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Background:

Let $x_t = Ax_{t-1} + w_t$ be a discrete linear time invariant system where:

  • $x_t \in \mathbb{R}^d$ for all time samples $t$ corresponds to the state vector
  • $A\in \mathbb{R}^{d\times d}$ is the system matrix
  • $w$ is noise vector (i.i.d. from a Gaussian with zero mean and identity covariance)

In this reference the authors try to put low bounds on the number of samples needed to estimate the matrix $A$. They say they can consider a change-of-measurement argument (I am unfamiliar with) and use some $A'\neq A$ such as to compute the log-likelihood:

\begin{align} L_t = \log \left(\frac{f_A(x_1,\ldots,x_t)}{f_{A'}(x_1,\ldots,x_t)}\right) \end{align} where $$ f_A = \prod_{s=1}^t f_A(x_s|x_{s-1}) $$ is the density of the state distribution at time $t$ given that at time $t-1$ the state was $x_{t-1}$ and this density is Gaussian: $$ \nu_{A,x} = \mathcal{Ax, Id}\\ \nu_{A',x} = \mathcal{Ax', Id}\\ $$

Confusion:

I am struggling to understand the following derivation as presented in their paper.

\begin{aligned} &\mathbb{E}_{A}\left[L_{t}\right]=\mathbb{E}_{A}\left[\sum_{s=1}^{t} \mathbb{E}_{A}\left[\log \left(\frac{f_{A}\left(x_{s} \mid x_{s-1}\right)}{f_{A^{\prime}}\left(x_{s} \mid x_{s-1}\right)}\right) \mid \mathcal{F}_{s-1}\right]\right] \\ &\quad=\mathbb{E}_{A}\left[\sum_{s=1}^{t} \mathrm{KL}\left(\nu_{A, x_{s-1}}\,, \nu_{A^{\prime}, x_{s-1}}\right)\right] \\ &\quad=\mathbb{E}_{A}\left[\sum_{s=0}^{t-1} \frac{1}{2} x_{s}^{\top}\left(A-A^{\prime}\right)^{\top}\left(A-A^{\prime}\right) x_{s}\right] \end{aligned}

Specifically:

  • In the first equality where does the second expectation value come from?
  • Why the inner expectation in the first equality is the same as the KL divergence of the two distributions?
  • More generally, what is the meaning and intuition of taking the expectation w.r.t. $A$ of either the function $L_t$?
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