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I wonder if there is a way to estimate the precision matrix when certain elements are restricted to be zero?

Suppose data are from $N(\mu,\Omega)$, where $\Omega=V^{-1}$, i.e. the precision matrix. Suppose we know that $\Omega_{i,j}=0$, $i\neq j\in\{1,2,\dots,p\}$, where $p$ is number of variables. Note, sample size $n$ may be less than $p$.

How do we estimate $\Omega$ when we restrict $\Omega_{i,j}$'s to be zero? Can this be done in R?

I understand that this post has a similar discussion. However, it penalizes $\Omega_{i,j}$ but not set it to be zero.

Tan
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  • As I understand, you question concerns sparse inverse covariance matrix estimation where the sparsity structure is known. There is a wealth of literature on this problem recast as structure learning of a Gaussian graphical model, but I couldn't find very much on how to encode a *known* sparsity structure, probably because it is arguably an easier problem. The following is a related question, but I can't vouch for the references: https://stats.stackexchange.com/questions/34502/estimating-a-sparse-inverse-covariance-matrix-with-known-sparsity – microhaus Aug 06 '21 at 15:49
  • The source in my view that addresses your problem directly are the following 2019 Wald Lecture slides given by Trevor Hastie here: https://web.stanford.edu/~hastie/TALKS/wald_III.pdf. See slide 11 onwards *"Solving for $\Theta$ when zero structure is known"*. In that case it looks like you just use constrained optimisation methods and some modification of co-ordinate descent. – microhaus Aug 06 '21 at 15:59

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Would "pathway graphical lasso" be suitable for your case? https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4497513/

a11msp
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