 Therefore, we can expand the factor formula_20 about the mean value formula_21, which will coincide with formula_22 for Gaussian fluctuations (i.e.average and most probable values coincide), and retaining lowest order terms result in
This is the Gaussian, or normal, distribution, which is defined by its first two moments.In general, one would need all the moments to specify the probability density, formula_24, which is referred to as the canonical, or posterior, density in contrast to the prior density formula_25, which is referred to as the 'structure' function.