深度学习中的13种概率分布

发布时间:2023年12月17日

1 概率分布概述

d3d991a320f84158872f0b73a7346cbe.png

  • 共轭意味着它有共轭分布的关系。

在贝叶斯概率论中,如果后验分布 p(θx)与先验概率分布 p(θ)在同一概率分布族中,则先验和后验称为共轭分布,先验称为似然函数的共轭先验。

  • 多分类表示随机方差大于 2。

  • n 次意味着我们也考虑了先验概率 p(x)。

2 分布概率与特征

2.1 均匀分布(连续)

均匀分布在 [a,b] 上具有相同的概率值,是简单概率分布。

示例代码:

import numpy as np
from matplotlib import pyplot as plt

def uniform(x, a, b):

    y = [1 / (b - a) if a <= val and val <= b
                    else 0 for val in x]

    return x, y, np.mean(y), np.std(y)

x = np.arange(-100, 100) # define range of x
for ls in [(-50, 50), (10, 20)]:
    a, b = ls[0], ls[1]
    x, y, u, s = uniform(x, a, b)
    plt.plot(x, y, label=r'$\mu=%.2f,\ \sigma=%.2f$' % (u, s))

plt.legend()
plt.show()

运行代码显示:

b5e207c3f6b546f9981f9560f7a5b4d9.png

2.2 伯努利分布(离散)

  • 先验概率 p(x)不考虑伯努利分布。因此,如果我们对最大似然进行优化,那么我们很容易被过度拟合。

  • 利用二元交叉熵对二项分类进行分类。它的形式与伯努利分布的负对数相同。

示例代码:

import random
import numpy as np
from matplotlib import pyplot as plt

def bernoulli(p, k):
    return p if k else 1 - p

n_experiment = 100
p = 0.6
x = np.arange(n_experiment)
y = []
for _ in range(n_experiment):
    pick = bernoulli(p, k=bool(random.getrandbits(1)))
    y.append(pick)

u, s = np.mean(y), np.std(y)
plt.scatter(x, y, label=r'$\mu=%.2f,\ \sigma=%.2f$' % (u, s))
plt.legend()
plt.show()

运行代码显示:

6638381a82cb4bd0b7204fb0360163b1.png

2.3 二项分布(离散)

  • 参数为 n 和 p 的二项分布是一系列 n 个独立实验中成功次数的离散概率分布。

  • 二项式分布是指通过指定要提前挑选的数量而考虑先验概率的分布。

示例代码:

import numpy as np
from matplotlib import pyplot as plt

import operator as op
from functools import reduce

def const(n, r):
    r = min(r, n-r)
    numer = reduce(op.mul, range(n, n-r, -1), 1)
    denom = reduce(op.mul, range(1, r+1), 1)
    return numer / denom

def binomial(n, p):
    q = 1 - p
    y = [const(n, k) * (p ** k) * (q ** (n-k)) for k in range(n)]
    return y, np.mean(y), np.std(y)

for ls in [(0.5, 20), (0.7, 40), (0.5, 40)]:
    p, n_experiment = ls[0], ls[1]
    x = np.arange(n_experiment)
    y, u, s = binomial(n_experiment, p)
    plt.scatter(x, y, label=r'$\mu=%.2f,\ \sigma=%.2f$' % (u, s))

plt.legend()
plt.show()

运行代码显示:

30555ee71170430fae90af850f5123ec.png

2.4?多伯努利分布,分类分布(离散)

  • 多伯努利称为分类分布。

  • 交叉熵和采取负对数的多伯努利分布具有相同的形式。

示例代码:

import random
import numpy as np
from matplotlib import pyplot as plt

def categorical(p, k):
    return p[k]

n_experiment = 100
p = [0.2, 0.1, 0.7]
x = np.arange(n_experiment)
y = []
for _ in range(n_experiment):
    pick = categorical(p, k=random.randint(0, len(p) - 1))
    y.append(pick)

u, s = np.mean(y), np.std(y)
plt.scatter(x, y, label=r'$\mu=%.2f,\ \sigma=%.2f$' % (u, s))
plt.legend()
plt.show()

运行代码显示:

7ee7365db9864858be3292a53422043b.png

2.5 多项式分布(离散)

多项式分布与分类分布的关系与伯努尔分布与二项分布的关系相同。

示例代码:

import numpy as np
from matplotlib import pyplot as plt

import operator as op
from functools import reduce

def factorial(n):
    return reduce(op.mul, range(1, n + 1), 1)

def const(n, a, b, c):
    """
        return n! / a! b! c!, where a+b+c == n
    """
    assert  a + b + c == n

    numer = factorial(n)
    denom = factorial(a) * factorial(b) * factorial(c)
    return numer / denom

def multinomial(n):
    """
    :param x : list, sum(x) should be `n`
    :param n : number of trial
    :param p: list, sum(p) should be `1`
    """
    # get all a,b,c where a+b+c == n, a<b<c
    ls = []
    for i in range(1, n + 1):
        for j in range(i, n + 1):
            for k in range(j, n + 1):
                if i + j + k == n:
                    ls.append([i, j, k])

    y = [const(n, l[0], l[1], l[2]) for l in ls]
    x = np.arange(len(y))
    return x, y, np.mean(y), np.std(y)

for n_experiment in [20, 21, 22]:
    x, y, u, s = multinomial(n_experiment)
    plt.scatter(x, y, label=r'$trial=%d$' % (n_experiment))

plt.legend()
plt.show()

