MCMC

Frequentist Approach

A random sample

$\displaystyle X_1,\ldots,X_n$

is regarded as independent and identically distributed (iid) random variables governed by an underlying probability density function $f(x; \theta)$ . A value $\theta$ represents the characteristics of this underlying distribution, and is called a parameter. A point estimate is a ``best guess'' for the true value $\theta$ . Suppose that the underlying distribution is the normal distribution with $(\mu,\sigma^2)$ . Then the sample mean

$\displaystyle \bar{X} = \frac{1}{n}\sum_{i=1}^n X_i$

is in some sense a best guess of the parameter $\mu$ .

Maximum likelihood estimate. Having observed a random sample $(X_1,\ldots,X_n) = (x_1,\ldots,x_n)$ from an underlying pdf $f(x; \theta)$ , we can construct the likelihood function

$\displaystyle L(\theta,\mathbf{x}) = \prod_{i=1}^n f(x_i; \theta),$

and consider it as a function of $\theta$ . Then the maximum likelihood estimate (MLE) $\hat{\theta}$ is the value of $\theta$ which ``maximizes'' the likelihood function $L(\theta,\mathbf{x})$ . It is usually easier to maximize the log likelihood

$\displaystyle \ln L(\theta,\mathbf{x}) = \sum_{i=1}^n \ln f(x_i; \theta).$

Bernoulli trials. Let be a random variable taking value only on 0 and 1. An experiment observing such a variable is called Bernoulli trial. It is determined by the parameter $\theta$ (which represents the probability that ). Suppose that $\mathbf{x} = (x_1,\ldots,x_n)$ are observed from independent Bernoulli trials. Then the joint density function is given by

$\displaystyle f(\mathbf{x};\theta) = \theta^{\sum_{i=1}^n x_i} (1-\theta)^{n-\sum_{i=1}^n x_i}$

By solving the equation

$\displaystyle \frac{\partial\ln L(\theta,\mathbf{x})}{\partial\theta} = {\text... ...a} - {\textstyle\left(n - \sum_{i=1}^n x_i\right)} \frac{1}{1 - \theta} = 0,$

we obtain $\theta^* = \sum_{i=1}^n x_i \big/ n$ . In fact, $\theta^*$ maximizes $\ln L(\theta,\mathbf{x})$ , and therefore, it is the MLE of $\theta$ .