#probabilistic-pca

Two noisy thermometers in the same room are modelled with probabilistic PCA: $\mathbf{x} = \boldsymbol\mu + \mathbf{w}z + \boldsymbol\epsilon$, with one latent temperature deviation $z \sim \mathcal{N}(0, 8)$, loading $\mathbf{w} = [1, 1]^\top/\sqrt{2}$ (the identity line) and independent sensor noise $\boldsymbol\epsilon \sim \mathcal{N}(\mathbf{0}, I)$. What correlation between the two readings does this model imply?