NB: I've asked a related question here, but did not get the answer I needed. I'm asking again with more detail in hopes that those details matter.

Inter-battery factor analysis (IBFA) is similar to probabilistic CCA (Bach and Jordan, 2006) except that it explicitly models a shared latent variable $\mathbf{z}_0$ and view-specific latent variables $\mathbf{z}_1$ and $\mathbf{z}_2$:

$$ \begin{aligned} \mathbf{z}_0 &\sim \mathcal{N}_{K_0}(\mathbf{0}, \mathbf{I}), \\ \mathbf{z}_1 &\sim \mathcal{N}_{K_1}(\mathbf{0}, \mathbf{I}), \\ \mathbf{z}_2 &\sim \mathcal{N}_{K_2}(\mathbf{0}, \mathbf{I}), \\ \mathbf{x}_1 \mid \mathbf{z}_0, \mathbf{z}_1, \mathbf{z}_2 &\sim \mathcal{N}_{P_1}(\mathbf{W}_1 \mathbf{z}_0 + \mathbf{B}_1 \mathbf{z}_1, \sigma_1^2 \mathbf{I}), \\ \mathbf{x}_2 \mid \mathbf{z}_0, \mathbf{z}_1, \mathbf{z}_2 &\sim \mathcal{N}_{P_2}(\mathbf{W}_2 \mathbf{z}_0 + \mathbf{B}_2 \mathbf{z}_2, \sigma_2^2 \mathbf{I}). \end{aligned} \tag{1} $$

In (Klami and Kaski 2007) (see Table 1 on p. 10), the authors propose EM updates for $\mathbf{W}_i$, $\mathbf{B}_i$, and $\sigma_i^2$ for $i \in \{1,2\}$. I can derive all the EM updates except for the EM update for $\sigma_i^2$.

To show you what I mean, we can find the optimal update for $\mathbf{W}_1$ and $\mathbf{W}_1$ by integrating out the view-specific latent variables. For example, to integrate out $\mathbf{z}_1$:

$$ \int p(\mathbf{x}_1, \mathbf{x}_2, \mathbf{z}_0, \mathbf{z}_1, \mathbf{z}_2) d \mathbf{z}_1 = p(\mathbf{x}_2 \mid \mathbf{z}_0, \mathbf{z}_2) p(\mathbf{z}_2) p(\mathbf{z}_0) \int p(\mathbf{x}_1 \mid \mathbf{z}_0, \mathbf{z}_1) p(\mathbf{z}_1) d \mathbf{z}_1. \tag{2} $$

Notice that $\mathbf{z}_0$ is a constant in the integration. Let $\tilde{\mathbf{x}}_1 = \mathbf{x}_1 - \mathbf{W}_1 \mathbf{z}_0$. Then we can easily integrate

$$ \int p(\tilde{\mathbf{x}}_1 \mid \mathbf{z}_1) p(\mathbf{z}_1) d\mathbf{z}_1 \tag{3} $$

since both densities are Gaussian:

$$ \begin{aligned} \tilde{\mathbf{x}}_1 \mid \mathbf{z}_0 &\sim \mathcal{N}_{P_1}(\mathbf{0}, \mathbf{B}_1 \mathbf{B}_1^{\top} + \sigma_1^2 \mathbf{I}), \\ &\Downarrow \\ \mathbf{x}_1 \mid \mathbf{z}_0 &\sim \mathcal{N}_{P_1}(\mathbf{W}_1 \mathbf{z}_0, \mathbf{B}_1 \mathbf{B}_1^{\top} + \sigma_1^2 \mathbf{I}). \end{aligned} \tag{4} $$

Notice that if $\mathbf{B}_1 \mathbf{B}_1^{\top} + \sigma_1^2 \mathbf{I}$ were full rank, we could write it as $\boldsymbol{\Psi}_1$ as in probabilistic CCA. If we applied this same logic to $\mathbf{x}_2$, we would get the same generative model as probabilistic CCA:

$$ \begin{aligned} \mathbf{z}_0 &\sim \mathcal{N}_{K_0}(\mathbf{0}, \mathbf{I}), \\ \mathbf{x}_1 \mid \mathbf{z}_0 &\sim \mathcal{N}_{P_1}(\mathbf{W}_1 \mathbf{z}_0, \mathbf{B}_1 \mathbf{B}_1^{\top} + \sigma_1^2 \mathbf{I}), \\ \mathbf{x}_2 \mid \mathbf{z}_0 &\sim \mathcal{N}_{P_2}(\mathbf{W}_2 \mathbf{z}_0, \mathbf{B}_2 \mathbf{B}_2^{\top} + \sigma_2^2 \mathbf{I}). \end{aligned} \tag{5} $$

Thus, the optimal updates for $\mathbf{W}_1$ and $\mathbf{W}_2$ are found in Section 4.1 ("EM algorithm") in (Bach and Jordan, 2006).

Furthermore, to find the optimal $\mathbf{B}_1$ and $\mathbf{B}_2$, we can integrate out of the shared latent variable. Let $\hat{\mathbf{x}}_1 = \mathbf{x}_1 - \mathbf{B}_1 \mathbf{z}_1$, then we apply the same trick as before to get:

$$ \mathbf{x}_1 \mid \mathbf{z}_1 \sim \mathcal{N}_{P_1}(\mathbf{B}_1 \mathbf{z}_1, \mathbf{W}_1 \mathbf{W}_1^{\top} + \sigma_1^2 \mathbf{I}). \tag{6} $$

Since we've integrated out the dependencies between the two models, we essentially have two probabilistic PCA/factor analysis models. Now the EM update for both $\mathbf{B}_i$ is the same as for probabilistic PCA. See equation $27$ in (Tipping and Bishop, 1999).

I've confirmed that everything so far matches what's in Klami's paper.

Question: I don't know how to derive the EM update for $\sigma_i^2$. As I mentioned in my previous post, if the covariance matrix in Eq. $6$ were just $\sigma^2 \mathbf{I}$, then the MLE would just be what you'd get for probabilistic PCA. However, we have to deal with this term:

$$ (\mathbf{WW}^{\top} + \sigma^2 \mathbf{I})^{-1}. \tag{7} $$

I don't know how to either compute the derivative of this term w.r.t. to $\sigma^2$ or even how to isolate $\sigma^2$, since the Woodbury matrix formula will keep $\sigma^2$ inside the inverse. In the other post, the accepted answer claims there is no closed form solution for $\sigma_i^2$. I'm hoping that by providing the full modeling problem, someone can see something I have overlooked.

Klami's MLE update for $\sigma_i^2$ is almost the MLE update for $\sigma^2$ in probabilistic PCA (see Eq. $28$ in (Tipping and Bishop, 1999)). However, he subtracts $\mathbf{W} \mathbf{W}^{\top}$, which suggests to me that he's somehow transforming Eq. $6$ before applying the probabilistic PCA updates.

  • $\begingroup$ just based on a quick skim of that paper, it doesn't seem like the authors solved for $\sigma_x$ purely in terms of other quantities. The updating equation for $\sigma_x$ has terms in it that depend on the current value of $\sigma_x$, like $\mathbf{\Psi}_x$ $\endgroup$ – jld Aug 20 at 4:21
  • $\begingroup$ This is similar to probabilistic PCA. In that paper, Eq. 28 depends on Eq. 26, which has a term $\mathbf{M}$ which includes $\sigma^2$. I don't show it here, but Tipping and Bishop's Eq. 28 is equivalent to everything in Klami's update except the $-W_x W_x^{\top}$ term. $\endgroup$ – gwg Aug 20 at 11:45

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