运行代码显示:

21b9679b43de462694db82542372a452.png

2.6 β分布(连续)

  • β分布与二项分布和伯努利分布共轭。

  • 利用共轭,利用已知的先验分布可以更容易地得到后验分布。

  • 当β分布满足特殊情况(α=1,β=1)时,均匀分布是相同的。

示例代码:

import numpy as np
from matplotlib import pyplot as plt

def gamma_function(n):
    cal = 1
    for i in range(2, n):
        cal *= i
    return cal

def beta(x, a, b):

    gamma = gamma_function(a + b) / \
            (gamma_function(a) * gamma_function(b))
    y = gamma * (x ** (a - 1)) * ((1 - x) ** (b - 1))
    return x, y, np.mean(y), np.std(y)

for ls in [(1, 3), (5, 1), (2, 2), (2, 5)]:
    a, b = ls[0], ls[1]

    # x in [0, 1], trial is 1/0.001 = 1000
    x = np.arange(0, 1, 0.001, dtype=np.float)
    x, y, u, s = beta(x, a=a, b=b)
    plt.plot(x, y, label=r'$\mu=%.2f,\ \sigma=%.2f,'
                         r'\ \alpha=%d,\ \beta=%d$' % (u, s, a, b))
plt.legend()
plt.show()

运行代码显示:

3626bfcc60bc403e8de5a04e80d28254.png

2.7?Dirichlet 分布(连续)

  • dirichlet 分布与多项式分布是共轭的。

  • 如果 k=2,则为β分布。

示例代码:

from random import randint
import numpy as np
from matplotlib import pyplot as plt

def normalization(x, s):
    """
    :return: normalizated list, where sum(x) == s
    """
    return [(i * s) / sum(x) for i in x]

def sampling():
    return normalization([randint(1, 100),
            randint(1, 100), randint(1, 100)], s=1)

def gamma_function(n):
    cal = 1
    for i in range(2, n):
        cal *= i
    return cal

def beta_function(alpha):
    """
    :param alpha: list, len(alpha) is k
    :return:
    """
    numerator = 1
    for a in alpha:
        numerator *= gamma_function(a)
    denominator = gamma_function(sum(alpha))
    return numerator / denominator

def dirichlet(x, a, n):
    """
    :param x: list of [x[1,...,K], x[1,...,K], ...], shape is (n_trial, K)
    :param a: list of coefficient, a_i > 0
    :param n: number of trial
    :return:
    """
    c = (1 / beta_function(a))
    y = [c * (xn[0] ** (a[0] - 1)) * (xn[1] ** (a[1] - 1))
         * (xn[2] ** (a[2] - 1)) for xn in x]
    x = np.arange(n)
    return x, y, np.mean(y), np.std(y)

n_experiment = 1200
for ls in [(6, 2, 2), (3, 7, 5), (6, 2, 6), (2, 3, 4)]:
    alpha = list(ls)

    # random samping [x[1,...,K], x[1,...,K], ...], shape is (n_trial, K)
    # each sum of row should be one.
    x = [sampling() for _ in range(1, n_experiment + 1)]

    x, y, u, s = dirichlet(x, alpha, n=n_experiment)
    plt.plot(x, y, label=r'$\alpha=(%d,%d,%d)$' % (ls[0], ls[1], ls[2]))

plt.legend()
plt.show()

运行代码显示:

a2cb19e178b54e59884804f8d84c3334.png

2.8?伽马分布(连续)

  • 如果 gamma(a,1)/gamma(a,1)+gamma(b,1)与 beta(a,b)相同,则 gamma 分布为β分布。

  • 指数分布和卡方分布是伽马分布的特例。

代码示例:

import numpy as np
from matplotlib import pyplot as plt

def gamma_function(n):
    cal = 1
    for i in range(2, n):
        cal *= i
    return cal

def gamma(x, a, b):
    c = (b ** a) / gamma_function(a)
    y = c * (x ** (a - 1)) * np.exp(-b * x)
    return x, y, np.mean(y), np.std(y)

for ls in [(1, 1), (2, 1), (3, 1), (2, 2)]:
    a, b = ls[0], ls[1]

    x = np.arange(0, 20, 0.01, dtype=np.float)
    x, y, u, s = gamma(x, a=a, b=b)
    plt.plot(x, y, label=r'$\mu=%.2f,\ \sigma=%.2f,'
                         r'\ \alpha=%d,\ \beta=%d$' % (u, s, a, b))
plt.legend()
plt.show()

