Cauchy

class paddle.distribution. Cauchy ( loc, scale, name=None ) [源代码]

柯西分布也叫柯西-洛伦兹分布，它是以奥古斯丁·路易·柯西与亨德里克·洛伦兹名字命名的连续概率分布。其在自然科学中有着非常广泛的应用。

柯西分布的概率密度函数（PDF）:

\[{ f(x; loc, scale) = \frac{1}{\pi scale \left[1 + \left(\frac{x - loc}{ scale}\right)^2\right]} = { 1 \over \pi } \left[ { scale \over (x - loc)^2 + scale^2 } \right], }\]

参数

  • loc (float|Tensor) - 定义分布峰值位置的位置参数。数据类型为 float32 或 float64。

  • scale (float|Tensor) - 最大值一半处的一半宽度的尺度参数。数据类型为 float32 或 float64。必须为正值。

  • name (str，可选) - 操作的名称，一般无需设置，默认值为 None，具体用法请参见 Name。

代码示例

import paddle
from paddle.distribution import Cauchy

# init Cauchy with float
rv = Cauchy(loc=0.1, scale=1.2)
print(rv.entropy())
# Tensor(shape=[], dtype=float32, place=Place(cpu), stop_gradient=True,
#        2.71334577)

# init Cauchy with N-Dim tensor
rv = Cauchy(loc=paddle.to_tensor(0.1), scale=paddle.to_tensor([1.0, 2.0]))
print(rv.entropy())
# Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
#        [2.53102422, 3.22417140])

属性

mean

柯西分布的均值

返回

ValueError，柯西分布没有均值

variance

柯西分布的方差

返回

ValueError，柯西分布没有方差

stddev

柯西分布的标准差

返回

ValueError，柯西分布没有标准差

方法

sample(shape, name=None)

生成指定维度的样本。

注解

sample 方法没有梯度，如果需要的话，请使用 rsample 方法代替。

参数

  • shape (Sequence[int]) - 指定生成样本的维度。

  • name (str，可选) - 操作的名称，一般无需设置，默认值为 None，具体用法请参见 Name。

返回

Tensor，样本，其维度为 \(\text{sample shape} + \text{batch shape} + \text{event shape}\)。

代码示例

import paddle
from paddle.distribution import Cauchy

# init Cauchy with float
rv = Cauchy(loc=0.1, scale=1.2)
print(rv.sample([10]).shape)
# [10]

# init Cauchy with 0-Dim tensor
rv = Cauchy(loc=paddle.full((), 0.1), scale=paddle.full((), 1.2))
print(rv.sample([10]).shape)
# [10]

# init Cauchy with N-Dim tensor
rv = Cauchy(loc=paddle.to_tensor(0.1), scale=paddle.to_tensor([1.0, 2.0]))
print(rv.sample([10]).shape)
# [10, 2]

# sample 2-Dim data
rv = Cauchy(loc=0.1, scale=1.2)
print(rv.sample([10, 2]).shape)
# [10, 2]

rv = Cauchy(loc=paddle.to_tensor(0.1), scale=paddle.to_tensor([1.0, 2.0]))
print(rv.sample([10, 2]).shape)
# [10, 2, 2]

rsample(shape, name=None)

重参数化采样，生成指定维度的样本。

参数

  • shape (Sequence[int]) - 指定生成样本的维度。

  • name (str，可选) - 操作的名称，一般无需设置，默认值为 None，具体用法请参见 Name。

返回

Tensor，样本，其维度为 \(\text{sample shape} + \text{batch shape} + \text{event shape}\)。

代码示例

import paddle
from paddle.distribution import Cauchy

# init Cauchy with float
rv = Cauchy(loc=0.1, scale=1.2)
print(rv.rsample([10]).shape)
# [10]

# init Cauchy with 0-Dim tensor
rv = Cauchy(loc=paddle.full((), 0.1), scale=paddle.full((), 1.2))
print(rv.rsample([10]).shape)
# [10]

# init Cauchy with N-Dim tensor
rv = Cauchy(loc=paddle.to_tensor(0.1), scale=paddle.to_tensor([1.0, 2.0]))
print(rv.rsample([10]).shape)
# [10, 2]

# sample 2-Dim data
rv = Cauchy(loc=0.1, scale=1.2)
print(rv.rsample([10, 2]).shape)
# [10, 2]

rv = Cauchy(loc=paddle.to_tensor(0.1), scale=paddle.to_tensor([1.0, 2.0]))
print(rv.rsample([10, 2]).shape)
# [10, 2, 2]

prob(value)

value 的概率密度函数。

\[{ f(x; loc, scale) = \frac{1}{\pi scale \left[1 + \left(\frac{x - loc}{ scale}\right)^2\right]} = { 1 \over \pi } \left[ { scale \over (x - loc)^2 + scale^2 } \right], }\]

参数

  • value (Tensor) - 输入 Tensor。

返回

Tensor， value 的概率密度函数。

代码示例

import paddle
from paddle.distribution import Cauchy

# init Cauchy with float
rv = Cauchy(loc=0.1, scale=1.2)
print(rv.prob(paddle.to_tensor(1.5)))
# Tensor(shape=[], dtype=float32, place=Place(cpu), stop_gradient=True,
#        0.11234467)

