Multivariate Gaussian detection with  yields the LRT statistic proportional to:

If you impose a Gaussian prior on weights in a linear-Gaussian model, the MAP estimate corresponds to:

For Gaussian binary detection with equal priors and equal costs, the LRT threshold simplifies to

For an inhomogeneous Poisson process with intensity , the point-process likelihood over has the form:

Poisson-to-Gaussian approximation is most defensible when:

Under the same model, with , the decision can be written as:

The GLRT statistic for unknown amplitude (Gaussian) reduces to a normalized matched-filter energy of the form:

In the inhomogeneous Poisson log-likelihood ratio between and , the spike-time terms appear as plus an integral “baseline correction.”

In Gaussian conjugate models, the posterior mean can be expressed as a precision-weighted combination of the prior mean and the sample mean.

For univariate Gaussian detection , the false-alarm probability has the generic form: