We know that in probabilistic PCA, where $$x = Wz + \mu + \epsilon, \epsilon \sim N(0, I)$$ The posterior can be derived to be $$z|x \sim N\left(M^{-1}W^T(x-\mu), \sigma^{-1}M\right)$$ where $M = W^TW + \sigma^2I$. In Bishop's Pattern recognition, it said that when $\sigma^2>0$ the posterior mean shifted towards the origin relative to the orthogonal projection. Could someone help me understand this statement? When $\sigma^2 > 0$, isn't there no longer an orthogonal projection, so what is this actually comparing, what does shift toward the origin mathematically mean in this case? Thanks!



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