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Binomial distribution.
Inherits From: Distribution
tf.contrib.distributions.Binomial(
total_count, logits=None, probs=None, validate_args=False, allow_nan_stats=True,
name='Binomial'
)
This distribution is parameterized by probs, a (batch of) probabilities for
drawing a 1 and total_count, the number of trials per draw from the
Binomial.
Mathematical Details
The Binomial is a distribution over the number of 1's in total_count
independent trials, with each trial having the same probability of 1, i.e.,
probs.
The probability mass function (pmf) is,
pmf(k; n, p) = p**k (1 - p)**(n - k) / Z
Z = k! (n - k)! / n!
where:
total_count = n,probs = p,Zis the normalizing constant, and,n!is the factorial ofn.
Examples
Create a single distribution, corresponding to 5 coin flips.
dist = Binomial(total_count=5., probs=.5)
Create a single distribution (using logits), corresponding to 5 coin flips.
dist = Binomial(total_count=5., logits=0.)
Creates 3 distributions with the third distribution most likely to have successes.
p = [.2, .3, .8]
# n will be broadcast to [4., 4., 4.], to match p.
dist = Binomial(total_count=4., probs=p)
The distribution functions can be evaluated on counts.
# counts same shape as p.
counts = [1., 2, 3]
dist.prob(counts) # Shape [3]
# p will be broadcast to [[.2, .3, .8], [.2, .3, .8]] to match counts.
counts = [[1., 2, 1], [2, 2, 4]]
dist.prob(counts) # Shape [2, 3]
# p will be broadcast to shape [5, 7, 3] to match counts.
counts = [[...]] # Shape [5, 7, 3]
dist.prob(counts) # Shape [5, 7, 3]
Args | |
|---|---|
total_count
|
Non-negative floating point tensor with shape broadcastable
to [N1,..., Nm] with m >= 0 and the same dtype as probs or
logits. Defines this as a batch of N1 x ... x Nm different Binomial
distributions. Its components should be equal to integer values.
|
logits
|
Floating point tensor representing the log-odds of a
positive event with shape broadcastable to [N1,..., Nm] m >= 0, and
the same dtype as total_count. Each entry represents logits for the
probability of success for independent Binomial distributions. Only one
of logits or probs should be passed in.
|
probs
|
Positive floating point tensor with shape broadcastable to
[N1,..., Nm] m >= 0, probs in [0, 1]. Each entry represents the
probability of success for independent Binomial distributions. Only one
of logits or probs should be passed in.
|
validate_args
|
Python bool, default False. When True distribution
parameters are checked for validity despite possibly degrading runtime
performance. When False invalid inputs may silently render incorrect
outputs.
|
allow_nan_stats
|
Python bool, default True. When True, statistics
(e.g., mean, mode, variance) use the value "NaN" to indicate the
result is undefined. When False, an exception is raised if one or
more of the statistic's batch members are undefined.
|
name
|
Python str name prefixed to Ops created by this class.
|
Attributes | |
|---|---|
allow_nan_stats
|
Python bool describing behavior when a stat is undefined.
Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined. |
batch_shape
|
Shape of a single sample from a single event index as a TensorShape.
May be partially defined or unknown. The batch dimensions are indexes into independent, non-identical parameterizations of this distribution. |
dtype
|
The DType of Tensors handled by this Distribution.
|
event_shape
|
Shape of a single sample from a single batch as a TensorShape.
May be partially defined or unknown. |
logits
|
Log-odds of drawing a 1.
|
name
|
Name prepended to all ops created by this Distribution.
|
parameters
|
Dictionary of parameters used to instantiate this Distribution.
|
probs
|
Probability of drawing a 1.
|
reparameterization_type
|
Describes how samples from the distribution are reparameterized.
Currently this is one of the static instances
|
total_count
|
Number of trials. |
validate_args
|
Python bool indicating possibly expensive checks are enabled.
|
Methods
batch_shape_tensor
batch_shape_tensor(
name='batch_shape_tensor'
)
Shape of a single sample from a single event index as a 1-D Tensor.
The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.
| Args | |
|---|---|
name
|
name to give to the op |
| Returns | |
|---|---|
batch_shape
|
Tensor.
|
cdf
cdf(
value, name='cdf'
)
Cumulative distribution function.
Given random variable X, the cumulative distribution function cdf is:
cdf(x) := P[X <= x]
| Args | |
|---|---|
value
|
float or double Tensor.
|
name
|
Python str prepended to names of ops created by this function.
|
| Returns | |
|---|---|
cdf
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype.
|
copy
copy(
**override_parameters_kwargs
)
Creates a deep copy of the distribution.
| Args | |
|---|---|
**override_parameters_kwargs
|
String/value dictionary of initialization arguments to override with new values. |
| Returns | |
|---|---|
distribution
|
A new instance of type(self) initialized from the union
of self.parameters and override_parameters_kwargs, i.e.,
dict(self.parameters, **override_parameters_kwargs).
