Download Two-Way Independent ANOVA (GLM 3)

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) =
•
•
•
•
Slide 10
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
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