Bayersian Parameter Estimation - Rhea

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:

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:

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:

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,

Furthermore, since

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 $ is defined as:

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 by \textbf{Duda's},

Where $ f(\sigma, \sigma_n) $ is defined as:

Hence, $ p(x|D) $ is normally distributed as:

\subsection{Experiment of Bayesian Parameter Estimation}

\paragraph{Design}

Assume n samples were obtained from the class $ \mathcal{D} $ of unknown mean $ \mu $ (known $ \sigma $). Assume,

$ p(x|\mu) \sim N(\mu, \sigma^2) $

$ p(\mu) \sim N(\mu_0, \sigma_0^2) $

While $ \sigma = \sigma_0 = constant $, and $ \mu_0 = 0 $ (It does not matter what $ \mu_0 $ it was assumed to be, this will be verified shortly after). Based on the sample data $ x_i \in \mathcal{D}, i = 1,2,3,...,n $, $ \mu $ is desired to be estimated.

The following results will be obtained: \begin{enumerate} \item The impact of $\mu_0$ on estimated $\hat{\mu}$ \item The impact of sample size $n$ have on estimation accuracy \end{enumerate}

\paragraph{Results} \begin{center} \includegraphics[scale=1]{BPE_1.png}

Figure 21. The impact of $\mu_0$ on estimated $\hat{\mu}$ averaged over 50 samples

\includegraphics[scale=1]{BPE_2.png}

Figure 22. The impact of $\mu_0$ on the variance of estimated $\hat{\mu}$ over 50 samples

\end{center}

The estimated mean is shifting up with $\mu_0$ increasing. \textbf{Based on the experiment it can be concluded that the most 'accurate' estimate could be obtained if $ \mu_0 = \mu $. But, according to the plot, even if the $\mu_0$ is different different from $\mu$, the error of estimation is still acceptable. (In our case, within [-0.1,+0.06] region)} However, the variance of estimated mean could be assumed to be identical as the \textbf{real empirical mean}.

\begin{center} \includegraphics[scale=0.7]{e23456.png}

Figure 23. The impact of sample size $n$ have on estimation shape accuracy (sample sizes = 2,3,4,5,6)

\includegraphics[scale=0.6]{ece662_14.png}

Figure 24. The impact of sample size $n$ have on estimation shape accuracy (sample sizes = 4,10,20,50,100)

\paragraph{Conclusion} Figure 23. and Figure 24. have demonstrated that with insufficient sample size the result would be really poor regarding prediction of points distribution.

Fig 3: Summary diagram of whitening and coloring process.

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.