12.5 Summary

A procedure for modelling and inference:

1. 1

   Subject-matter question needs answering.

2. 2

   Data are, or become, available to address this question.

3. 3

   Look at the data – exploratory analysis.

4. 4

   Propose a model.

5. 5

   Check the model fits.

6. 6

   Use the model to address the question.

• 1

  The likelihood function is the probability of the observed data for instances of a parameter. Often we use the log-likelihood function as it is easier to work with. The likelihood is a function of an unknown parameter.

• 2

  The maximum likelihood estimator (MLE) is the value of the parameter that maximises the likelihood. This is intuitively appealing, and later we will show it is a theoretically justified choice. The MLE should be found using an appropriate maximisation technique.

• 3

  If the parameter is continuous, we can often (but not always) find the MLE by considering the derivative of the log-likelihood. If the parameter is discrete, we usually evaluate the likelihood at a range of possible values.

  DON’T JOIN UP POINTS WHEN PLOTTING THE LIKELIHOOD FOR A DISCRETE PARAMETER.

  DO NOT DIFFERENTIATE LIKELIHOODS OF DISCRETE PARAMETERS!