Maximum Likelihood Find the parameters of a model that best fit the data… Forms the foundation of Bayesian inference Slide 1

Distributions of Discrete Variables Random variables (the observed data) Discrete Are integer values Example: Binomial Multinomial Poisson Negative binomial

Distributions of continuous Variables Random variables are continuous Example: Gaussian (normal) Log normal Gamma Beta

PMF of Poisson Probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value P(Yi = k | rate parameter = r) = 𝑒 −𝑟 𝑟 𝑘 𝑘!

PMF of Poisson In one unit of time we predict that Yi = k P(Yi = k | rate parameter = r) = 𝑒 −𝑟 𝑟 𝑘 𝑘!

Likelihood P(Yi | p) Probability distribution of observing data Yi, given a particular parameter value, p Subscript on Y indicates that there are many possible outcomes but only one possible parameter. Slide 7

Likelihood P(Yi = k | rate parameter = r) = 𝑒 −𝑟 𝑟 𝑘 𝑘! This expression is the probability of “data” given the hypothesis. Data are k events in one unit time Hypothesis is that the rate parameter is r Slide 8

Likelihood P(Yi = k | rate parameter = r) = 𝑒 −𝑟 𝑟 𝑘 𝑘! After collection of the data, the data are known. Alternative hypotheses are different values of r. Given the data, how likely are the possible hypotheses? Slide 9

Likelihood P(Yi = k | rate parameter = r) = 𝑒 −𝑟 𝑟 𝑘 𝑘! Introduce symbol: “L” likelihood L(data | hypothesis), L(Y | pm) Shift in thinking – m alternative parameters… One set of data Slide 10

Likelihood P(Yi = k | rate parameter = r) = 𝑒 −𝑟 𝑟 𝑘 𝑘! Difference in likelihood and probability: Probability: the hypothesis is known, data are unknown Likelihood: data are known, hypothesis is not known Slide 11

Likelihood in practice Generate data Determine range of parameter values that are alternative hypotheses Determine the probability that the data came from a distribution with a given parameter value

Likelihood in practice Generate data

Likelihood in practice Determine range of parameter values that are alternative hypotheses best.guess.mu <- seq(15,25,by = 0.1) best.guess.sig <- 5

Likelihood in practice Determine the probability that the data came from a distribution with a given parameter value