frequentism-and-bayesianism-chs

频率主义和贝叶斯主义——一个实用的介绍

此notebook最初出现在博客Pythonic Perambulations文章。BSD licensed。

这个系列共4个部分:中文版Part I Part II Part III Part IV,英文版Part I Part II Part III Part IV

 

科学工作者学习统计学的第一件事儿是要知道有两种不同的方法论:频率主义和贝叶斯主义。尽管这点很重要,但很多科学工作者从来没有机会认识两者的差异,及其产生结果的不同。本文的目的是综合频率主义和贝叶斯主义的理论与实践内容,让科学工作者在数据分析之前做出更好的选择。

开始我将通过理论来论述两者区别,之后用Python代码演示两者在应用中的差异。

频率主义 VS 贝叶斯主义: 理论之争

频率主义与贝叶斯主义的本质之争是对概率定义的差异。

频率主义认为概率是表示有限次的重复测量 。假设我多次测量某个星星的亮度(光量子通量)(F) 时(假设星星通量不会变化),由于测量误差每次得到不同的测量值。在多次测量的前提下,某个变量的频率 指的是测量变量的概率。频率主义认为概率本质上是与频率相关的事件 。因此,一个纯粹的频率主义者会认为,讨论星星真实亮度的概率是没啥意义的,因为真实亮度其实是固定值,也就是说讨论一个固定值的频率分布是瞎扯淡。

贝叶斯理论认为,概率的概念应该被扩展为对一个命题信任的程度 。贝叶斯理论会说测量概率为P(F)的星星亮度F:虽然概率确实能够通过大量重复实验获取的频率测得,但是这并非频率的本质。概率是一个由个人经验决定测量结果的主观命题。贝叶斯理论认为概率本质上与我们对事件的掌握程度有关 。贝叶斯理论认为人们可以在一个固定的范围内讨论星星真实亮度的概率。概率也会调整我们对先验信息和/或已有数据的认识。(概率会改变常识)

奇怪的是理论争议不大,但是在实际的数据分析中会产生极大差异。下面我会给出一个实际案例的两种理论实现,用Python演示。

频率理论与贝叶斯理论的实际应用: 光量子计量

让我们看一个简单问题,然后比较一下两种方法的差异。这儿得有点儿数学公式,不过我不会入得特别深哈。如果你想深入,你可以考虑一下练练教材4-5章的内容。

研究对象: 简易光量子计量方法

想象一下我们拿着望远镜,观察遥远的星星传来的无尽星光。首先,让我们假设星星的真实亮度是固定值,不会随时间变化。用常量Ftrue 表示(我们还将忽略大气层的影响,以及其他系统性误差)。我们假设用望远镜进行了N次测量,第ith 次测量报道了观察到的亮度Fi 和误差ei。

问题是有了这些测量值D={Fi,ei},真实亮度Ftrue的最佳估计值是多少?

(关于测量误差:如果假设测量误差是按正态分布。频率主义者的观点是ei是在大量重复**实验**条件下单次测量的标准差。而贝叶斯理论认为ei是按照我们以往的测量经验描述的(正态)概率分布的标准差。)

我们用Python随机生成一些数据来演示两种理论的差异。由于测量是按次计数的离散事件,我们可以使用泊松分布做演示:

In [1]:
# Generating some simple photon count data
import numpy as np
from scipy import stats
np.random.seed(1) # for repeatability F_true = 1000 # true flux, say number of photons measured in 1 second
N = 50 # number of measurements
F = stats.poisson(F_true).rvs(N) # N measurements of the flux
e = np.sqrt(F) # errors on Poisson counts estimated via square root
 

现在让我们为”测量“值做个简单的图形:

In [2]:
get_ipython().magic(u'matplotlib inline')
import matplotlib.pyplot as plt fig, ax = plt.subplots()
ax.errorbar(F, np.arange(N), xerr=e, fmt='ok', ecolor='gray', alpha=0.5)
ax.vlines([F_true], 0, N, linewidth=5, alpha=0.2)
ax.set_xlabel("Flux");ax.set_ylabel("measurement number");
 
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XJyMtauXcs3NoKJXkv3ZR3dqCY4wHkNXZb/R2R82h4gz/yEE83i0LVrV3tRaGhowKZNm7zasjsu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%0AAABJRU5ErkJggg==" alt="" />
 

用标准平方根法则对泊松分布进行分析,可以看到每次测量值的误差都不同。上面的例子中我们是先知道真实值Ftrue的,但是实际问题是:只给测量值和误差,怎么找到真实值的最佳估计值?

