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Exercise - Variational Mean Field Approximation for Univariate Gaussian

Table of Contents

Introduction

[TODO]

In order to detect errors in your own code, execute the notebook cells containing assert or assert_almost_equal.

Requirements

Knowledge

[TODO]

Python Modules

%matplotlib inline
import numpy as np
from matplotlib import pyplot as plt
import scipy.stats
np.random.seed(40)

Data

Data: $ X \sim \mathcal N(\mu, \frac{1}{\tau}) $

Probability Density Function: $ p(X \mid \mu, \tau) = \sqrt{\frac{\tau}{2\pi}} \exp\left( -\frac{\tau (X-\mu)^2 }{2} \right) $

with
-$ \mu $: mean -$ \sigma^2 $: variance -$ \tau =\frac{1}{\sigma^2} $ : precision

# generation of observed data
N = 10
mu = 10.
sigma = 2.
X = np.random.normal(mu, sigma, N)
x = np.arange(3,18,0.01)
p_x = scipy.stats.norm.pdf(x, loc=mu, scale=sigma)
plt.plot(x, p_x, label="true Gaussian")
plt.plot(X, np.zeros_like(X), "ro", label="Data")
plt.title("")
plt.xlabel("x")
plt.ylabel("p(x)")
plt.legend();

Exercises

Exercise - Mean Field Approximation of the Posterior

Task:

Find the mean field approximation of the posterior:

$ p(\mu, \tau \mid X) \approx q(\mu)q(\tau) $.

  • The observed data was sampled from a Gaussian distribution:$ X \sim \mathcal N(\mu, \frac{1}{\tau}) $.
  • Use a constant prior for the mean and the precision$ \tau = \frac{1}{\sigma^2} $: $ p(\mu, \tau) = const. \quad \text{ for } \tau > 0 $ $ \theta = (\theta_0, \theta_1) = (\mu, \tau) $

Note: Typically the mean-field approximation in closed form is done with the conjugate distributions. But here we use the constant (improper) prior because it's a little bit easier.

Recap: Mean field approximation

Loop until convergence: $ \log q({\theta}_k) = \mathbb E_{q_{-k}} \left[ \log{\hat p( {\theta} \mid {\mathcal D} )} \right] + const. $

Hint:

Use the "Sum of difference of the mean": $ \sum_{i=1}^n (x_i-\mu)^2 = \sum_{i=1}^n(x_i-\bar{x})^2 + n(\bar{x} -\mu)^2 $ with $ \bar{x} = \frac{1}{n}\sum_{i=1}^n x_i $

Exercise - Proof

Task:

Show that: $ q(\mu) = \mathcal N(\bar{X}, \frac{1}{\gamma_1}) $ with$ \gamma_1 = m \mathbb E_{q_{\tau}}[\tau] $

and

$ q(\tau) = \text{Gamma}(\frac{m}{2}+1, \frac{2}{\gamma_2}) $

with $ \gamma_2= \sum_{i=1}^m \left(X_i^2\right)-m\bar X^2 + m\text{var}_{q_{\mu}}(\mu) $

Gamma distribution

$ \text{Gamma}(k, \theta') = p(x) = x^{k-1}\frac{e^{-x/\theta'}}{\theta'^k\Gamma(k)} $ -$ k $ is the shape -$ \theta' $ the scale, -$ \Gamma(.) $ is the Gamma function

Exercise - Implementation of the Mean Field Approximation

Task:

Implement the mean field approximation. Like always, you are free to implement as many helper functions as you want.

If everything is correct, executing the cells below should plot figures similar to these:

def optim(X=X,  mean_mu=1, sigma_quare_mu=1, loc_tau=1.,  scale_tau=1.):
    raise NotImplementedError()
mean_mu, sigma_quare_mu, loc_tau, scale_tau = optim(X)
mean_mu, sigma_quare_mu, loc_tau, scale_tau
plt.figure(figsize=(12,4))

x = np.arange(5,15,0.01)
p_mu = scipy.stats.norm.pdf(x, loc=mean_mu, scale=np.sqrt(sigma_quare_mu))
ax = plt.subplot(121)
ax.plot(x, p_mu)
ax.set_xlabel("$\mu$")
ax.set_ylabel("q($\mu$)")
ax.set_title("Mean: q($\\mu$)")
print("true mu: ", mu)

x = np.arange(0,1,0.01)
p_tau = scipy.stats.gamma.pdf(x, a=loc_tau, scale=scale_tau)
ax = plt.subplot(122)
ax.plot(x, p_tau)
ax.set_xlabel("$\\tau$")
ax.set_ylabel("q($\\tau$)")
ax.set_title("Precision: q($\\tau$)")
print("true tau: ", 1/sigma**2)

Literature

[TODO]

Licenses

Notebook License (CC-BY-SA 4.0)

The following license applies to the complete notebook, including code cells. It does however not apply to any referenced external media (e.g., images).

Exercise - Variational Mean Field Approximation for Univariate Gaussian
by Christian Herta
is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at https://gitlab.com/deep.TEACHING.

Code License (MIT)

The following license only applies to code cells of the notebook.

Copyright 2018 Christian Herta

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.