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<center><math>p(x|\mathcal{D}) = \int p(x|\mu)p(\mu|\mathcal{D})d\mu</math></center> | <center><math>p(x|\mathcal{D}) = \int p(x|\mu)p(\mu|\mathcal{D})d\mu</math></center> | ||
| + | <center><math>p(x|\mathcal{D}) = \int \frac{1}{\sqrt{2 \pi } \sigma} \exp[{-\frac{1}{2} (\frac{x-\mu}{\sigma})^2}] \frac{1}{\sqrt{2 \pi } \sigma_n} \exp[{-\frac{1}{2} (\frac{\mu-\mu_n}{\sigma_n})^2}] d\mu</math></center> | ||
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<center><math>p(x|D) \sim N(\mu_n, \sigma^2 + \sigma_n^2)</math></center> | <center><math>p(x|D) \sim N(\mu_n, \sigma^2 + \sigma_n^2)</math></center> | ||
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Latest revision as of 07:31, 29 April 2014
Bayesian Parameter Estimation: Gaussian Case
A slecture by ECE student Shaobo Fang
Loosely based on the ECE662 Spring 2014 lecture material of Prof. Mireille Boutin.
Contents
Introduction: Bayesian Estimation
According to Chapter #3.3 (Duda's book), although the answers we get by BPE will generally be nearly identical to those obtained by maximum likelihood estimation, the conceptual difference is significant. For maximum likelihood estimation, the parameter $ \theta $ is a fixed while in Bayersian estimation $ \theta $ is considered to be a random variable.
By definition, given samples class $ \mathcal{D} $, Bayes' formula then becomes:
$ P(w_i|x,D) = \frac{p(x|w_i,D)P(w_i|D)}{\sum_{j = 1}^c p(x|w_j,D)P(w_j|D)} $
As the above equation suggests, we can use the information provided by the training data to help determine both the class-conditional densities and the priori probabilities.
Furthermore, since we are treating supervised case, we can separate the training samples by class into c subsets $ \mathcal{D}_1, \mathcal{D}_2, ..., \mathcal{D}_c $, accordingly:
$ P(w_i|x,D) = \frac{p(x|w_i,D_i)P(w_i)}{\sum_{j = 1}^c p(x|w_j,D_j)P(w_j)} $
Now, assume $ p(x) $ has a parameter form. We are given a set of $ N $ independent samples $ \mathcal{D} = \{x_1, x_2, ... , x_N \} $. View $ \theta $ as a random variable. Consider more specifically in continuous case:
$ p(x|D) $ can be computed as:
$ p(x|D) = \int p(x, \theta|D)d\theta = \int p(x|\theta)p(\theta|D)d\theta $
Bayesian Parameter Estimation: General Theory
We first start with a generalized approach which can be applied to any situation in which the unknown density can be parameterized. The basic assumptions are as follows:
1. The form of the density $ p(x|\theta) $ is assumed to be known, but the value of the parameter vector $ \theta $ is not known exactly.
2. The initial knowledge about $ \theta $ is assumed to be contained in a known a priori density $ p(\theta) $.
3. The rest of the knowledge about $ \theta $ is contained in a set $ \mathcal{D} $ of n samples $ x_1, x_2, ... , x_n $ drawn independently according to the unknown probability density $ p(x) $.
Accordingly, already know:
and By Bayes Theorem,
Now, since we are attempting to transform the equation to be based on samples $ x_k $, by independent assumption,
Hence, if a sample $ \mathcal{D} $ has n samples, we can denote the sample space as: $ \mathcal{D}^n = \{x_1, x_2, ... x_n\} $.
Combine the sample space definition with the equation above:
Using this equation, we can transform the Bayesian Parameter Estimation to:
Bayesian Parameter Estimation: Gaussian Case
The Univariate Case: $ p(\mu|\mathcal{D}) $
Consider the case where $ \mu $ is the only unknown parameter. For simplicity we assume:
From the previous section, the following expression could be easily obtained using Bayes' formula:
Where $ \alpha $ is a factorization factor independent of $ \mu $.
Now, substitute $ p(x_k|\mu) $ and $ p(u) $ with:
The equation has now become:
Update the scaling factor to $ \alpha' $ and $ \alpha'' $ correspondingly,
With the knowledge of Gaussian distribution:
Finally, the estimate of $ u_n $ can be obtained:
Where $ \bar{x_n} $ is defined as sample means and $ n $ is the sample size.
In order to form a Gaussian distribution, the variance $ \sigma_n^2 $ associated with $ \mu_n $ could also be obtained correspondingly as:
Observation:
$ p(\mu|D) $ becomes more sharply peaked around $ \mu_D $
The Univariate Case: $ p(x|\mathcal{D}) $
Having obtained the posteriori density for the mean $ u_n $ of set $ \mathcal{D} $, the remaining of the task is to estimate the "class-conditional" density for $ p(x|D) $.
Based on the text Duda's chatpter #3.4.2 and Prof. Mimi's notes:
Where $ f(\sigma, \sigma_n) $ is defined as:
Hence, $ p(x|D) $ is normally distributed as:
References
[1]. Mireille Boutin, "ECE662: Statistical Pattern Recognition and Decision Making Processes," Purdue University, Spring 2014.
[2]. R. Duda, P. Hart, Pattern Classification. Wiley-Interscience. Second Edition, 2000.
