Exercise - Estimating Mean and Standard Deviation of Normal Distribution with pyro

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

import numpy as np

import scipy.stats
from scipy.stats import norm

from matplotlib import pyplot as plt
from IPython.core.pylabtools import figsize

%matplotlib inline
import torch
from torch.distributions import constraints

import pyro
import pyro.infer
import pyro.optim
import pyro.distributions as dist

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

dtype=torch.float32
torch.manual_seed(101);
np.random.seed(10)
# generate observed data
N = 10
mu_ = 10.
sigma_=2.
X = np.random.normal(mu_, sigma_, N)
X = np.array(X, dtype=np.float32)
X
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 - Model

Task:

Build the following model with pyro

  • The observed data should come from a Gaussian distribution $ \mathcal N(\mu, \sigma^2) $ .
  • Use a Uniform prior for the mean: $ \text{Uniform}(-25,25) $
  • Use a constant $ \tau=1/4 $ for the precision.

Task:

  • Implement the "Guide".
  • Use as variational distribution also a Gaussian. $ \mu \sim \mathcal N(mean_{\mu}, scale_{\mu}^2) $

Exercise - Estimate Mean

Task:

Optimize the variational parameters, i.e. find vales for $ mean_{\mu}, scale_{\mu} $

X
plt.xlabel("# iteration")
plt.ylabel("MC-Estimate of ELBO")
plt.plot(range(len(losses)), losses)

Exercise - Estimate Precision $ \tau $ (and Mean)

Task:

Extend the model and the Guide by using additionally a variational distribution for $ \tau $ :

  • Use a Uniform distribution for $ \tau \sim \text{Uniform}(0.01, 2) $
  • Use a Gamma distribution as variational distribution for $ \tau $ : $ \text{Gamma}(a, b) $
  • Find the parameters $ a, b $ (and $ mean_{\mu}, scale_{\mu} $ ) via optimization.

If your extensions are correct, executing the cells below should plot figures similar to these:

# Adjust the strings according to your names for
# the parameters "mu_mean", etc...
mu_mean_param = pyro.param("guide_mu_mean")
mu_scale_param = pyro.param("guide_mu_scale")
mu_mean_param, mu_scale_param
# Adjust the strings according to your names for
# the parameters "mu_mean", etc...
tau_concentration_param = pyro.param("guide_tau_concentration")
tau_rate_param = pyro.param("guide_tau_rate")
tau_concentration_param, tau_rate_param
plt.figure(figsize=(12,4))

mu_mean = mu_mean_param.detach().numpy()
mu_scale = mu_scale_param.detach().numpy()

x = np.arange(5,15,0.01)
p_mu = scipy.stats.norm.pdf(x, loc=mu_mean, scale=np.sqrt(mu_scale))
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_)

tau_concentration =tau_concentration_param.detach().numpy()
tau_rate = tau_rate_param.detach().numpy()

x = np.arange(0,1,0.01)
p_tau = scipy.stats.gamma.pdf(x, a=tau_concentration, scale=1/tau_rate)
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.