What would be the learning outcome from this slecture?

• Basic Theory behind Maximum Likelihood Estimation (MLE)
• Derivations for Maximum Likelihood Estimates for parameters of Exponential Distribution, Geometric Distribution, Binomial Distribution, Poisson Distribution, and Uniform Distribution

Outline of the slecture

1. Introduction
2. Derivation for Maximum Likelihood Estimates (MLE) for parameters of:

Exponential Distribution
Geometric Distribution
Binomial Distribution
Poisson Distribution
Uniform Distribution

1. Summary
2. References

Introduction

The maximum likelihood estimate (MLE) is the value $ \hat{\theta} $ which maximizes the function L(θ) given by L(θ) = f (X1,X2,...,Xn | θ) where 'f' is the probability density function in case of continuous random variables and probability mass function in case of discrete random variables and 'θ' is the parameter being estimated.

In other words,$ \hat{\theta} $ = arg max_θ L(θ), where $ \hat{\theta} $ is the best estimate of the parameter 'θ' . Thus, we are trying to maximize the probability density (in case of continuous random variables) or the probability of the probability mass (in case of discrete random variables)

If the random variables X1,X2,...,Xn $ \epsilon $ R are Independent Identically Distributed (I.I.D.) then,

L(θ) = _i=1 ∏ ⁿ f(xi | θ) . We need to find the value $ \hat{\theta} \epsilon $ θ that maximizes this function.

In the course, Purdue ECE 662, Pattern Recognition and Decision Taking Processes, we have already looked at the MLE of the Normal Distribution and found that to be:

$ \widehat{\mu} = \frac{\sum_{i=1}^{n}x_{i}}{n} $

$ \hat{\sigma{}^{2}} = \frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\widehat{\mu})^{2} $

where, Xi's are the Normal (Gaussian) Random Variables $ \epsilon $ R , 'n' is the number of samples, and $ \widehat{\mu} $ and $ \hat{\sigma{}^{2}} $ are the estimated Mean and estimated Standard Deviation.

Maximum Likelihood Estimate (MLE)for :-

1. Exponential Distribution

Let X1,X2,...,Xn $ \epsilon $

 R be a random sample from the exponential distribution with p.d.f.

f(x,θ)=(1|θ)*exp(−x|θ)

The likelihood function L(θ) is a function of x1, x2, x3,...,xn

L(θ)=(1|θ)*exp(−x1|θ)*(1|θ)*exp(−x2|θ)*...*(1|θ)*exp(−xn|θ)

L(θ)= (1|θⁿ) * exp( _i=1∑ⁿ -xi|θ)

We need to maximize L(θ) . The logarithm of this function will be easier to maximize.

ln[L(θ)] = -n . ln(θ) - (1|θ) _i=1 ∑ ⁿ xi

Setting, its derivative to zero, we have:

(d|dθ) ln[L(θ)] = -n|θ + _i=1∑ⁿ (-xi| θ²) = 0

which implies that $ \hat{\theta} = \frac{\sum_{i=1}^{n}x_{i}}{n} = \overline{X} $

 = Mean of x1, x2, x3,...,xnand this is the maximum likelihood estimate.

2. Geometric Distribution

Let X1,X2,...,Xn $ \epsilon $

 R be a random sample from the geometric distribution with p.d.f. 

f(x,p)=(1−p)^x-1.p

where x=1,2,3,.... and $ 0 \leq  p \leq  1 $ 

The likelihood function is given by:

L(p)=(1−p)^x1-1.p.(1−p)^x2-1.p.(1−)^x3-1.p...(1−p)^xn-1.p

L(p)=pⁿ.(1-p)^{i=1 ∑ n xi-n}}

The log-likelihood is:

ln L(p)=n . ln(p) + _i=1∑ⁿxi-n . ln(1-p)

Setting its derivative to zero, we have:

(d|dp) ln. L(p) = (n|p) - (_i=1∑ⁿ xi-n|(1-p)) = 0

which implies that

$ \hat{p} = \frac{n}{\sum_{i=1}^{n}x_{i}} = \frac{1}{\overline{X}} $

The likelihood function is a function of x1,x2,...,xn,

which is the maximum likelihood estimate.

This can be intuitively checked as well. Since, Geometric Distribution is used to model a random variable X which is the number of trials before the first success is obtained. So for random variables X1,X2,...,Xn, these contain n

 successes in X1+ X2 +...+ Xn trials. Thus, the estimate of p
 is the number of successes divided by the total number of trials.

3. Binomial Distribution

Let X1,X2,...,XN $ \epsilon $

 R be the Binomially Distributed Random Variables.

Binomial Distribution is used to model x

 successes in n
 Bernoulli trials. Its p.d.f. is given by:

$ f(x) = \frac{n!}{x!(n-x)!} p^{^{x}} (1-p)^{n-x} $

The likelihood function L(p) is given by:

$ L(p) = \prod_{i=1}^{n} f(x_{i}) = \prod_{i=1}^{N} \frac{n!}{x_{i}!(n-xi)!} p^{x_{i}} (1-p)^{n-x_{i}} $

The log-likelihood is:

$ lnL(p) = \sum_{i=1}^{N} ln (n!) - \sum_{i=1}^{N} ln (x_{i}! ) - \sum_{i=1}^{N} (n-x_{i}! ) + \sum_{i=1}^{N} xi.ln(p) + (n - \sum_{i=1}^{N} xi) . ln(1-p) $

Setting its derivative with respect to p

 to zero,

$ \frac{d}{dp} lnL(p) = \frac{1}{p} . \sum_{i=1}^{N} xi - \frac{1}{1-p} \sum_{i=1}^{N} (n - xi) = 0 $

which implies,

$ \frac{1}{p} . \sum_{i=1}^{N} xi = (\frac{1}{1-p})( N .n - \sum_{i=1}^{N} xi) $

which gives,

$ \hat{p} = \frac{1}{N} (\frac{\sum_{i=1}^{N}x_{i}}{n} ) = \frac{1}{N} (\frac{X1}{n} + \frac{X2}{n} + ... + \frac{XN}{n} $

); which is the maximum likelihood estimate.

