Difference between revisions of "Lecture 7 - MLE and BPE OldKiwi" - Rhea

Revision as of 20:24, 16 March 2008

ECE662 Main Page

Class Lecture Notes

LECTURE THEME :

    - Maximum Likelihood Estimation and Bayesian Parameter Estimation

• • See also:** [Comparison of MLE and Bayesian Parameter Estimation]

• • Parametric Estimation** of Class Conditional Density

.. |classcond1| image:: tex

  :alt: tex: p(\vec{x}|w_i)

.. |vectheta1| image:: tex

  :alt: tex: \vec{\theta}

The class conditional density |classcond1| can be estimated using training data. We denote the parameter of estimation as |vectheta1|. There are two methods of estimation discussed.

MLE ([Maximum Likelihood Estimation])

BPE ([Bayesian Parameter Estimation])

.. |Dsample1k| image:: tex

  :alt: tex: D_1, \ldots, D_c

.. |classes1k| image:: tex

  :alt: tex: \omega_1, \ldots, \omega_c

.. |Di| image:: tex

  :alt: tex: D_i

.. |Dj| image:: tex

  :alt: tex: D_j, i \neq j

• • Maximum Likelihood Estimation**

Let "c" denote the number of classes. D, the entire collection of sample data. |Dsample1k| represent the classification of data into classes |classes1k|. It is assumed that:

  - Samples in |Di| give no information about the samples in |Dj|, and
  - Each sample is drawn independently.

.. |vec_thetai| image:: tex

  :alt: tex: \vec{\theta_i}

.. |X_normal| image:: tex

  :alt: tex: X ~ N(\mu,\sigma^2)

.. |theta1| image:: tex

  :alt: tex: \vec{\theta}=[\mu,\sigma^2]

Example: The class conditional density |classcond1| depends on parameter |vec_thetai|. If |X_normal| denotes the class conditional density; then |theta1|.

.. |D_collecX| image:: tex

  :alt: tex: D=\{\vec{X_1}, \ldots, \vec{X_n}\}

Let n be the size of training sample, and |D_collecX|. Then,

.. |XgivenOmTh| image:: tex

  :alt: tex: p(\vec{X}|\omega_i,\vec{\theta_i})

.. |XgivenTh| image:: tex

  :alt: tex: p(\vec{X}|\vec{\theta})

.. |likelihood1| image:: tex

  :alt: tex: p(D|\vec{\theta})=\displaystyle \prod_{k=1}^n p(\vec{X_k}|\vec{\theta})

|XgivenOmTh| equals |XgivenTh| for a single class.

The **Likelihood Function** is, then, defined as |likelihood1|

which needs to be maximized for obtaining the parameter.

.. |loglikelihood1| image:: tex

  :alt: tex: l(\vec{\theta})=log p(D|\vec{\theta})=\displaystyle log(\prod_{k=1}^n p(\vec{X_k}|\vec{\theta}))=\displaystyle \sum_{k=1}^n log(p(\vec{X_k}|\vec{\theta}))

Since logarithm is a monotonic function, maximizing the Likelihood is same as maximizing log of Likelihood which is defined as |loglikelihood1|.

"l" is the log likelihood function.

Maximize log likelyhood function with respect to |jinha_theta|

.. |jinha_theta| image:: tex

  :alt: tex: \vec{\theta}

|jinha_est_theta|

.. |jinha_est_theta| image:: tex

  :alt: tex: \rightarrow \hat{\theta} = argmax \left( l (\vec{\theta}) \right)

If |jinha_ltheta| is a differentiable function

.. |jinha_ltheta| image:: tex

  :alt: tex: l(\vec{\theta})

Let |jinha_vectheta| be 1 by p vector, then

.. |jinha_vectheta| image:: tex

  :alt: tex: \vec{\theta} = \left[ \theta_1, \theta_2, \cdots , \theta_p \right]

|jinha_nabia|

.. |jinha_nabia| image:: tex

  :alt: tex: \nabla_{\vec{\theta}} = \left[ \frac{\partial}{\partial\theta_1} \\ \frac{\partial}{\partial\theta_2} \\ \cdots \\ \frac{\partial}{\partial\theta_p} \right]^{t}

