when estimating gradients, whereas in the first version Pyro must assume they are dependent (even though the normals are in fact conditionally independent). This is analogous to d-separation in graphical models: it is always safe to add edges and assume variables may be dependent (i.e. to widen the model class), but it is unsafe to assume
independence when variables are actually dependent (i.e. narrowing the model class so the true model lies outside of the class, as in mean field). In practice Pyro’s SVI inference algorithm uses reparameterized gradient estimators for Normal distributions so both gradient estimators have the same performance.
Declaring independent dims with plate¶
Pyro models can use the context manager pyro.plate to declare that certain batch dimensions are independent. Inference algorithms can then take advantage of this independence to e.g. construct lower variance gradient estimators or to enumerate in linear space rather than exponential space. An example of an independent dimension is the index over data in a minibatch: each datum should be independent of all others.
The simplest way to declare a dimension as independent is to declare the rightmost batch dimension as independent via a simple
with pyro.plate("my_plate"):
# within this context, batch dimension -1 is independent
We recommend always providing an optional size argument to aid in debugging shapes
with pyro.plate("my_plate", len(my_data)):
# within this context, batch dimension -1 is independent
Starting with Pyro 0.2 you can additionally nest plates, e.g. if you have per-pixel independence:
with pyro.plate("x_axis", 320):
# within this context, batch dimension -1 is independent
with pyro.plate("y_axis", 200):
# within this context, batch dimensions -2 and -1 are independent
Note that we always count from the right by using negative indices like -2, -1.
Finally if you want to mix and match plates for e.g. noise that depends only on x, some noise that depends only on y, and some noise that depends on both, you can declare multiple plates and use them as reusable context managers. In this case Pyro cannot automatically allocate a dimension, so you need to provide a dim argument (again counting from the right):
x_axis = pyro.plate("x_axis", 3, dim=-2)
y_axis = pyro.plate("y_axis", 2, dim=-3)
with x_axis:
# within this context, batch dimension -2 is independent
with y_axis:
# within this context, batch dimension -3 is independent
with x_axis, y_axis:
# within this context, batch dimensions -3 and -2 are independent
Let’s take a closer look at batch sizes within plates.
[7]:
def model1():
a = pyro.sample("a", Normal(0, 1))
b = pyro.sample("b", Normal(torch.zeros(2), 1).to_event(1))
with pyro.plate("c_plate", 2):
c = pyro.sample("c", Normal(torch.zeros(2), 1))
with pyro.plate("d_plate", 3):
d = pyro.sample("d", Normal(torch.zeros(3,4,5), 1).to_event(2))
assert a.shape == () # batch_shape == () event_shape == ()
assert b.shape == (2,) # batch_shape == () event_shape == (2,)
assert c.shape == (2,) # batch_shape == (2,) event_shape == ()
assert d.shape == (3,4,5) # batch_shape == (3,) event_shape == (4,5)
x_axis = pyro.plate("x_axis", 3, dim=-2)
y_axis = pyro.plate("y_axis", 2, dim=-3)
with x_axis:
x = pyro.sample("x", Normal(0, 1))
with y_axis:
y = pyro.sample("y", Normal(0, 1))
with x_axis, y_axis:
xy = pyro.sample("xy", Normal(0, 1))
z = pyro.sample("z", Normal(0, 1).expand([5]).to_event(1))
assert x.shape == (3, 1) # batch_shape == (3,1) event_shape == ()
assert y.shape == (2, 1, 1) # batch_shape == (2,1,1) event_shape == ()
assert xy.shape == (2, 3, 1) # batch_shape == (2,3,1) event_shape == ()
assert z.shape == (2, 3, 1, 5) # batch_shape == (2,3,1) event_shape == (5,)
test_model(model1, model1, Trace_ELBO())
It is helpful to visualize the .shapes of each sample site by aligning them at the boundary between batch_shape and event_shape: dimensions to the right will be summed out in .log_prob() and dimensions to the left will remain.
