# Expectation of latent variables in Factor analysis Model

I am going to through the theory behind factor analysis models given here

Let's say our model is \begin{align} y_i = \mathcal \Lambda x_i +\epsilon, \end{align}

where $y_i$ is the $p$-dimensional observation and $x_i \in \mathcal N (0,I_q)$ is the q-dimensional underlying latent variable. $\Lambda$ is the loading matrix. $\epsilon \in \mathcal N(0,\Psi)$ is the error term.

Now I want to get the conditional expectation of latent variable $x$ given {$y,\Lambda,\Psi$}, i.e. \begin{align} p(x|y,\Lambda,\Psi) \end{align}

Here is my attempt: \begin{align} \Lambda x & = y - \epsilon \\ \implies \Lambda^{'}\Lambda x & = \Lambda^{'}(y - \epsilon) \\ \implies x & = (\Lambda^{'}\Lambda)^{-1}(\Lambda^{'}(y - \epsilon)) \\\end{align}

If I take the conditional expected value of $x$ now, I get \begin{align} E(x|y,\Lambda,\Psi)&= E((\Lambda^{'}\Lambda)^{-1}(\Lambda^{'}(y - \epsilon))) \\ E(x|y,\Lambda,\Psi)&= ((\Lambda^{'}\Lambda)^{-1}(\Lambda^{'}(y - E(\epsilon)))) \\ E(x|y,\Lambda,\Psi)&= (\Lambda^{'}\Lambda)^{-1}(\Lambda^{'}y) \\\end{align}

But, the expression given on Slide 18 in the above slides, the conditional expected value of $x$ reads something likes $Λ (ΛΛ^T+ Ψ)^ {−1}y$

Can you please point out the mistake I am doing in my derivation. Thanks in advance!

• I just updated to explain what went wrong with your proof. Basically, your method is also a perfectly fine way to do it, but you made the mistake of saying $E(\varepsilon | y_i ) = 0$ when that's not actually the case – jld Jul 13 '18 at 18:33
• That’s brilliant! I think this answers my doubt completely!! – kasa Jul 14 '18 at 1:30
• this was a sneaky error, I also set $E(\varepsilon | y) = 0$ the first time i went through it without even thinking about it :) – jld Jul 14 '18 at 16:41