Then, we can compute the first derivatives of log likelyhood function,

|jinha_fd_ltheta|

.. |jinha_fd_ltheta| image:: tex

  :alt: tex: \rightarrow \nabla_{\vec{\theta}} ( l (\vec{\theta}) ) = \sum_{k=1}^{n} \nabla_{\vec{\theta}} \left[  log(p(\vec{x_k} | \vec{\theta})) \right]

and equate this first derivative to be zero

|jinha_fd_0|

.. |jinha_fd_0| image:: tex

  :alt: tex: \rightarrow \nabla_{\vec{\theta}} ( l (\vec{\theta}) ) = 0

• • Example of Guassian case**

Assume that covariance matrix are know.

|jinha_mult_normal|

.. |jinha_mult_normal| image:: tex

  :alt: tex: p(\vec{x_k} | \vec{\mu}) = \frac{1}{ \left( (2\pi)^{d} |\Sigma| \right)^{\frac{1}{2}}} exp \left[ - \frac{1}{2} (\vec{x_k} - \vec{\mu})^{t} \Sigma^{-1} (\vec{x_k} - \vec{\mu}) \right]

• • Step 1: Take log**

|jinha_log_normal|

.. |jinha_log_normal| image:: tex

  :alt: tex: log p(\vec{x_k} | \vec{\mu}) = -\frac{1}{2} log \left( (2\pi)^d |\Sigma| \right) - \frac{1}{2} (\vec{x_k} - \vec{\mu})^{t} \Sigma^{-1} (\vec{x_k} - \vec{\mu})

• • Step 2: Take derivative**

|jinha_fd_log_normal|

.. |jinha_fd_log_normal| image:: tex

  :alt: tex: \frac{\partial}{\partial\vec{\mu}} \left( log p(\vec{x_k} | \vec{\mu}) \right) = \frac{1}{2} \left[ (\vec{x_k} - \vec{\mu})^t \Sigma^{-1}\right]^t + \frac{1}{2} \left[ \Sigma^{-1} (\vec{x_k} - \vec{\mu}) \right] = \Sigma^{-1} (\vec{x_k} - \vec{\mu})

• • Step 3: Equate to 0**

|jinha_eqtozero|

.. |jinha_eqtozero| image:: tex

  :alt: tex: \sum_{k=1}^{n} \Sigma^{-1} (\vec{x_k} - \vec{\mu}) = 0

|jinha_eqtozero2|

.. |jinha_eqtozero2| image:: tex

  :alt: tex: \rightarrow \Sigma^{-1} \sum_{k=1}^{n} (\vec{x_k} - \vec{\mu}) = 0

|jinha_eqtozero3|

.. |jinha_eqtozero3| image:: tex

  :alt: tex: \rightarrow \Sigma^{-1} \left[ \sum_{k=1}^{n} \vec{x_k} - n \vec{\mu}\right] = 0

|jinha_eqtozero4|

.. |jinha_eqtozero4| image:: tex

  :alt: tex: \Longrightarrow \hat{\vec{\mu}} = \frac{1}{n} \sum_{k=1}^{n} \vec{x_k}

This is the sample mean for a sample size n.

[MLE Examples: Exponential and Geometric Distributions]

[MLE Examples: Binomial and Poisson Distributions]

Advantages of MLE:

- Simple
- Converges
- Asymptotically unbiased (though biased for small N)

• • Bayesian Parameter Estimation**

For a given class, let |x_khh| be feature vector of the class and |theta_khh| be parameter of pdf of |x_khh| to be estimated.