batch dims | event dims
-----------+-----------
| a = sample("a", Normal(0, 1))
|2 b = sample("b", Normal(zeros(2), 1)
| .to_event(1))
| with plate("c", 2):
2| c = sample("c", Normal(zeros(2), 1))
| with plate("d", 3):
3|4 5 d = sample("d", Normal(zeros(3,4,5), 1)
| .to_event(2))
|
| x_axis = plate("x", 3, dim=-2)
| y_axis = plate("y", 2, dim=-3)
| with x_axis:
3 1| x = sample("x", Normal(0, 1))
| with y_axis:
2 1 1| y = sample("y", Normal(0, 1))
| with x_axis, y_axis:
2 3 1| xy = sample("xy", Normal(0, 1))
2 3 1|5 z = sample("z", Normal(0, 1).expand([5])
| .to_event(1))
To examine the shapes of sample sites in a program automatically, you can trace the program and use the Trace.format_shapes() method, which prints three shapes for each sample site: the distribution shape (both site["fn"].batch_shape and site["fn"].event_shape), the value shape (site["value"].shape), and if log probability has been computed also the log_prob shape (site["log_prob"].shape):
[8]:
trace = poutine.trace(model1).get_trace()
trace.compute_log_prob() # optional, but allows printing of log_prob shapes
print(trace.format_shapes())
Trace Shapes:
Param Sites:
Sample Sites:
a dist |
value |
log_prob |
b dist | 2
value | 2
log_prob |
c_plate dist |
value 2 |
log_prob |
c dist 2 |
value 2 |
log_prob 2 |
d_plate dist |
value 3 |
log_prob |
d dist 3 | 4 5
value 3 | 4 5
log_prob 3 |
x_axis dist |
value 3 |
log_prob |
y_axis dist |
value 2 |
log_prob |
x dist 3 1 |
value 3 1 |
log_prob 3 1 |
y dist 2 1 1 |
value 2 1 1 |
log_prob 2 1 1 |
xy dist 2 3 1 |
value 2 3 1 |
log_prob 2 3 1 |
z dist 2 3 1 | 5
value 2 3 1 | 5
log_prob 2 3 1 |
Subsampling tensors inside a plate¶
One of the main uses of plate is to subsample data. This is possible within a plate because data are conditionally independent, so the expected value of the loss on, say, half the data should be half the expected loss on the full data.
To subsample data, you need to inform Pyro of both the original data size and the subsample size; Pyro will then choose a random subset of data and yield the set of indices.
[9]:
data = torch.arange(100.)
def model2():
mean = pyro.param("mean", torch.zeros(len(data)))
with pyro.plate("data", len(data), subsample_size=10) as ind:
assert len(ind) == 10 # ind is a LongTensor that indexes the subsample.
batch = data[ind] # Select a minibatch of data.
mean_batch = mean[ind] # Take care to select the relevant per-datum parameters.
# Do stuff with batch:
x = pyro.sample("x", Normal(mean_batch, 1), obs=batch)
assert len(x) == 10
test_model(model2, guide=lambda: None, loss=Trace_ELBO())
Broadcasting to allow parallel enumeration¶
Pyro 0.2 introduces the ability to enumerate discrete latent variables in parallel. This can significantly reduce the variance of gradient estimators when learning a posterior via SVI.
To use parallel enumeration, Pyro needs to allocate tensor dimension that it can use for enumeration. To avoid conflicting with other dimensions that we want to use for plates, we need to declare a budget of the maximum number of tensor dimensions we’ll use. This budget is called max_plate_nesting and is an argument to SVI (the argument is simply passed through to
TraceEnum_ELBO). Usually Pyro can determine this budget on its own (it runs the (model,guide) pair once and record what happens), but in case of dynamic model structure you may need to declare max_plate_nesting manually.
