Link function
A function used to transform the data from a non-continuous distribution into "real space" so it can be used for regression analysis.
Real space means there are no bounds to the numbers - they range from
It is denoted with
| Observation Model | Observation model | Typical Link Function |
|---|---|---|
| Gaussian Observation Model | $$p(y_i\textbar x_i)={1\over\sqrt{2\pi\sigma^2}}e^{-(y_i-\mu_i)^2/\sigma^2}$$ | None (already in real space) |
| Poisson Observation Model | $$p(y_i\textbar E_i\lambda_i)={(E_i\lambda_i)^{y_i}e^{E_i\lambda_i}\over y_i!}$$ | $$g(\lambda_i)=\log(\lambda_i)$$ |
| Bernoulli observation model | $$p(y_i \textbar \theta_i)=\theta_i^{y_i}(1-\theta_i)^{1-y_i}$$ | $$g(\theta_i)=\log\left( {\theta_i\over1-\theta_i}\right)$$ |
| Binomial observation model | $$p(y_i \textbar \theta_i)={n_i \choose y_i}\theta_i^{y_i}(1-\theta_i)^{n_i-y_i}$$ | $$g(\theta_i)=\log\left( {\theta_i\over1-\theta_i}\right)$$ |