This is intuitively correct too, as this is the average of the ratio \frac{X_{i}}{n}

 for each X_{i}
, which is intuitively average of 'p'
 for each X_{i}

4.Poisson Distribution

Let X1,X2,...,Xn $ \epsilon $

 R be a random sample from a Poisson distribution

The p.d.f. of a Poisson Distribution is :

$ f(x) = \frac{\lambda^{x}e^{-\lambda}}{x!} $

 where x =0,1,2,...

The likelihood function is: $ L(\lambda ) = \prod_{i=1}^{n} \frac{\lambda^{x_{i}}e^{-\lambda}}{x_{i}!} = e^{-\lambda n} \frac{\lambda\sum_{i=1}^{n}x_{i}}{\prod_{i=1}^{n}x_{i}} $

The log-likelihood is:

$ ln L(\lambda ) = - \lambda n + \sum_{i=1}^{n} xi . ln(\lambda ) - ln( \prod_{i=1}^{n} xi) $

Setting its derivative to zero, we have:

$ \frac{d}{d\lambda} ln L(\lambda ) = -n + \sum_{i=1}^{n} xi .\frac{1}{\lambda} = 0 $

$ \widehat{\lambda} = \frac{\sum_{i=1}^{n}x_{i}}{n} = \overline{X} $

 which is the maximum likelihood estimate

5.Uniform Distribution

For Uniformly Distributed random variables X1,X2,...,Xn $ \epsilon $

 R, the density function is given by:

$ f(xi) = \frac{1}{\theta} if 0 \leq xi \leq \theta $

$ f(x) = 0 $ otherwise

If the uniformly distributed random variables are arranged in the following order

$ X1\leq X2 \leq X3 ... \leq Xn and 0 \leq X1\leq X2 \leq X3 ... \leq Xn \leq \theta $

, 

Now, the likelihood function is given as:

$ L(\theta ) = \prod_{i=1}^{n} f(xi) = \prod_{i=1}^{n} \frac{1}{\theta} = \theta{}^{-n} $

The log-likelihood is:

$ ln L(\theta ) = -n ln (\theta ) $

Setting its derivative to zero, we get:

$ \frac{d}{d\theta} ln L(\theta ) = \frac{-n}{\theta} which is < 0 for \theta >0 $

Hence, L($ \theta $

) is a decreasing function and it is maximized at $ \theta $
 = xn. The maximum likelihood estimate is thus,

$ \hat{\theta} = xn $

Summary

Using the usual notations and symbols,

1) Normal Distribution:

$ f(x,\mu ,\sigma ) = \frac{1}{\sigma \sqrt(2\pi)} exp(-\frac{1}{2}(\frac{x-\mu}{\sigma})^{2} ) $

$ \widehat{\mu} = \frac{\sum_{i=1}^{n}x_{i}}{n} $

$ \hat{\sigma{}^{2}} = \frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\widehat{\mu})^{2} $

2) Exponential Distribution:

f(x,$ \lambda $)=(1|$ \lambda $)*exp(−x|$ \lambda $)$ \epsilon $ R

$ \widehat{\lambda} = \frac{\sum_{i=1}^{n}x_{i}}{n} = \overline{X} $

3) Geometric Distribution:

f(x, p) = (1−p)x-1. p ; X1,X2,...,Xn $ \epsilon $

R

$ \hat{p} = \frac{n}{\sum_{i=1}^{n}x_{i}} = \frac{1}{\overline{X}} $

4) Binomial Distribution:

$ f(x) = \frac{n!}{x!(n-x)!} p^{^{x}} (1-p)^{n-x} $

; X1,X2,...,XN $ \epsilon $
R

$ \hat{p} = \frac{1}{N} (\frac{\sum_{i=1}^{N}x_{i}}{n} ) = \frac{1}{N} (\frac{X1}{n} + \frac{X2}{n} + ... + \frac{XN}{n} $

5) Poisson Distribution:

$ f(x) = \frac{\lambda^{x}e^{-\lambda}}{x!} ; X1,X2,...,Xn\epsilon R \widehat{\lambda} = \frac{\sum_{i=1}^{n}x_{i}}{n} = \overline{X} $

6) Uniform Distribution:

$ For, X1,X2,...,Xn\epsilon R f(xi) = \frac{1}{\theta} if 0 \leq xi\leq \theta f(xi) = 0, otherwise, and 0 \leq X1\leq X2 \leq X3 ... \leq Xn \leq \theta \hat{\theta} = Xn $

References

1) A module on Maximum Likelihood Estimation - Examples by Ewa Paszek

2) Lecture on Maximum Likelihood Estimation by Dr. David Levin, Assistant Professor, Univeristy of Utah

3) Partially based on Dr. Mireille Boutin lecture notes for Purdue ECE 662 - Pattern Recognition and Decision Making Processes