To understand max_plate_nesting and how Pyro allocates dimensions for enumeration, let’s revisit model1() from above. This time we’ll map out three types of dimensions: enumeration dimensions on the left (Pyro takes control of these), batch dimensions in the middle, and event dimensions on the right.
max_plate_nesting = 3
|<--->|
enumeration|batch|event
-----------+-----+-----
|. . .| a = sample("a", Normal(0, 1))
|. . .|2 b = sample("b", Normal(zeros(2), 1)
| | .to_event(1))
| | with plate("c", 2):
|. . 2| c = sample("c", Normal(zeros(2), 1))
| | with plate("d", 3):
|. . 3|4 5 d = sample("d", Normal(zeros(3,4,5), 1)
| | .to_event(2))
| |
| | x_axis = plate("x", 3, dim=-2)
| | y_axis = plate("y", 2, dim=-3)
| | with x_axis:
|. 3 1| x = sample("x", Normal(0, 1))
| | with y_axis:
|2 1 1| y = sample("y", Normal(0, 1))
| | with x_axis, y_axis:
|2 3 1| xy = sample("xy", Normal(0, 1))
|2 3 1|5 z = sample("z", Normal(0, 1).expand([5]))
| | .to_event(1))
Note that it is safe to overprovision max_plate_nesting=4 but we cannot underprovision max_plate_nesting=2 (or Pyro will error). Let’s see how this works in practice.
[10]:
@config_enumerate
def model3():
p = pyro.param("p", torch.arange(6.) / 6)
locs = pyro.param("locs", torch.tensor([-1., 1.]))
a = pyro.sample("a", Categorical(torch.ones(6) / 6))
b = pyro.sample("b", Bernoulli(p[a])) # Note this depends on a.
with pyro.plate("c_plate", 4):
c = pyro.sample("c", Bernoulli(0.3))
with pyro.plate("d_plate", 5):
d = pyro.sample("d", Bernoulli(0.4))
e_loc = locs[d.long()].unsqueeze(-1)
e_scale = torch.arange(1., 8.)
e = pyro.sample("e", Normal(e_loc, e_scale)
.to_event(1)) # Note this depends on d.
# enumerated|batch|event dims
assert a.shape == ( 6, 1, 1 ) # Six enumerated values of the Categorical.
assert b.shape == ( 2, 1, 1, 1 ) # Two enumerated Bernoullis, unexpanded.
assert c.shape == ( 2, 1, 1, 1, 1 ) # Only two Bernoullis, unexpanded.
assert d.shape == (2, 1, 1, 1, 1, 1 ) # Only two Bernoullis, unexpanded.
assert e.shape == (2, 1, 1, 1, 5, 4, 7) # This is sampled and depends on d.
assert e_loc.shape == (2, 1, 1, 1, 1, 1, 1,)
assert e_scale.shape == ( 7,)
test_model(model3, model3, TraceEnum_ELBO(max_plate_nesting=2))
Let’s take a closer look at those dimensions. First note that Pyro allocates enumeration dims starting from the right at max_plate_nesting: Pyro allocates dim -3 to enumerate a, then dim -4 to enumerate b, then dim -5 to enumerate c, and finally dim -6 to enumerate d. Next note that samples only have extent (size > 1) in the new enumeration dimension. This helps keep tensors small and computation cheap. (Note that the log_prob shape will be broadcast up to contain both
enumeratin shape and batch shape, so e.g. trace.nodes['d']['log_prob'].shape == (2, 1, 1, 1, 5, 4).)
We can draw a similar map of the tensor dimensions:
max_plate_nesting = 2
|<->|
enumeration batch event
------------|---|-----
6|1 1| a = pyro.sample("a", Categorical(torch.ones(6) / 6))
2 1|1 1| b = pyro.sample("b", Bernoulli(p[a]))
| | with pyro.plate("c_plate", 4):
2 1 1|1 1| c = pyro.sample("c", Bernoulli(0.3))
| | with pyro.plate("d_plate", 5):
2 1 1 1|1 1| d = pyro.sample("d", Bernoulli(0.4))
2 1 1 1|1 1|1 e_loc = locs[d.long()].unsqueeze(-1)
| |7 e_scale = torch.arange(1., 8.)