And let |D_khh| , where |xx_khh| are training samples of the class

.. |x_khh| image:: tex

  :alt: tex: \bf{x}

.. |D_khh| image:: tex

  :alt: tex: D= \{  \mathbf{x}_1, \mathbf{x}_2, \cdots , \mathbf{x}_n  \} \\

.. |xx_khh| image:: tex

 :alt: tex:  \mathbf{x}_1, \mathbf{x}_2, \cdots , \mathbf{x}_n 

Note that |theta_khh| is random variable with probability density |p_theta_khh|

.. |theta_khh| image:: tex

  :alt: tex: \bf{ \theta }

.. |p_theta_khh| image:: tex

 :alt: tex: p( \bf { \theta } )

.. image:: tex

 :alt: tex: \qquad p(\mathbf{x} \vert D)=\displaystyle \int p(\mathbf{x} \vert \mathbf{\theta} ) p(\mathbf{\theta} \vert D) d \mathbf{\theta }

where

.. image:: tex

 :alt: tex: \qquad p(\mathbf{\theta} \vert D)=\frac {\displaystyle p(D \vert \mathbf{\theta} ) p(\mathbf{\theta} )} {\displaystyle  \int p(D \vert \mathbf{\theta} ) p(\mathbf{\theta}  ) d \mathbf{\theta } }

- Example

          Here is a good example: 
          http://www-ccrma.stanford.edu/~jos/bayes/Bayesian_Parameter_Estimation.html

• • EXAMPLE: Bayesian Inference for Gaussian Mean**

The univariate case. The variance is assumed to be known.

Here's a summary of results:

  * Univariate Gaussian density |bi_gm_1|

  * Prior density of the mean |bi_gm_2|

  * Posterior density of the mean |bi_gm_3|

where

  * |bi_gm_4|

  * |bi_gm_5|

  * |bi_gm_6|

Finally, the class conditional density is given by

|bi_gm_7|

.. |bi_gm_1| image:: tex

  :alt: tex:p(x|\mu)\sim N(\mu,\sigma^{2})

.. |bi_gm_2| image:: tex

  :alt: tex:p(\mu)\sim N(\mu_{0},\sigma_{0}^{2})

.. |bi_gm_3| image:: tex

  :alt: tex:p(\mu|D)\sim N(\mu_{n},\sigma_{n}^{2})

.. |bi_gm_4| image:: tex

  :alt: tex:\mu_{n}=\left(\frac{n\sigma_{0}^{2}}{n\sigma_{0}^{2}+\sigma^{2}}\right)\hat{\mu}_{n}+\frac{\sigma^{2}}{n\sigma_{0}^{2}+\sigma^{2}}\mu_{0}

.. |bi_gm_5| image:: tex

  :alt: tex:\sigma_{n}^{2}=\frac{\sigma_{0}^{2}\sigma^{2}}{n\sigma_{0}^{2}+\sigma^{2}}

.. |bi_gm_6| image:: tex

  :alt: tex:\hat{\mu}_{n}=\frac{1}{n}\sum_{k=1}^{n}x_{k} 

.. |bi_gm_7| image:: tex

  :alt: tex: p(x|D)\sim N(\mu_{n},\sigma^{2}+\sigma_{n}^{2}) 

.. |hsantos_sigma1| image:: tex

  :alt: tex: \sigma^{2}

.. |hsantos_sigma2| image:: tex

  :alt: tex: \sigma_{n}^{2}

The above formular can be interpreted as: in making prediction for a single new observatioin, the variance of the estimate will have two components: 1) |hsantos_sigma1| - the inherent variance within the distribution of x, i.e. the variance that would never be eliminated even with perfect information about the underlying distribution model; 2) |hsantos_sigma2| - the variance introduced from the estimation of the mean vector "mu", this component can be eliminated given exact prior information or very large training set ( N goes to infinity);

.. image:: BayesianInference_GaussianMean_small.jpg

The above figure illustrates the Bayesian inference for the mean of a Gaussian distribution, for which the variance is assumed to be known. The curves show the prior distribution over 'mu' (the curve labeled N=0), which in this case is itself Gaussian, along with the posterior distributions for increasing number N of data points. The figure makes clear that as the number of data points increase, the posterior distribution peaks around the true value of the mean. This phenomenon is known as *Bayesian learning*.

• • For more information:**

[Parametric Estimators]