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covariance
covariance(
name='covariance'
)
Covariance.
Covariance is (possibly) defined only for non-scalar-event distributions.
For example, for a length-k, vector-valued distribution, it is calculated
as,
Cov[i, j] = Covariance(X_i, X_j) = E[(X_i - E[X_i]) (X_j - E[X_j])]
where Cov is a (batch of) k x k matrix, 0 <= (i, j) < k, and E
denotes expectation.
Alternatively, for non-vector, multivariate distributions (e.g.,
matrix-valued, Wishart), Covariance shall return a (batch of) matrices
under some vectorization of the events, i.e.,
Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]
where Cov is a (batch of) k' x k' matrices,
0 <= (i, j) < k' = reduce_prod(event_shape), and Vec is some function
mapping indices of this distribution's event dimensions to indices of a
length-k' vector.
| Args | |
|---|---|
name
|
Python str prepended to names of ops created by this function.
|
| Returns | |
|---|---|
covariance
|
Floating-point Tensor with shape [B1, ..., Bn, k', k']
where the first n dimensions are batch coordinates and
k' = reduce_prod(self.event_shape).
|
cross_entropy
cross_entropy(
other, name='cross_entropy'
)
Computes the (Shannon) cross entropy.
Denote this distribution (self) by P and the other distribution by
Q. Assuming P, Q are absolutely continuous with respect to
one another and permit densities p(x) dr(x) and q(x) dr(x), (Shanon)
cross entropy is defined as:
H[P, Q] = E_p[-log q(X)] = -int_F p(x) log q(x) dr(x)
where F denotes the support of the random variable X ~ P.
| Args | |
|---|---|
other
|
tfp.distributions.Distribution instance.
|
name
|
Python str prepended to names of ops created by this function.
|
| Returns | |
|---|---|
cross_entropy
|
self.dtype Tensor with shape [B1, ..., Bn]
representing n different calculations of (Shanon) cross entropy.
|
entropy
entropy(
name='entropy'
)
Shannon entropy in nats.
event_shape_tensor
event_shape_tensor(
name='event_shape_tensor'
)
Shape of a single sample from a single batch as a 1-D int32 Tensor.
| Args | |
|---|---|
name
|
name to give to the op |
| Returns | |
|---|---|
event_shape
|
Tensor.
|
is_scalar_batch
is_scalar_batch(
name='is_scalar_batch'
)
Indicates that batch_shape == [].
| Args | |
|---|---|
name
|
Python str prepended to names of ops created by this function.
|
| Returns | |
|---|---|
is_scalar_batch
|
bool scalar Tensor.
|
is_scalar_event
is_scalar_event(
name='is_scalar_event'
)
Indicates that event_shape == [].
| Args | |
|---|---|
name
|
Python str prepended to names of ops created by this function.
|
| Returns | |
|---|---|
is_scalar_event
|
bool scalar Tensor.
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kl_divergence
kl_divergence(
other, name='kl_divergence'
)
Computes the Kullback--Leibler divergence.
Denote this distribution (self) by p and the other distribution by
q. Assuming p, q are absolutely continuous with respect to reference
measure r, the KL divergence is defined as:
KL[p, q] = E_p[log(p(X)/q(X))]
= -int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
= H[p, q] - H[p]
where F denotes the support of the random variable X ~ p, H[., .]
denotes (Shanon) cross entropy, and H[.] denotes (Shanon) entropy.
| Args | |
|---|---|
other
|
tfp.distributions.Distribution instance.
|
name
|
Python str prepended to names of ops created by this function.
|
| Returns | |
|---|---|
kl_divergence
|
self.dtype Tensor with shape [B1, ..., Bn]
representing n different calculations of the Kullback-Leibler
divergence.
|
log_cdf
log_cdf(
value, name='log_cdf'
)
Log cumulative distribution function.
Given random variable X, the cumulative distribution function cdf is:
log_cdf(x) := Log[ P[X <= x] ]
Often, a numerical approximation can be used for log_cdf(x) that yields
a more accurate answer than simply taking the logarithm of the cdf when
x << -1.
| Args | |
|---|---|
value
|
float or double Tensor.
|
name
|
Python str prepended to names of ops created by this function.
|
| Returns | |
|---|---|
logcdf
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype.
|
log_prob
log_prob(
value, name='log_prob'
)
Log probability density/mass function.
Additional documentation from Binomial:
For each batch member of counts value, P[value] is the probability that
after sampling self.total_count draws from this Binomial distribution, the
number of successes is value. Since different sequences of draws can result in
the same counts, the probability includes a combinatorial coefficient.
| Args | |
|---|---|
value
|
float or double Tensor.
|
name
|
Python str prepended to names of ops created by this function.
|
| Returns | |
|---|---|
log_prob
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype.
|
log_survival_function
log_survival_function(
value, name='log_survival_function'
)
Log survival function.