2 1 1 1|5 4|7 e = pyro.sample("e", Normal(e_loc, e_scale)
| | .to_event(1))
To automatically examine this model with enumeration semantics, we can create an enumerated trace and then use Trace.format_shapes():
[11]:
trace = poutine.trace(poutine.enum(model3, first_available_dim=-3)).get_trace()
trace.compute_log_prob() # optional, but allows printing of log_prob shapes
print(trace.format_shapes())
Trace Shapes:
Param Sites:
p 6
locs 2
Sample Sites:
a dist |
value 6 1 1 |
log_prob 6 1 1 |
b dist 6 1 1 |
value 2 1 1 1 |
log_prob 2 6 1 1 |
c_plate dist |
value 4 |
log_prob |
c dist 4 |
value 2 1 1 1 1 |
log_prob 2 1 1 1 4 |
d_plate dist |
value 5 |
log_prob |
d dist 5 4 |
value 2 1 1 1 1 1 |
log_prob 2 1 1 1 5 4 |
e dist 2 1 1 1 5 4 | 7
value 2 1 1 1 5 4 | 7
log_prob 2 1 1 1 5 4 |
Writing parallelizable code¶
It can be tricky to write Pyro models that correctly handle parallelized sample sites. Two tricks help: broadcasting and ellipsis slicing. Let’s look at a contrived model to see how these work in practice. Our aim is to write a model that works both with and without enumeration.
[12]:
width = 8
height = 10
sparse_pixels = torch.LongTensor([[3, 2], [3, 5], [3, 9], [7, 1]])
enumerated = None # set to either True or False below
def fun(observe):
p_x = pyro.param("p_x", torch.tensor(0.1), constraint=constraints.unit_interval)
p_y = pyro.param("p_y", torch.tensor(0.1), constraint=constraints.unit_interval)
x_axis = pyro.plate('x_axis', width, dim=-2)
y_axis = pyro.plate('y_axis', height, dim=-1)
# Note that the shapes of these sites depend on whether Pyro is enumerating.
with x_axis:
x_active = pyro.sample("x_active", Bernoulli(p_x))
with y_axis:
y_active = pyro.sample("y_active", Bernoulli(p_y))
if enumerated:
assert x_active.shape == (2, 1, 1)
assert y_active.shape == (2, 1, 1, 1)
else:
assert x_active.shape == (width, 1)
assert y_active.shape == (height,)
# The first trick is to broadcast. This works with or without enumeration.
p = 0.1 + 0.5 * x_active * y_active
if enumerated:
assert p.shape == (2, 2, 1, 1)
else:
assert p.shape == (width, height)
dense_pixels = p.new_zeros(broadcast_shape(p.shape, (width, height)))
# The second trick is to index using ellipsis slicing.
# This allows Pyro to add arbitrary dimensions on the left.
for x, y in sparse_pixels:
dense_pixels[..., x, y] = 1
if enumerated:
assert dense_pixels.shape == (2, 2, width, height)
else:
assert dense_pixels.shape == (width, height)
with x_axis, y_axis:
if observe:
pyro.sample("pixels", Bernoulli(p), obs=dense_pixels)
def model4():
fun(observe=True)
def guide4():
fun(observe=False)
# Test without enumeration.
enumerated = False
test_model(model4, guide4, Trace_ELBO())
# Test with enumeration.
enumerated = True
test_model(model4, config_enumerate(guide4, "parallel"),
TraceEnum_ELBO(max_plate_nesting=2))
Automatic broadcasting inside pyro.plate¶
Note that in all our model/guide specifications, we have relied on pyro.plate to automatically expand sample shapes to satisfy the constraints on batch shape enforced by pyro.sample statements. However this broadcasting is equivalent to hand-annotated .expand() statements.
We will demonstrate this using model4 from the previous section. Note the following changes to the code from earlier:
For the purpose of this example, we will only consider “parallel” enumeration, but broadcasting should work as expected without enumeration or with “sequential” enumeration.