运行代码显示:

1009001a4a754f21947db41db2d12c0b.png

2.9 指数分布(连续)

指数分布是 α 为 1 时 γ 分布的特例。

import numpy as np
from matplotlib import pyplot as plt

def exponential(x, lamb):
    y = lamb * np.exp(-lamb * x)
    return x, y, np.mean(y), np.std(y)

for lamb in [0.5, 1, 1.5]:

    x = np.arange(0, 20, 0.01, dtype=np.float)
    x, y, u, s = exponential(x, lamb=lamb)
    plt.plot(x, y, label=r'$\mu=%.2f,\ \sigma=%.2f,'
                         r'\ \lambda=%d$' % (u, s, lamb))
plt.legend()
plt.show()

运行代码显示

c97ec2c0d4f44ade97e214e78cf70650.png

2.10?高斯分布(连续)

高斯分布是一种非常常见的连续概率分布。

示例代码:

import numpy as np
from matplotlib import pyplot as plt

def gaussian(x, n):
    u = x.mean()
    s = x.std()

    # divide [x.min(), x.max()] by n
    x = np.linspace(x.min(), x.max(), n)

    a = ((x - u) ** 2) / (2 * (s ** 2))
    y = 1 / (s * np.sqrt(2 * np.pi)) * np.exp(-a)

    return x, y, x.mean(), x.std()

x = np.arange(-100, 100) # define range of x
x, y, u, s = gaussian(x, 10000)

plt.plot(x, y, label=r'$\mu=%.2f,\ \sigma=%.2f$' % (u, s))
plt.legend()
plt.show()

运行代码显示:

72168899350446a3b6e2748244a26d6f.png

2.11 标准正态分布(连续)

标准正态分布为特殊的高斯分布,平均值为 0,标准差为 1。

import numpy as np
from matplotlib import pyplot as plt

def normal(x, n):
    u = x.mean()
    s = x.std()

    # normalization
    x = (x - u) / s

    # divide [x.min(), x.max()] by n
    x = np.linspace(x.min(), x.max(), n)

    a = ((x - 0) ** 2) / (2 * (1 ** 2))
    y = 1 / (s * np.sqrt(2 * np.pi)) * np.exp(-a)

    return x, y, x.mean(), x.std()

x = np.arange(-100, 100) # define range of x
x, y, u, s = normal(x, 10000)

plt.plot(x, y, label=r'$\mu=%.2f,\ \sigma=%.2f$' % (u, s))
plt.legend()
plt.show()

运行代码显示:

1d47e619ea3145dba3d6b45ef956ea93.png

2.12?卡方分布(连续)

  • k 自由度的卡方分布是 k 个独立标准正态随机变量的平方和的分布。

  • 卡方分布是 β 分布的特例

示例代码:

import numpy as np
from matplotlib import pyplot as plt

def gamma_function(n):
    cal = 1
    for i in range(2, n):
        cal *= i
    return cal

def chi_squared(x, k):

    c = 1 / (2 ** (k/2)) * gamma_function(k//2)
    y = c * (x ** (k/2 - 1)) * np.exp(-x /2)

    return x, y, np.mean(y), np.std(y)

for k in [2, 3, 4, 6]:
    x = np.arange(0, 10, 0.01, dtype=np.float)
    x, y, _, _ = chi_squared(x, k)
    plt.plot(x, y, label=r'$k=%d$' % (k))

plt.legend()
plt.show()

运行代码显示

45847e204210461a94099453422c31d2.png

2.13?t 分布(连续)

t 分布是对称的钟形分布,与正态分布类似,但尾部较重,这意味着它更容易产生远低于平均值的值。

示例代码:

import numpy as np
from matplotlib import pyplot as plt

def gamma_function(n):
    cal = 1
    for i in range(2, n):
        cal *= i
    return cal

def student_t(x, freedom, n):

    # divide [x.min(), x.max()] by n
    x = np.linspace(x.min(), x.max(), n)

    c = gamma_function((freedom + 1) // 2) \
        / np.sqrt(freedom * np.pi) * gamma_function(freedom // 2)
    y = c * (1 + x**2 / freedom) ** (-((freedom + 1) / 2))

    return x, y, np.mean(y), np.std(y)

for freedom in [1, 2, 5]:

    x = np.arange(-10, 10) # define range of x
    x, y, _, _ = student_t(x, freedom=freedom, n=10000)
    plt.plot(x, y, label=r'$v=%d$' % (freedom))

plt.legend()
plt.show()

运行代码显示

84ec7b8d6794491fa4b904b23840fcd9.png

?

文章来源:https://blog.csdn.net/lsb2002/article/details/134937677
本文来自互联网用户投稿,该文观点仅代表作者本人,不代表本站立场。本站仅提供信息存储空间服务,不拥有所有权,不承担相关法律责任。