Given random variable X, the survival function is defined:
log_survival_function(x) = Log[ P[X > x] ]
= Log[ 1 - P[X <= x] ]
= Log[ 1 - cdf(x) ]
Typically, different numerical approximations can be used for the log
survival function, which are more accurate than 1 - cdf(x) when x >> 1.
| Args | |
|---|---|
value
|
float or double Tensor.
|
name
|
Python str prepended to names of ops created by this function.
|
| Returns | |
|---|---|
Tensor of shape sample_shape(x) + self.batch_shape with values of type
self.dtype.
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mean
mean(
name='mean'
)
Mean.
mode
mode(
name='mode'
)
Mode.
Additional documentation from Binomial:
Note that when (1 + total_count) * probs is an integer, there are
actually two modes. Namely, (1 + total_count) * probs and
(1 + total_count) * probs - 1 are both modes. Here we return only the
larger of the two modes.
param_shapes
@classmethodparam_shapes( sample_shape, name='DistributionParamShapes' )
Shapes of parameters given the desired shape of a call to sample().
This is a class method that describes what key/value arguments are required
to instantiate the given Distribution so that a particular shape is
returned for that instance's call to sample().
Subclasses should override class method _param_shapes.
| Args | |
|---|---|
sample_shape
|
Tensor or python list/tuple. Desired shape of a call to
sample().
|
name
|
name to prepend ops with. |
| Returns | |
|---|---|
dict of parameter name to Tensor shapes.
|
param_static_shapes
@classmethodparam_static_shapes( sample_shape )
param_shapes with static (i.e. TensorShape) shapes.
This is a class method that describes what key/value arguments are required
to instantiate the given Distribution so that a particular shape is
returned for that instance's call to sample(). Assumes that the sample's
shape is known statically.
Subclasses should override class method _param_shapes to return
constant-valued tensors when constant values are fed.
| Args | |
|---|---|
sample_shape
|
TensorShape or python list/tuple. Desired shape of a call
to sample().
|
| Returns | |
|---|---|
dict of parameter name to TensorShape.
|
| Raises | |
|---|---|
ValueError
|
if sample_shape is a TensorShape and is not fully defined.
|
prob
prob(
value, name='prob'
)
Probability density/mass function.
Additional documentation from Binomial:
For each batch member of counts value, P[value] is the probability that
after sampling self.total_count draws from this Binomial distribution, the
number of successes is value. Since different sequences of draws can result in
the same counts, the probability includes a combinatorial coefficient.
| Args | |
|---|---|
value
|
float or double Tensor.
|
name
|
Python str prepended to names of ops created by this function.
|
| Returns | |
|---|---|
prob
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype.
|
quantile
quantile(
value, name='quantile'
)
Quantile function. Aka "inverse cdf" or "percent point function".
Given random variable X and p in [0, 1], the quantile is:
quantile(p) := x such that P[X <= x] == p
| Args | |
|---|---|
value
|
float or double Tensor.
|
name
|
Python str prepended to names of ops created by this function.
|
| Returns | |
|---|---|
quantile
|
a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype.
|
sample
sample(
sample_shape=(), seed=None, name='sample'
)
Generate samples of the specified shape.
Note that a call to sample() without arguments will generate a single
sample.
| Args | |
|---|---|
sample_shape
|
0D or 1D int32 Tensor. Shape of the generated samples.
|
seed
|
Python integer seed for RNG |
name
|
name to give to the op. |
| Returns | |
|---|---|
samples
|
a Tensor with prepended dimensions sample_shape.
|
stddev
stddev(
name='stddev'
)
Standard deviation.
Standard deviation is defined as,
stddev = E[(X - E[X])**2]**0.5
where X is the random variable associated with this distribution, E
denotes expectation, and stddev.shape = batch_shape + event_shape.
| Args | |
|---|---|
name
|
Python str prepended to names of ops created by this function.
|
| Returns | |
|---|---|
stddev
|
Floating-point Tensor with shape identical to
batch_shape + event_shape, i.e., the same shape as self.mean().
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survival_function
survival_function(
value, name='survival_function'
)
Survival function.
Given random variable X, the survival function is defined:
survival_function(x) = P[X > x]
= 1 - P[X <= x]
= 1 - cdf(x).
| Args | |
|---|---|
value
|
float or double Tensor.
|
name
|
Python str prepended to names of ops created by this function.
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| Returns | |
|---|---|
Tensor of shape sample_shape(x) + self.batch_shape with values of type
self.dtype.
|
variance
variance(
name='variance'
)
Variance.
Variance is defined as,
Var = E[(X - E[X])**2]
where X is the random variable associated with this distribution, E
denotes expectation, and Var.shape = batch_shape + event_shape.
| Args | |
|---|---|
name
|
Python str prepended to names of ops created by this function.
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| Returns | |
|---|---|
variance
|
Floating-point Tensor with shape identical to
batch_shape + event_shape, i.e., the same shape as self.mean().
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