Fitting Poisson
Overview
Fitting the Poisson distribution to a dataset typically follows the principles of
Maximum Likelihood
The log-likelihood function is given as
{% \mathcal{L}(\mu,y) = \sum_{i=1}^n [y_i ln(\mu_i) - \mu_i - ln \Gamma(y_i + 1)] %}
Typically, the functional form of {% \mu %} is given as
{% \mu_i = exp[\vec{x}_i^T \vec{\beta}] %}
Next, an
optimization
algorithm is applied to find the maximum value of the log-likelihood.
Implementation
The following are different ways to run the optimization.
- Newtons Method
- uses a numerically calculated gradient and hessian matrix to run Newtons algoirthm
