6. Likelihood Functions and Maximum Likelihood Estimation

Topics, Class Notes and Code Assignments and Recommended Readings
  • Likelihood and Likelihood Function
    • Likelihood of observing data
    • Likelihhod as a function of parameters based single data value $x_i$: $L(\theta|x_i) \propto f(x_i|\theta)$ or $P(x_i|\theta)$
      • Continous Distribution: $f(x_i|\theta)$
      • Discrete Distribution: $P(x_i|\theta)$
  • Two Parts of Likelihood Principle
    • Law of likelihood: if $L(\theta_1|x > L(\theta|x))$, then $x$ supports $\theta_1$ over $\theta_2$
    • All evidence about $\theta$ is contained in $L(\theta|x)$
  • Maximum Likelihood Estimation with IID Data
    • Maximize likelihood /log-likelihood
    • Score equations (also called gradient)
  • Notes and code
  • Base R Functions for Finding MLE
  • A Technical Tutorial on Optimization [HTML]
    Good for those who want to have a big-picture view of various practical optimization methods.
  • Review Basic Rules of Rules
    • Rules for general functions [link]
    • Rules for logarithmic and exponential functions [link]
  • Written Assignment
    • Guidelines: [HTML]
    • Due: Wednesday, 3/3/2026