or what is called a joint probability, the probability of simultaneously observing all m events. Another way of writing this is
where I use product notation in the last step.
The likelihood
The joint probability function is a function of the data, the x-values. The model parameter λ is assumed to be a fixed value in nature. To fix this idea we use the following notation in which a semicolon is used to separate the quantities that are random (the data) from the quantity that is fixed (the parameter of the probability model).
This is the probability of our data. If we knew λ we could calculate the probability of obtaining any set of values x_{1}, x_{2}, ... , x_{m}. Furthermore, for fixed λ if we summed this expression over all possible values of x_{1}, x_{2}, ... , x_{m} we would get 1.
Viewed this way it's no longer a probability. (For fixed data if we sum over all possible values of λ we will not get 1.) So instead we call this function the likelihood function. Keep in mind that it is still the joint probability function for our data under the assumed probability model only by another name.
Jack Weiss Phone: (919) 962-5930 E-Mail: jack_weiss@unc.edu Address: Curriculum in Ecology, Box 3275, University of North Carolina, Chapel Hill, 27516 Copyright © 2007 Last Revised--Feb 18, 2007 URL: http://www.unc.edu/courses/2007spring/enst/562/001/docs/lectures/lecture14.htm |