The product of two Gaussians
Chapter 3 · Multiply the density N(x | μ₀, σ₀²) by the density N(x | μ₁, σ₁²), as functions of the same x. The result has a Gaussian shape: precisions (1/variance) add, and the mean is the precision-weighted average of μ₀ and μ₁. The product is not itself a normalised density, and its area is a constant that depends on how far apart the two means are.
Examples:
━ N(x | μ₀, σ₀²) ━ N(x | μ₁, σ₁²) ━ their product (filled, not normalised) ╍ the normalised Gaussian N(x | μ, σ²) with the same shape. The axes are fixed.
Precisions 1/σ₀² and 1/σ₁²
Sum of precisions 1/σ²
Product variance σ² = σ₀²σ₁²/(σ₀² + σ₁²)
Product mean μ
Area c, from the formula N(μ₀ | μ₁, σ₀² + σ₁²)
Area under the product, found numerically
Read N(x | μ₀, σ₀²) as a prior and N(x | μ₁, σ₁²) as a likelihood centred on a measured value μ₁. The product is then the unnormalised posterior. Its precision is the prior precision plus the data precision, and its mean sits between μ₀ and μ₁, closer to the more certain one (the smaller variance). The product is always narrower than both inputs. This is exactly what happens to the weights in Bayesian linear regression.