Maximum-likelihood fitted value

A maximum-likelihood fitted value is a response mean or probability evaluated at a maximum-likelihood estimate. In a parametric mean model ${\mu }_i(\theta )$ it is ${\hat{\mu }}_i={\mu }_i(\hat{\theta })$.

A maximum-likelihood fitted probability evaluates a model probability at a maximum-likelihood estimate: $\hat{p}(x)=p_{\hat{\theta }}(x)$. For independent Bernoulli trials with common success probability $p$, the fitted probability is $\hat{p}=K/n$, the observed success fraction. In logistic regression it instead varies with the predictor through $\hat{p}(x)=(1+e^{-x^T\hat{\beta }}{)}^{-1}$. It is an estimated event probability, rather than the probability that the fitted model is correct or a p-value.