Exercise 20.2 - Gaussian times Gaussian is Gaussian

Answers

For this exercise, we have:

\begin{aligned} 𝒩 (x | μ_{1}, λ_{1}^{- 1}) \times 𝒩 (x | μ_{2}, λ_{2}^{- 1}) \\ = & \frac{\sqrt{λ_{1} λ_{2}}}{2 π} \cdot \exp {- \frac{λ_{1}}{2} {(x - μ_{1})}^{2} - \frac{λ_{2}}{2} {(x - μ_{2})}^{2}} \\ = & \frac{\sqrt{λ_{1} λ_{2}}}{2 π} \cdot \exp {- \frac{λ_{1} + λ_{2}}{2} \cdot x^{2} + (λ_{1} μ_{1} + λ_{2} μ_{2}) \cdot x - \frac{λ_{1} μ_{1}^{2} + λ_{2} μ_{2}^{2}}{2}} . \end{aligned}

We now complete the square:

\begin{aligned} \exp {- \frac{λ_{1} + λ_{2}}{2} x^{2} + (λ_{1} μ_{1} + λ_{2} μ_{2}) x - \frac{λ_{1} μ_{1}^{2} + λ_{2} μ_{2}^{2}}{2}} \\ = & c \cdot \exp {- \frac{λ}{2} {(x - μ)}^{2}}, \end{aligned}

where:

λ = λ_{1} + λ_{2},

μ = λ^{- 1} (λ_{1} μ_{1} + λ_{2} μ_{2}) .

The constant factor $c$ is obtained by computing the constant terms inside the exponential, which ends up with (20.20).

solour_lfq

2021-03-24 13:42