Marginal calculation in linear gaussian model

Question

If we have a linear gaussian model where we have,

$P(Z) = \mathcal{N}(\mu_z,\sigma^2_z)$ and

$P(X|Z) = \mathcal{N}(w_o+zw,\sigma^2_x)$

How do we calculate the marginal

$P(X) = \int_Z P(Z)P(X|Z)$ ?

Answer 1

The models you wrote are equivalent to:

$$Z = μ_z + e_z \text{, where } e_z \text{ ~ } N(0,σz^2)$$

$$X = w_0 + w_Z + e_x \text{, where } e_x \text{ ~ } N(0,σx^2) \text{ and } e_x \text{ independent of } $$ $$e_z = w_0 + w μ_z + w e_z + e_z $$

The sum of the last two terms is a linear of combination of zero mean Gaussians.

The expected value of $X$ is

$$E(X) = w_0 + w μ_z$$

The variance of $X$ is

$$var(X) = var(w e_z) + var(e_x) = w^2 σz^2 + σx^2$$

since the variance of a sum of independent variables is the sum of the individual variances.

Finally, a linear combination of independent Gaussians is also Gaussian, proved easily by examining the characteristic function (e.g. ref *). Therefore the marginal distribution of X is:

$$X ~ N (w0 + w μ_z, w^2 σz^2 + σx^2)$$

• https://www.projectrhea.org/rhea/index.php/Linear_combinations_of_independent_gaussian_RVs