340 words (approximately 2 mins)

Maximum Likelihood Estimation

  • Maximum likelihood estimation – a method of estimating the parameters of a probability distribution by maximizing a likelihood function.
    • It finds the values of parameters (like mean and standard deviation for a normal distribution, or lambda for a Poisson distribution) that result in the curve that best fits the data.
  • Likelihood function – measures the fit (support) of a statistical model to a sample of data for given values of the unknown parameters.
    • It measures the evidential support provided by data for particular parameter values.
    • It is formed from the joint probability distribution of a sample.
    • It treats random variables as fixed at the observed values.
  • Parameter – a value that summarizes or describes an aspect of a statistical population.
  • Confidence interval – an estimated interval within which an unknown parameter may plausibly lie.
    • It describes ranges of parameter values that are consistent with data.

Probability model parameters (mean, SD, etc) can be estimated from the data via MLE

  • Estimating parameters is fitting the model to the data.
  • Parametric distributions, regression models, Markov chain models, etc. can all be fit by MLE.

Poisson rate (lambda) can be estimated from the data with MLE

  • n = arrivals
  • lambda = expected arrivals
  • dpois(n, lambda) = probability of n arrivals if lambda arrivals are expected.
  • Keep n fixed and vary lambda to find the maximum value.

If n is known (fixed), and lambda is varied, this returns a likelihood function.

  • For Poisson, the MLE of lambda = n / time.

Poisson Regression

  • Normally-distributed random variables can be extended to let their means depend on other variables via simple formulas: E(Y | x) = a + bx (expected value of Y given x)
  • Poisson probability models can be extended similarly: lambda = a + bx
  • This results in a Poisson regression model.
    • The parameters in this model are estimated from data by MLE.

Calculate Confidence Intervals for Mean of a Poisson Distribution

install.packages("DescTools")
library(DescTools)
PoissonCI(n)			# return the confidence interval for n where n is the mean of a Poisson distribution