# Normal¶

class paddle.distribution. Normal ( loc, scale, name=None ) [source]

The Normal distribution with location loc and scale parameters.

Mathematical details

The probability density function (pdf) is

$pdf(x; \mu, \sigma) = \frac{1}{Z}e^{\frac {-0.5 (x - \mu)^2} {\sigma^2} }$
$Z = (2 \pi \sigma^2)^{0.5}$

In the above equation:

• $$loc = \mu$$: is the mean.

• $$scale = \sigma$$: is the std.

• $$Z$$: is the normalization constant.

Parameters
• loc (int|float|list|tuple|numpy.ndarray|Tensor) – The mean of normal distribution.The data type is float32 and float64.

• scale (int|float|list|tuple|numpy.ndarray|Tensor) – The std of normal distribution.The data type is float32 and float64.

• name (str, optional) – Name for the operation (optional, default is None). For more information, please refer to Name.

Examples

>>> import paddle
>>> from paddle.distribution import Normal

>>> # Define a single scalar Normal distribution.
>>> dist = Normal(loc=0., scale=3.)
>>> # Define a batch of two scalar valued Normals.
>>> # The first has mean 1 and standard deviation 11, the second 2 and 22.
>>> dist = Normal(loc=[1., 2.], scale=[11., 22.])
>>> # Get 3 samples, returning a 3 x 2 tensor.
>>> dist.sample([3])

>>> # Define a batch of two scalar valued Normals.
>>> # Both have mean 1, but different standard deviations.
>>> dist = Normal(loc=1., scale=[11., 22.])

>>> # Complete example
>>> value_tensor = paddle.to_tensor([0.8], dtype="float32")

>>> normal_a = Normal([0.], [1.])
>>> normal_b = Normal([0.5], [2.])
>>> sample = normal_a.sample([2])
>>> # a random tensor created by normal distribution with shape: [2, 1]
>>> entropy = normal_a.entropy()
>>> print(entropy)
Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
[1.41893852])
>>> lp = normal_a.log_prob(value_tensor)
>>> print(lp)
Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
[-1.23893857])
>>> p = normal_a.probs(value_tensor)
>>> print(p)
Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
[0.28969154])
>>> kl = normal_a.kl_divergence(normal_b)
>>> print(kl)
Tensor(shape=[1], dtype=float32, place=Place(cpu), stop_gradient=True,
[0.34939718])

property batch_shape

Returns batch shape of distribution

Returns

batch shape

Return type

Sequence[int]

property event_shape

Returns event shape of distribution

Returns

event shape

Return type

Sequence[int]

property mean

Mean of multinomial distribuion.

Returns

mean value.

Return type

Tensor

prob ( value )

Probability density/mass function evaluated at value.

Parameters

value (Tensor) – value which will be evaluated

property variance

Variance of lognormal distribution.

Returns

variance value.

Return type

Tensor

sample ( shape=(), seed=0 )

Generate samples of the specified shape.

Parameters
• shape (Sequence[int], optional) – Shape of the generated samples.

• seed (int) – Python integer number.

Returns

Tensor, A tensor with prepended dimensions shape.The data type is float32.

rsample ( shape=() )

Generate reparameterized samples of the specified shape.

Parameters

shape (Sequence[int], optional) – Shape of the generated samples.

Returns

A tensor with prepended dimensions shape.The data type is float32.

Return type

Tensor

entropy ( )

Shannon entropy in nats.

The entropy is

$entropy(\sigma) = 0.5 \log (2 \pi e \sigma^2)$

In the above equation:

• $$scale = \sigma$$: is the std.

Returns

Tensor, Shannon entropy of normal distribution.The data type is float32.

log_prob ( value )

Log probability density/mass function.

Parameters

value (Tensor) – The input tensor.

Returns

log probability.The data type is same with value .

Return type

Tensor

probs ( value )

Probability density/mass function.

Parameters

value (Tensor) – The input tensor.

Returns

Tensor, probability. The data type is same with value .

kl_divergence ( other ) [source]

The KL-divergence between two normal distributions.

The probability density function (pdf) is

$KL\_divergence(\mu_0, \sigma_0; \mu_1, \sigma_1) = 0.5 (ratio^2 + (\frac{diff}{\sigma_1})^2 - 1 - 2 \ln {ratio})$
$ratio = \frac{\sigma_0}{\sigma_1}$
$diff = \mu_1 - \mu_0$

In the above equation:

• $$loc = \mu_0$$: is the mean of current Normal distribution.

• $$scale = \sigma_0$$: is the std of current Normal distribution.

• $$loc = \mu_1$$: is the mean of other Normal distribution.

• $$scale = \sigma_1$$: is the std of other Normal distribution.

• $$ratio$$: is the ratio of scales.

• $$diff$$: is the difference between means.

Parameters

other (Normal) – instance of Normal.

Returns

Tensor, kl-divergence between two normal distributions.The data type is float32.