## The Method of Maximum Likelihood

Suppose we are trying to measure the true mean of some quantity. We make repeated measurements Intuitively we say the true value of the mean is 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 made measurements of 2. Assume we know the probability distribution function that describes where a is the parameter who value we want to estimate.

3. The probability of measuring is the probability of measuring is the probability of measuring is 4. If the measurements are independent, the probability of getting the measurements is 5. We want to maximise and solve for We may do this by differentiation. The value of that gives the maximum for also gives the maximum for For ease of calculation we may take logs and convert the product into a sum. Either way we solve for Example: Let be given by a Gaussian distribution., let be the mean of the Gaussian. We want the best estimate of labelled from our set of measurements  so Taking natural logs gives and differentiating this gives (notice the first term vanishes because it contains no occurrences of )  