For any likelihood that allows a second-order Taylor series expansion, we can approximate $\ell(\bt \vert \x)$ with the log-likelihood from a multivariate normal distribution:

where $\u$ and $\oI$ are the score and information evaluated at $\btt$, the point about which we are taking the Taylor series approximation, and $\y = \btt - \H^{-1}\u = \btt + \oI^{-1}\u$ is the “pseudoresponse”.

Note here that $\u$ is the random variable; in reality, the point $\btt$ that we take the approximation about is often random as well, but this is not accounted for here. In particular, if $\btt$ is the MLE, then $\y=\bth$ and we have the typical Wald approximation.