# broadcast to value
rv = Cauchy(loc=0.1, scale=1.2)
print(rv.prob(paddle.to_tensor([1.5, 5.1])))
# Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
#        [0.11234467, 0.01444674])

# init Cauchy with N-Dim tensor
rv = Cauchy(loc=paddle.to_tensor([0.1, 0.1]), scale=paddle.to_tensor([1.0, 2.0]))
print(rv.prob(paddle.to_tensor([1.5, 5.1])))
# Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
#        [0.10753712, 0.02195240])

# init Cauchy with N-Dim tensor with broadcast
rv = Cauchy(loc=paddle.to_tensor(0.1), scale=paddle.to_tensor([1.0, 2.0]))
print(rv.prob(paddle.to_tensor([1.5, 5.1])))
# Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
#        [0.10753712, 0.02195240])

log_prob(value)

对数概率密度函数

参数

  • value (Tensor) - 输入 Tensor。

返回

Tensor， value 的对数概率密度函数。

代码示例

import paddle
from paddle.distribution import Cauchy

# init Cauchy with float
rv = Cauchy(loc=0.1, scale=1.2)
print(rv.log_prob(paddle.to_tensor(1.5)))
# Tensor(shape=[], dtype=float32, place=Place(cpu), stop_gradient=True,
#        -2.18618369)

# broadcast to value
rv = Cauchy(loc=0.1, scale=1.2)
print(rv.log_prob(paddle.to_tensor([1.5, 5.1])))
# Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
#        [-2.18618369, -4.23728657])

# init Cauchy with N-Dim tensor
rv = Cauchy(loc=paddle.to_tensor([0.1, 0.1]), scale=paddle.to_tensor([1.0, 2.0]))
print(rv.log_prob(paddle.to_tensor([1.5, 5.1])))
# Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
#        [-2.22991920, -3.81887865])

# init Cauchy with N-Dim tensor with broadcast
rv = Cauchy(loc=paddle.to_tensor(0.1), scale=paddle.to_tensor([1.0, 2.0]))
print(rv.log_prob(paddle.to_tensor([1.5, 5.1])))
# Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
#        [-2.22991920, -3.81887865])

cdf(value)

value 的累积分布函数 （CDF）

\[{ \frac{1}{\pi} \arctan\left(\frac{x-loc}{ scale}\right)+\frac{1}{2}\! }\]

参数

  • value (Tensor) - 输入 Tensor。

返回

Tensor， value 的累积分布函数。

代码示例

import paddle
from paddle.distribution import Cauchy

# init Cauchy with float
rv = Cauchy(loc=0.1, scale=1.2)
print(rv.cdf(paddle.to_tensor(1.5)))
# Tensor(shape=[], dtype=float32, place=Place(cpu), stop_gradient=True,
#        0.77443725)

# broadcast to value
rv = Cauchy(loc=0.1, scale=1.2)
print(rv.cdf(paddle.to_tensor([1.5, 5.1])))
# Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
#        [0.77443725, 0.92502367])

# init Cauchy with N-Dim tensor
rv = Cauchy(loc=paddle.to_tensor([0.1, 0.1]), scale=paddle.to_tensor([1.0, 2.0]))
print(rv.cdf(paddle.to_tensor([1.5, 5.1])))
# Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
#        [0.80256844, 0.87888104])

# init Cauchy with N-Dim tensor with broadcast
rv = Cauchy(loc=paddle.to_tensor(0.1), scale=paddle.to_tensor([1.0, 2.0]))
print(rv.cdf(paddle.to_tensor([1.5, 5.1])))
# Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
#        [0.80256844, 0.87888104])

entropy()

柯西分布的信息熵。

\[{ \log(4\pi scale)\! }\]

返回

Tensor，柯西分布的信息熵。

代码示例

import paddle
from paddle.distribution import Cauchy

# init Cauchy with float
rv = Cauchy(loc=0.1, scale=1.2)
print(rv.entropy())
# Tensor(shape=[], dtype=float32, place=Place(cpu), stop_gradient=True,
#        2.71334577)

# init Cauchy with N-Dim tensor
rv = Cauchy(loc=paddle.to_tensor(0.1), scale=paddle.to_tensor([1.0, 2.0]))
print(rv.entropy())
# Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
#        [2.53102422, 3.22417140])

kl_divergence(other)

两个柯西分布之间的 KL 散度。

注解

[1] Frédéric Chyzak, Frank Nielsen, A closed-form formula for the Kullback-Leibler divergence between Cauchy distributions, 2019

参数

  • other (Cauchy) - Cauchy 的实例。

返回

Tensor，两个柯西分布之间的 KL 散度。

代码示例

import paddle
from paddle.distribution import Cauchy

rv = Cauchy(loc=0.1, scale=1.2)
rv_other = Cauchy(loc=paddle.to_tensor(1.2), scale=paddle.to_tensor([2.3, 3.4]))
print(rv.kl_divergence(rv_other))
# Tensor(shape=[2], dtype=float32, place=Place(cpu), stop_gradient=True,
#        [0.19819736, 0.31532931])