Difference between revisions of "Parametric Estimators OldKiwi" - Rhea

Revision as of 12:50, 7 April 2008

Course Topics

This page and its subtopics discusses about Parametric Estimators

Lectures discussing Parametric Estimators: Lecture 7_OldKiwi and Lecture 8_OldKiwi

• [MORE ON THE MLE]: Issues related to the properties and computational efficiency of the Maximum Likelihood Estimator

The MLE estimator is probably the most important parameter estimator in classical statistics. The reason is that the MLE estimator is asymptotically efficient. That is to say that given a large enough data sample, the estimator will be efficient. Furthermore if $ \hat \theta $ is the MLE estimator of the parameter $ \theta $ , then $ \sqrt{n}({\hat \theta}-\theta) $ will asymptotically converges to the distribution $ \mathcal{N}(0,v(\theta)) $ where $ v(\theta) $ is the Cramer Rao Bound(http://en.wikipedia.org/wiki/Cram%C3%A9r-Rao_inequality).

But what is an efficient estimator? An estimator $ {\hat \theta} $ is efficient if:

1. $ \hat \theta $ here is an unbiased estimator.

1. $ \hat \theta $ achieves the Cramer-Rao Lower Bound(CRLB).

The CRLB is the minimum variance achievable by any unbiased estimator for a parameter. The estimator that is unbiased and achieves the CRLB is referred to as the Minimum Variance Unbiased Estimator(MVUE).

So the MLE is an important estimator because: 1. If an MVUE exists, the MLE procedure will find it. 2. If an MVUE does not exist, the MLE will asymptotically converge to it.

Therefore if the pdf of the model is known the MLE is often a good candidate estimator since it can be computed (although this might not be an easy task) and it is "optimal" for a large enough data set ( although how large is large enough is not always easily answered). The MLE does have some disadvantages in practice: 1. It is not the best method for small data and can give highly erroneous results in some cases. 2. The computation can be extremely difficult and sometimes leads to a plethora of numerical methods such as:

1. Brute Force Method (i.e compute the pdf on a very fine grid and try to get the maximum). Although it can be done, this is very computationally inefficiently.

2. Iterative Methods (i.e. Newton-Raphson which does not guarantee convergence. In fact good initial guess is needed here)

3. Scoring Method

4. Expectation-Maximization: Guarantees convergence to at least a local maximum. A good method for the complicated vector-parameter cases.

There are two methods for estimating parameters that are the center of a heated debate within the pattern recognition community. These methods are Maximum Likelihood Estimation (MLE) and Bayesian parameter estimation. Despite the difference in theory between these two methods, they are quite similar when they are applied in practice.

• [COMPARISON OF MLE AND BAYESIAN PARAMETER ESTIMATION]

Maximum Likelihood (ML) and Bayesian parameter estimation make very different assumptions. Here the assumptions are contrasted briefly:

MLE:

+ Deterministic (single, non-random) estimate of parameters, theta_ML
+ Determining probability of a new point requires one calculation: P(x|theta)
+ No "prior knowledge"
+ Estimate of variance and other parameters is often biased
+ Overfitting solved with regularization parameters

Bayes:

+ Probabilistic (probability density) estimate of parameters, p(theta | Data)
+ Determining probability of a new point requires integration over the parameter space
+ Prior knowledge is necessary
+ With certain specially-designed priors, leads naturally to unbiased estimate of variance
+ Overfitting is solved by the selection of the prior

In the end, both methods require a parameter to avoid overfitting. The parameter used in Maximum Likelihood Estimation is not as intellectually satisfying, because it does not arise as naturally from the derivation. However, even with Bayesian likelihood it is difficult to justify a given prior. For example, what is a "typical" standard deviation for a Gaussian distribution?