# #StackBounty: #covariance #gaussian-process #conditional-expectation #prediction-interval Are the conditional expectation values of y a…

### Bounty: 50

Suppose $$y$$ is a Gaussian process given by $$y sim f + epsilon$$, where $$epsilon$$ is a Gaussian noise model with zero mean, and $$f$$ is a deterministic yet unknown mean function (or a Gaussian process independent of $$epsilon$$). Therefore, one would find that $$mathbb{E}[y] = mathbb{E}[f]$$ since $$mathbb{E}[epsilon] = 0$$. But my question is: does $$mathbb{E}[{{ bf y}_b vert { bf y}_a}] = mathbb{E}[{{ bf f}_b vert { bf y}_a}]$$? Namely, are the conditional means of $$bf f_b$$ and $$bf y_b$$ equivalent?

The reason I ask is because we know $$text{Var}[y] neq text{Var}[f]$$ and $$text{Var}[{{ bf y}_b vert { bf y}_a}] neq text{Var}[{{ bf f}_b vert { bf y}_a}]$$. Additionally, the covariance matrix of $$y$$ is given by: $$Sigma_y(x_1,x_2) = k(x_1,x_2) + sigma^2 (x_1) delta(x_1 – x_2),$$ while the covariance matrix of $$f$$ is given by (c.f. the lines below equations 5.8 or below 2.30): $$Sigma_f(x_1,x_2) = k(x_1,x_2),$$ i.e. $$y$$ has an additional (possibly) heteroscedastic noise model, $$sigma$$, added along the diagonal of covariance matrix to represent the variance of the noise, $$epsilon$$. But after observing a set of measurements, $$boldsymbol y_a$$ at inputs $$boldsymbol x_a$$, the conditional mean of $$boldsymbol y_b$$ is given by:

$$mathbb{E}[{ bf y}_b vert { bf y}_a] = boldsymbolmu_b+Sigma_y(boldsymbol x_b,boldsymbol x_a){Sigma_y(boldsymbol x_a,boldsymbol x_a)}^{-1}({boldsymbol y_a}-boldsymbolmu_a)$$

but the conditional mean of $$boldsymbol f_b$$ is given by:

$$mathbb{E}[{ bf f}_b vert { bf y}_a] = boldsymbolmu_b+Sigma_f(boldsymbol x_b,boldsymbol x_a){Sigma_f(boldsymbol x_a,boldsymbol x_a)}^{-1}({boldsymbol y_a}-boldsymbolmu_a)$$

Therefore since $$Sigma_y(x_1,x_2)$$ does not necessarily equal $$Sigma_f(x_1,x_2)$$, is it accurate to state that $$mathbb{E}[{{ bf y}_b vert { bf y}_a}]$$ does not necessarily equal $$mathbb{E}[{{ bf f}_b vert { bf y}_a}]$$?

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