Introduction to Likelihood Statistics

Introduction to Likelihood Statistics

Edward L. Robinson* Department of Astronomy and

McDonald Observatory University of Texas at Austin

1. The likelihood function. 2. Use of the likelihood function to model data. 3. Comparison to standard frequentist and Bayesean statistics.

*Look for: "Data Analysis for Scientists and Engineers" Princeton University Press, Sept 2016.

The Likelihood Function

? Let a probability distribution function for have m + 1

parameters aj

(, , , ? ? ? , ) = (, ~ ), f a0 a1 am f a

The joint probability distribution for n samples of is

( , , ? ? ? , , , , ? ? ? , ) = (~, ~ ).

f1 2

n a0 a1 am f a

?

Now make measurements. For each variable i there is a measured

value xi.

?

(~ , ~ )

To obtain the likelihood function L x a , replace each variable i with

the numerical value of the corresponding data point xi:

(~ , ~ ) (~ , ~ ) = ( , , ? ? ? , , ~ ). L x a f x a f x1 x2 xn a

~

~

In the likelihood function the x are known and fixed, while the a are

the variables.

A Simple Example

? Suppose the probability distribution for the data is (, ) = 2 a. f a ae

? Measure a single data point. It turns out to be x = 2.

? The likelihood function is ( = , ) = 2 2a. L x 2 a 2a e

A Somewhat More Realistic Example

Suppose we have n independent data points, each drawn from the same probability distribution

(, ) = 2 a. f a ae

The joint probability distribution is

(~, ) = ( , ) ( , ) ? ? ? ( , ) f a f 1 af 2 a f n a

Yn

=

2 ai.

a ie

i=1

The data points are xi. The resulting likelihood function is

Yn

(~ , ) =

2

axi .

Lx a

a xie

= i1

What Is the Likelihood Function? ? 1

The likelihood function is not a probability distribution. ? It does not transform like a probability distribution. ? Normalization is not defined.

How do we deal with this?

Traditional approach: Use the Likelihood Ratio.

To compare the likelihood of two possible sets of parameters ~a1 and ~a2, construct the likelihood ratio:

LR

=

L(~x, ~a1) L(~x, ~a2)

=

f f

(~x, (~x,

~a1) ~a2)

.

This is the ratio of the probabilities that data ~x would be produced by parameter values ~a1 and ~a2.

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