The Method of Maximum Likelihood

Suppose we are trying to measure the true meanof some quantity. We make repeated measurementsIntuitively we say the true value of the meanis likely to be close to the mean of our measurements,

The maximum likelihood method is a general method for estimating parameters of interest from data.

1. Assume we have mademeasurements of

2. Assume we know the probability distribution function that describeswhere a is the parameter who value we want to estimate.

3. The probability of measuringisthe probability of measuringisthe probability of measuringis

4. If the measurements are independent, the probability of getting the measurementsis

5. We want to maximiseand solve forWe may do this by differentiation. The value of that gives the maximum foralso gives the maximum forFor ease of calculation we may take logs and convert the product into a sum. Either way we solvefor

Example: Letbe given by a Gaussian distribution., letbe the mean of the Gaussian. We want the best estimate oflabelledfrom our set ofmeasurements

so

Taking natural logs givesand differentiating this gives (notice the first term vanishes because it contains no occurrences of)