Maximum likelihood

Gist

Given the data we have, what parameter best describe what we observe. One way to fit model.

Here, we have a likelihood function:

L (θ; Y)

Where Theta is the parameter and Y is the given observations. This is also the same as the joint probability of the observations.

Note that we often work with the log likelihood as the product becomes a sum.

Be wary of using it for fitting incidence data

When I was fitting RSV, negative binomial penalty was weird because of the variacne

If you assume that the data is normally distributed (see Distribution - Normal, this is like finding the Least squares

Given a

Estimate on the unconstrained scale. Lot of the biological rates in the model such as in the SIR is positive. For example, $β$ in the SIR model should be $l o g β$ . Back transofrm at the end.
Return a large finite value on failure, not NA: If the ODE solver struggles at some parameter combination, 1e12 tells the optimizer to go the other way