Why lasso cannot be arbitrarily applied? Consider any log likelihood function $f(\theta|x)$ where $x$ is data. I can consider $f(\theta|x)+\lambda||\theta||_1$ where $||\theta||_1$ is the standard $L_1$ norm.
It seems that I cannot apply this sort of argument naively to obtain sparsity. I found in the following article that conditional "independence" is required. I think conditional independence in this case is equivalent to sparsity. https://tibshirani.su.domains/ftp/graph.pdf
(pg 1)"The basic model for continuous data assumes that the observations have a
multivariate Gaussian distribution with mean $\mu$ and covariance matrix $\Sigma$. If
the $ij$th component of $\Sigma^{-1}$ is zero, then variables $i$ and $j$ are conditionally independent, given the other variables. Thus it makes sense to impose an $L_1$ penalty for the estimation of $\Sigma^{-1}$, to increase sparsity."
I think the paper is used to estimate $\Sigma^{-1}$ which is sparse. Why conditional independence is required to make sense for sparsity by $L_1$? It seems that variables $i$ and $j$ are conditional independent if and only if $ij$th component of $\Sigma^{-1}$ is zero. That is reason why sparsity is expected by conditional independence.


I found in the following article that conditional "independence" is required.

This condition relates to a different issue which is specific to the application in that article.
It is not a condition that is necessary in order to be able to apply a L1 norm to obtain sparsity.
That idea of independence relates to the model

The basic model for continuous data assumes that the observations have a multivariate Gaussian distribution with mean $\mu$ and covariance matrix $\Sigma$

For this model independence between components $i$ and $j$ is equivalent to zero covariance $\Sigma_{ij}$ (and those covariance terms is what is being estimated in this specific case).
The zero's $\Sigma_{ij}$ are a motivation to use estimation of the $\hat\Sigma_{ij}$ with an L1-norm regularisation. But for different problems, where not the parameters $\Sigma_{ij}$ are being estimated but something else, it will not be necessary and we would expect the sparsity to be present in some other way.
