ContinuousBernoulli¶

class paddle.distribution. ContinuousBernoulli ( probs: float | Tensor, lims: tuple[float] = (0.499, 0.501) ) [source]

The Continuous Bernoulli distribution with parameter: probs characterizing the shape of the density function. The Continuous Bernoulli distribution is defined on [0, 1], and it can be viewed as a continuous version of the Bernoulli distribution.

The continuous Bernoulli: fixing a pervasive error in variational autoencoders.

Mathematical details

The probability density function (pdf) is

\[p(x;\lambda) = C(\lambda)\lambda^x (1-\lambda)^{1-x}\]

In the above equation:

$x$: is continuous between 0 and 1
$probs = \lambda$: is the probability.
$C(\lambda)$: is the normalizing constant factor

\[\begin{split}C(\lambda) = \left\{ \begin{aligned} &2 & \text{ if $\lambda = \frac{1}{2}$} \\ &\frac{2\tanh^{-1}(1-2\lambda)}{1 - 2\lambda} & \text{ otherwise} \end{aligned} \right.\end{split}\]

Parameters

probs (int|float|Tensor) – The probability of Continuous Bernoulli distribution between [0, 1], which characterize the shape of the pdf. If the input data type is int or float, the data type of probs will be convert to a 1-D Tensor the paddle global default dtype.
lims (tuple) – Specify the unstable calculation region near 0.5, where the calculation is approximated by talyor expansion. The default value is (0.499, 0.501).

Examples

>>> import paddle
>>> from paddle.distribution import ContinuousBernoulli
>>> paddle.set_device("cpu")
>>> paddle.seed(100)

>>> rv = ContinuousBernoulli(paddle.to_tensor([0.2, 0.5]))

>>> print(rv.sample([2]))
Tensor(shape=[2, 2], dtype=float32, place=Place(cpu), stop_gradient=True,
[[0.38694882, 0.20714243],
 [0.00631948, 0.51577556]])

>>> print(rv.mean)
Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
[0.38801414, 0.50000000])

>>> print(rv.variance)
Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
[0.07589778, 0.08333334])

>>> print(rv.entropy())
Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
[-0.07641457,  0.        ])

>>> print(rv.cdf(paddle.to_tensor(0.1)))
Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
[0.17259926, 0.10000000])

>>> print(rv.icdf(paddle.to_tensor(0.1)))
Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
[0.05623737, 0.10000000])

>>> rv1 = ContinuousBernoulli(paddle.to_tensor([0.2, 0.8]))
>>> rv2 = ContinuousBernoulli(paddle.to_tensor([0.7, 0.5]))
>>> print(rv1.kl_divergence(rv2))
Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
[0.20103608, 0.07641447])

probs ( value: Tensor ) → Tensor probs¶: Probability density/mass function.

Note

This method will be deprecated in the future, please use prob instead.

property mean : Tensor

Mean of Continuous Bernoulli distribution.

Returns: mean value.
Return type: Tensor

property variance : Tensor

Variance of Continuous Bernoulli distribution.

Returns: variance value.
Return type: Tensor

sample ( shape: Sequence[int] = [] ) → Tensor sample¶

Generate Continuous Bernoulli samples of the specified shape. The final shape would be sample_shape + batch_shape.

Parameters: shape (Sequence[int], optional) – Prepended shape of the generated samples.
Returns: Tensor, Sampled data with shape sample_shape + batch_shape.

rsample ( shape: Sequence[int] = [] ) → Tensor rsample¶

Generate Continuous Bernoulli samples of the specified shape. The final shape would be sample_shape + batch_shape.

Parameters: shape (Sequence[int], optional) – Prepended shape of the generated samples.
Returns: Tensor, Sampled data with shape sample_shape + batch_shape.

log_prob ( value: Tensor ) → Tensor log_prob¶

Log probability density function.

Parameters: value (Tensor) – The input tensor.
Returns: log probability. The data type is the same as self.probs.
Return type: Tensor

prob ( value: Tensor ) → Tensor prob¶

Probability density function.

Parameters: value (Tensor) – The input tensor.
Returns: probability. The data type is the same as self.probs.
Return type: Tensor

property batch_shape : Sequence[int]

Returns batch shape of distribution

Returns: batch shape
Return type: Sequence[int]

entropy ( ) → Tensor entropy¶

Shannon entropy in nats.

The entropy is

\[\mathcal{H}(X) = -\log C + \left[ \log (1 - \lambda) -\log \lambda \right] \mathbb{E}(X) - \log(1 - \lambda)\]

In the above equation:

$\Omega$: is the support of the distribution.

Returns: Tensor, Shannon entropy of Continuous Bernoulli distribution.

property event_shape : Sequence[int]

Returns event shape of distribution

Returns: event shape
Return type: Sequence[int]

cdf ( value: Tensor ) → Tensor cdf¶

Cumulative distribution function

\[\begin{split}{ P(X \le t; \lambda) = F(t;\lambda) = \left\{ \begin{aligned} &t & \text{ if $\lambda = \frac{1}{2}$} \\ &\frac{\lambda^t (1 - \lambda)^{1 - t} + \lambda - 1}{2\lambda - 1} & \text{ otherwise} \end{aligned} \right. }\end{split}\]

Parameters: value (Tensor) – The input tensor.
Returns: quantile of value. The data type is the same as self.probs.
Return type: Tensor

icdf ( value: Tensor ) → Tensor icdf¶

Inverse cumulative distribution function

\[\begin{split}{ F^{-1}(x;\lambda) = \left\{ \begin{aligned} &x & \text{ if $\lambda = \frac{1}{2}$} \\ &\frac{\log(1+(\frac{2\lambda - 1}{1 - \lambda})x)}{\log(\frac{\lambda}{1-\lambda})} & \text{ otherwise} \end{aligned} \right. }\end{split}\]

Parameters: value (Tensor) – The input tensor, meaning the quantile.
Returns: the value of the r.v. corresponding to the quantile. The data type is the same as self.probs.
Return type: Tensor

kl_divergence ( other: ContinuousBernoulli ) → Tensor [source] kl_divergence¶

The KL-divergence between two Continuous Bernoulli distributions with the same batch_shape.

The probability density function (pdf) is

\[KL\_divergence(\lambda_1, \lambda_2) = - H - \{\log C_2 + [\log \lambda_2 - \log (1-\lambda_2)] \mathbb{E}_1(X) + \log (1-\lambda_2) \}\]

Parameters: other (ContinuousBernoulli) – instance of Continuous Bernoulli.
Returns: Tensor, kl-divergence between two Continuous Bernoulli distributions.

ContinuousBernoulli¶

probs¶

sample¶

rsample¶

log_prob¶

prob¶

entropy¶

cdf¶

icdf¶

kl_divergence¶