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

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

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

4. 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!

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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. Likelihood in practice
Generate data

14. 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

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