下面让我们来看看两种理论的不同解决方案。

简易光量子计量的频率主义方法

我们用频率主义中经典的似然估计方法开始。对应某一次测量值Di=(Fi,ei),以前面假设的正态分布误差ei为标准差,该事件在真实值Ftrue已知的条件下发生的概率分布满足:

P(Di | Ftrue)=12πe2i−−−−√exp[−(Fi−Ftrue)22e2i]

公式应该读作“在Ftrue条件下事件Di发生的概率...”,是个均值Ftrue、标准差为ei的正态分布。

我们通过计算每个观测值概率的乘积来构建似然函数

L(D | Ftrue)=∏i=1NP(Di | Ftrue)

其中,D={Di}表示整个测量事件。因为似然估计值可能非常小,通常更方便的做法是取对数。

logL=−12∑i=1N[log(2πe2i)+(Fi−Ftrue)2e2i]

我们这么做是为了确定Ftrue的极大似然估计值。由于问题很简单,所以最大值也容易算(比如通过微分求极值dlogL/dFtrue=0)。下面是Ftrue的极大似然估计值:

Fest=∑wiFi∑wi;  wi=1/e2i

特殊情况:当所有的测量误差ei相等时,可以推出

Fest=1N∑i=1NFi

也就是说,当误差相等时,Fest等于观测值的期望,这与直觉一致。

我们可以进一步研究下这么估计的误差。按照频率主义方法,可以通过李雅普诺夫定理将最大似然函数的曲线进行正态近似;在这个简单的例子中也可以容易解决。正态近似的标准差是:

σest=(∑i=1Nwi)−1/2

这个结果很容易计算;让我们估计一些我们的数据:

In [3]:
w = 1. / e ** 2
print("""
F_true = {0}
F_est = {1:.0f} +/- {2:.0f} (based on {3} measurements)
""".format(F_true, (w * F).sum() / w.sum(), w.sum() ** -0.5, N))
 
      F_true = 1000
F_est = 998 +/- 4 (based on 50 measurements)
 

可以看出,以50次测量得到的观测值为输入,得出我们的估计误差在千分之4。

简易光量子计量的贝叶斯方法

你可能认为,如果按照贝叶斯方法,通篇都得使用概率计算。我们真正想计算的其实是我们对参数的主观认识,在本例中就是:

P(Ftrue | D)

可以看出这与频率主义完成相反,要对Ftrue这样的固定值建模求解,概率做为模型参数没有任何意义。然而,贝叶斯主义却完全可以接受。

贝叶斯主义使用贝叶斯定理这一基本概率论公式:

P(Ftrue | D)=P(D | Ftrue) P(Ftrue)P(D)

虽然贝叶斯定理这一法则本身没有争议,贝叶斯主义的名字就从那儿来,但是贝叶斯主义对概率有争议的解释却是暗含在法则的P(Ftrue | D)之中。

让我们仔细看看公式中的术语:

  • P(Ftrue | D): 后验概率,是在已获得的数据条件下模型参数的概率:这是我们想要计算的。
  • P(D | Ftrue): 相似度,与上面频率主义的似然估计L(D | Ftrue)成正比。
  • P(Ftrue): 先验概率,在观察数据D之前我们对事件不确定性的认识。
  • P(D): 标准化常量,对实际数量进行简单标准化。

如果我们设置P(Ftrue)∝1(扁平先验,flat prior),那么会发现

P(Ftrue|D)∝L(D|Ftrue)

也就是说贝叶斯主义的概率是频率主义似然估计结果的最大值!因此,遑论两种理论的差异,我们会发现(至少是简单问题)两种估计是等价的。

什么叫先验概率?

你会发现,我为了图省事儿,先验概率P(Ftrue)没解释清楚。先验概率允许在计算中包含其他有用的信息,实际应用中用多种测量方法组合对某一事件进行评估是非常有用的(比如宇宙成因的参数估计)。然而,确实先验概率的必要性也是贝叶斯分析的争议之一。

当先验事实不存在时,频率主义者会说先验概率不靠谱。尽管用一个未知先验概率更简单,像前面提到那个先验概率常量,但是也会有一些意外的微妙之处。很多实验表明,真正的未知先验概率是不存在的!频率主义者认为主观选择先验概率会偏离结果,不能进行统计数据分析。

贝叶斯主义者认为频率主义解决不了问题,只想简单的避开。在确定(隐含的)先验概率这点上,频率主义经常被简单地看作贝叶斯主义理论的特殊情形:贝叶斯主义会要求最好让隐含变成明确,即使这种选择可能包含主观性。

光量子计量:贝叶斯方法

离开这些理论之争,让我们看看贝叶斯主义的实际计算结果。因为只考虑一个参数,可以将后验概率P(Ftrue | D)当作Ftrue的函数计算:这个分布反映了我们对Ftrue分观布主的认识。但是随着模型规模的增长,这种方法就变得难以把握。因此,贝叶斯主义计算方法常常依赖于抽样方法,如蒙特卡罗马尔可夫链(MCMC)

这里不细究MCMC的理论知识,我用Dan Foreman-Mackey的Python emcee包做一个简单的应用。目标是产生一堆满足后验概率分布的点,用这些点来确定我们问题的答案。

要演示MCMC,我们用Python定义先验概率P(Ftrue),似然估计值P(D | Ftrue),以及后验概率P(Ftrue | D)。我们的模型是一维的,但是为了处理多维模型,我们要定义参数数组θ,在我们的例子中就是θ=[Ftrue]:

In [4]:
def log_prior(theta):
return 1 # flat prior def log_likelihood(theta, F, e):
return -0.5 * np.sum(np.log(2 * np.pi * e ** 2)
+ (F - theta[0]) ** 2 / e ** 2) def log_posterior(theta, F, e):
return log_prior(theta) + log_likelihood(theta, F, e)
 

现在我们开始解决问题,产生随机数来猜测这些马尔可夫链上的点。

In [5]:
ndim = 1  # number of parameters in the model
nwalkers = 50 # number of MCMC walkers
nburn = 1000 # "burn-in" period to let chains stabilize
nsteps = 2000 # number of MCMC steps to take # we'll start at random locations between 0 and 2000
starting_guesses = 2000 * np.random.rand(nwalkers, ndim) import emcee
sampler = emcee.EnsembleSampler(nwalkers, ndim, log_posterior, args=[F, e])
sampler.run_mcmc(starting_guesses, nsteps) sample = sampler.chain # shape = (nwalkers, nsteps, ndim)
sample = sampler.chain[:, nburn:, :].ravel() # discard burn-in points
 

如果都能正常运行,数值sample会包含满足后验概率分布的50000个点。让我们画出来看看:

In [6]:
# plot a histogram of the sample
plt.hist(sample, bins=50, histtype="stepfilled", alpha=0.3, normed=True) # plot a best-fit Gaussian
F_fit = np.linspace(975, 1025)
pdf = stats.norm(np.mean(sample), np.std(sample)).pdf(F_fit) plt.plot(F_fit, pdf, '-k')
plt.xlabel("F"); plt.ylabel("P(F)")
Out[6]:
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fpxdnaGXC7HyZMnuY5CiFmwumOdkLFWXn4a5eVaaLV1cHf3RHn5BTg4yLiOha4uHvLyjoDH48HL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%0A8N2BAAAAAElFTkSuQmCC" alt="" />
 

我们从(正态)后验概率分布得到一样本。后验概率的期望和标准差是由前面的频率主义极大似然估计得来的:

In [7]:
print("""
F_true = {0}
F_est = {1:.0f} +/- {2:.0f} (based on {3} measurements)
""".format(F_true, np.mean(sample), np.std(sample), N))
 
      F_true = 1000
F_est = 998 +/- 4 (based on 50 measurements)
 

可以看出,在简单情况下,贝叶斯主义和频率主义两种方法的结果相同!

分析

现在,你可能认为贝叶斯主义的复杂性没啥必要,这个例子已经充分证明了。其实,用一个仿射不变MCMC综合样本发生器来模拟这样一个简单的一维正态分布,实在是杀鸡用牛刀,但是这么做的目的是要表明该方法同样适用于多维度的复杂后验概率情形,也能在复杂条件下提供似然估计不可能提供的优美的结果。

另外,你还可能注意到一个小伎俩:最后,我们用来频率主义方法来描述后验概率样本!当我们计算样本期望和标准差的时候,我们使用了一个标准的频率主义方法来刻画后验概率分布。用纯贝叶斯理论分析这个问题的结果可能要包含后验概率本身(比如,它的类似样本)。也就是说,在纯贝叶斯主义的结果中,问题不是一个简单的数值和误差,而是后验概率分布和模型参数!

增加一个维度:研究更复杂的模型

让我们看一个更复杂的情况,再比较一下频率主义和贝叶斯主义的结果。前面我们假设星星是静止的,现在假设我们观察的天体是一个随机变量;就是说它会随着时间变化(就是类星体那种)。

我们将用双参数正态分布来表示这个天体:θ=[μ,σ],其中,μ是期望值,σ是天体本征变量的标准差。于是我们每次观察时真实亮度概率的模型:

Ftrue∼12πσ2−−−−√exp[−(F−μ)22σ2]

现在,我们再考虑N次观察里每次的误差。可以这么产生:

In [8]:
np.random.seed(42)  # for reproducibility
N = 100 # we'll use more samples for the more complicated model
mu_true, sigma_true = 1000, 15 # stochastic flux model F_true = stats.norm(mu_true, sigma_true).rvs(N) # (unknown) true flux
F = stats.poisson(F_true).rvs() # observed flux: true flux plus Poisson errors.
e = np.sqrt(F) # root-N error, as above
 

可变条件下的光量子计量:频率主义方法

似然估计结果是本征分布与误差分布的卷积,可以得出

L(D | θ)=∏i=1N12π(σ2+e2i)−−−−−−−−−√exp[−(Fi−μ)22(σ2+e2i)]

与前面类似,我们可以通过求微分解出似然估计的最大值,即μ的最佳估计:

μest=∑wiFi∑wi;  wi=1σ2+e2i

这里有个问题:参数μ的值依赖于σ的值。两者具有相关性,因此我们不能直接使用频率主义的解析方法求解。

然而,我们能用数值分析计算来解极大似然估计值。这里我们用Scipy里面的最优化工具optimize

In [9]:
def log_likelihood(theta, F, e):
return -0.5 * np.sum(np.log(2 * np.pi * (theta[1] ** 2 + e ** 2))
+ (F - theta[0]) ** 2 / (theta[1] ** 2 + e ** 2)) # maximize likelihood <--> minimize negative likelihood
def neg_log_likelihood(theta, F, e):
return -log_likelihood(theta, F, e) from scipy import optimize
theta_guess = [900, 5]
theta_est = optimize.fmin(neg_log_likelihood, theta_guess, args=(F, e))
print("""
Maximum likelihood estimate for {0} data points:
mu={theta[0]:.0f}, sigma={theta[1]:.0f}
""".format(N, theta=theta_est))
 
Optimization terminated successfully.
Current function value: 502.839505
Iterations: 58
Function evaluations: 114 Maximum likelihood estimate for 100 data points:
mu=999, sigma=19
 

Scipy给出了μ和σ的极大似然估计值的最佳组合。但是问题才解决一半:我们需要确定答案的置信区间,也就是我们要计算μ和σ的误差。

在频率主义模式中有不同的方法来确定误差。如前所述,我们可以用正态近似去拟合最大似然估计和协方差矩阵(这里我们应该用数值分析而不是解析方法)。另外,我们可以计算统计量如χ2和χ2dof,然后使用卡方检验来决定置信区间,还需要假设似然估计具有正态性。我们也可以使用随机抽样方法如Jackknife或者Bootstrap,为了研究结果的确定程度,使输入数据的随机样本的似然估计最大化。

这些方法都行,但是每种方法都有各自的假设前提和特点。我们用Python的astroML包里面的basic bootstrap resampler:

In [10]:
from astroML.resample import bootstrap

def fit_samples(sample):
# sample is an array of size [n_bootstraps, n_samples]
# compute the maximum likelihood for each bootstrap.
return np.array([optimize.fmin(neg_log_likelihood, theta_guess,
args=(F, np.sqrt(F)), disp=0)
for F in sample]) samples = bootstrap(F, 1000, fit_samples) # 1000 bootstrap resamplings
 

和前面的贝叶斯后验概率的MCMC方法类似,我们将计算样本期望值和标准差来确定参数误差。

In [11]:
mu_samp = samples[:, 0]
sig_samp = abs(samples[:, 1]) print " mu = {0:.0f} +/- {1:.0f}".format(mu_samp.mean(), mu_samp.std())
print " sigma = {0:.0f} +/- {1:.0f}".format(sig_samp.mean(), sig_samp.std())
 
 mu    = 999 +/- 4
sigma = 18 +/- 5
 

我得说明bootstrap resampling的细究有大堆资料可供参考,这里粗略地说明一下这个方法的特点。最明显的地方就是误差需要是相关的、非正态的,但是这两个特点都不能通过简单的计算每个模型参数的均值和方差反映出来。然而,我相信这给频率主义提供的基本的解题思路。

可变条件下的光量子计量:贝叶斯主义方法

贝叶斯主义对这个问题的解决方法可之前那个问题基本雷同,我们可以把上面的代码简单改改就行。

In [12]:
def log_prior(theta):
# sigma needs to be positive.
if theta[1] <= 0:
return -np.inf
else:
return 0 def log_posterior(theta, F, e):
return log_prior(theta) + log_likelihood(theta, F, e) # same setup as above:
ndim, nwalkers = 2, 50
nsteps, nburn = 2000, 1000 starting_guesses = np.random.rand(nwalkers, ndim)
starting_guesses[:, 0] *= 2000 # start mu between 0 and 2000
starting_guesses[:, 1] *= 20 # start sigma between 0 and 20 sampler = emcee.EnsembleSampler(nwalkers, ndim, log_posterior, args=[F, e])
sampler.run_mcmc(starting_guesses, nsteps) sample = sampler.chain # shape = (nwalkers, nsteps, ndim)
sample = sampler.chain[:, nburn:, :].reshape(-1, 2)
 

现在有了样本,我们可以利用astroML包方便的功能来画出轨迹图,并用等高线描述1、2标准差的情况:

In [13]:
from astroML.plotting import plot_mcmc
fig = plt.figure()
ax = plot_mcmc(sample.T, fig=fig, labels=[r'$\mu$', r'$\sigma$'], colors='k')
ax[0].plot(sample[:, 0], sample[:, 1], ',k', alpha=0.1)
ax[0].plot([mu_true], [sigma_true], 'o', color='red', ms=10);
 
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fdd3HTTTXbH7Qj0ej00Gs01FXpcyF6lS+dxNTY2ksDAQDJw4ECycOFCQgghHh4e7Pempia7v4V0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" alt="" />
 

红点显示在真实值附近(在前述问题条件下),等高线显示了1、2个标准差的置信区间情况(68%和95%置信水平)。

这里要注意σ=0与两个标准差的情况一致:也就是说,依赖于你自己对确定性的临界值感觉,我们的数据不足以保证一个定点数据源概率的置信区间。

另外,后验概率确实是正态的:可以通过垂直方向的对称性来看。也就意味着用频率主义的正态近似不会将真实的不确定性反映到结果中。对频率主义者来说这不是啥事儿(频率主义模式有方法会考虑到非正态性),但是频率主义方法的绝大多数使用者都直接或间接地使用了正态分布的假设。贝叶斯方法一般不需要这个假设。

(先验概率:有些观点在确定σ的扁平先验时与本文的计算方式有轻微差异:它们认为,在处理像σ这样的比例因子时,扁平先验不是必须的。[Jeffreys 先验](http://en.wikipedia.org/wiki/Jeffreys_prior)的一些有趣的论述更容易接受。我认为Jeffreys不太适合,因为σ不是一个真实的比例因子(比如,正态分布也会影响ei)。关于这个问题,我将参考其他专业人士的观点。我个人认为,这些小问题往往是贝叶斯主义与频率主义争论的焦点)。

结论

我希望我已经通过这篇文章展示了频率主义与贝叶斯主义根本的区别,导致两者本质上对同样问题有着不同的解释,然而彼此却经常可以产生相似甚至相同的结果。

两者差异总结如下:

  • 频率主义认为概率与事实或假想事件的频率有关。
  • 贝叶斯主义认为概率是对事件认识程度的度量。
  • 频率主义通常使用点估计和极大似然估计方法进行分析。
  • 贝叶斯主义通常直接的计算后验概率,或者通过一些MCMC抽样的方法计算。

处理简单问题时,两种方法会产生近似的结果。但当数据和模型越来越复杂时,两种方法会有天壤之别。在以后的文章中,我会显示一两个更复杂的案例。继续关注噢!

Update: 看下一篇文章:频率主义与贝叶斯主义 II: 当结果不同时

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