用 PyMC 解决逆问题 [英] Solving inverse problems with PyMC

问题描述

假设我们有一个 X 的先验(例如 X ~ Gaussian)和一个前向运算符 y = f(x).进一步假设我们通过实验观察到 y 并且这个实验可以无限重复.假设输出 Y 是高斯 (Y ~ Gaussian) 或无噪声 (Y ~ Delta(observation)).

Suppose we're given a prior on X (e.g. X ~ Gaussian) and a forward operator y = f(x). Suppose further we have observed y by means of an experiment and that this experiment can be repeated indefinitely. The output Y is assumed to be Gaussian (Y ~ Gaussian) or noise-free (Y ~ Delta(observation)).

如何根据观察结果持续更新我们对 X 的主观知识程度?我已经用 PyMC 尝试了以下模型，但似乎我遗漏了一些东西:

How to consistently update our subjective degree of knowledge about X given the observations? I've tried the following model with PyMC, but it seems I'm missing something:

from pymc import *

xtrue = 2                        # this value is unknown in the real application
x = rnormal(0, 0.01, size=10000) # initial guess

for i in range(5):
    X = Normal('X', x.mean(), 1./x.var())
    Y = X*X                        # f(x) = x*x
    OBS = Normal('OBS', Y, 0.1, value=xtrue*xtrue+rnormal(0,1), observed=True)
    model = Model([X,Y,OBS])
    mcmc = MCMC(model)
    mcmc.sample(10000)

    x = mcmc.trace('X')[:]       # posterior samples

后验没有收敛到xtrue.

推荐答案

@user1572508 所设计的功能现在是 PyMC 的一部分，名称为 stochastic_from_data() 或 Histogram()代码>.该线程的解决方案则变为:

The functionality purposed by @user1572508 is now part of PyMC under the name stochastic_from_data() or Histogram(). The solution to this thread then becomes:

from pymc import *
import matplotlib.pyplot as plt

xtrue = 2 # unknown in the real application
prior = rnormal(0,1,10000) # initial guess is inaccurate
for i in range(5):
  x = stochastic_from_data('x', prior)
  y = x*x
  obs = Normal('obs', y, 0.1, xtrue*xtrue + rnormal(0,1), observed=True)

  model = Model([x,y,obs])
  mcmc = MCMC(model)
  mcmc.sample(10000)

  Matplot.plot(mcmc.trace('x'))
  plt.show()

  prior = mcmc.trace('x')[:]

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