We have separated out the sampling function which returns the tensors corresponding to the active pixels. Modularizing the model code into components is a common practice, and helps with maintainability of large models.
We would also like to use the
pyro.plateconstruct to parallelize the ELBO estimator over num_particles. This is done by wrapping the contents of model/guide inside an outermostpyro.platecontext.
[13]:
num_particles = 100 # Number of samples for the ELBO estimator
width = 8
height = 10
sparse_pixels = torch.LongTensor([[3, 2], [3, 5], [3, 9], [7, 1]])
def sample_pixel_locations_no_broadcasting(p_x, p_y, x_axis, y_axis):
with x_axis:
x_active = pyro.sample("x_active", Bernoulli(p_x).expand([num_particles, width, 1]))
with y_axis:
y_active = pyro.sample("y_active", Bernoulli(p_y).expand([num_particles, 1, height]))
return x_active, y_active
def sample_pixel_locations_full_broadcasting(p_x, p_y, x_axis, y_axis):
with x_axis:
x_active = pyro.sample("x_active", Bernoulli(p_x))
with y_axis:
y_active = pyro.sample("y_active", Bernoulli(p_y))
return x_active, y_active
def sample_pixel_locations_partial_broadcasting(p_x, p_y, x_axis, y_axis):
with x_axis:
x_active = pyro.sample("x_active", Bernoulli(p_x).expand([width, 1]))
with y_axis:
y_active = pyro.sample("y_active", Bernoulli(p_y).expand([height]))
return x_active, y_active
def fun(observe, sample_fn):
p_x = pyro.param("p_x", torch.tensor(0.1), constraint=constraints.unit_interval)
p_y = pyro.param("p_y", torch.tensor(0.1), constraint=constraints.unit_interval)
x_axis = pyro.plate('x_axis', width, dim=-2)
y_axis = pyro.plate('y_axis', height, dim=-1)
with pyro.plate("num_particles", 100, dim=-3):
x_active, y_active = sample_fn(p_x, p_y, x_axis, y_axis)
# Indices corresponding to "parallel" enumeration are appended
# to the left of the "num_particles" plate dim.
assert x_active.shape == (2, 1, 1, 1)
assert y_active.shape == (2, 1, 1, 1, 1)
p = 0.1 + 0.5 * x_active * y_active
assert p.shape == (2, 2, 1, 1, 1)
dense_pixels = p.new_zeros(broadcast_shape(p.shape, (width, height)))
for x, y in sparse_pixels:
dense_pixels[..., x, y] = 1
assert dense_pixels.shape == (2, 2, 1, width, height)
with x_axis, y_axis:
if observe:
pyro.sample("pixels", Bernoulli(p), obs=dense_pixels)
def test_model_with_sample_fn(sample_fn):
def model():
fun(observe=True, sample_fn=sample_fn)
@config_enumerate
def guide():
fun(observe=False, sample_fn=sample_fn)
test_model(model, guide, TraceEnum_ELBO(max_plate_nesting=3))
test_model_with_sample_fn(sample_pixel_locations_no_broadcasting)
test_model_with_sample_fn(sample_pixel_locations_full_broadcasting)
test_model_with_sample_fn(sample_pixel_locations_partial_broadcasting)
In the first sampling function, we had to do some manual book-keeping and expand the Bernoulli distribution’s batch shape to account for the conditionally independent dimensions added by the pyro.plate contexts. In particular, note how sample_pixel_locations needs knowledge of num_particles, width and height and is accessing these variables from the global scope, which is not ideal.
The second argument to
pyro.plate, i.e. the optionalsizeargument needs to be provided for implicit broadasting, so that it can infer the batch shape requirement for each of the sample sites.The existing
batch_shapeof the sample site must be broadcastable with the size of thepyro.platecontexts. In our particular example,Bernoulli(p_x)has an empty batch shape which is universally broadcastable.
Note how simple it is to achieve parallelization via tensorized operations using pyro.plate! pyro.plate also helps in code modularization because model components can be written agnostic of the plate contexts in which they may subsequently get embedded in.
[ ]: