Learning the structure of Bayesian networks is a fundamental yet computationally intensive task, especially as the number of variables grows. Traditional algorithms require retraining from scratch when new variables are introduced, making them impractical for dynamic or large-scale applications. In this paper, we propose an extendable structure learning strategy that efficiently incorporates a new variable $Y$ into an existing Bayesian network graph $\mathcal{G}$ over variables $\mathcal{X}$, resulting in an updated P-map graph $\bar{\mathcal{G}}$ on $\bar{\mathcal{X}} = \mathcal{X} \cup \{Y\}$. By leveraging the information encoded in $\mathcal{G}$, our method significantly reduces computational overhead compared to learning $\bar{\mathcal{G}}$ from scratch. Empirical evaluations demonstrate runtime reductions of up to 1300x without compromising accuracy. Building on this approach, we introduce a novel iterative paradigm for structure learning over $\mathcal{X}$. Starting with a small subset $\mathcal{U} \subset \mathcal{X}$, we iteratively add the remaining variables using our extendable algorithms to construct a P-map graph over the full set. This method offers runtime advantages comparable to common algorithms while maintaining similar accuracy. Our contributions provide a scalable solution for Bayesian network structure learning, enabling efficient model updates in real-time and high-dimensional settings.

Noisy linear structural causal models (SCMs) in the presence of confounding variables are known to be identifiable if all confounding and noise variables are non-Gaussian and unidentifiable if all are Gaussian. The identifiability when only some are Gaussian remains concealed. We show that, in the presence of Gaussian noise, a linear SCM is uniquely identifiable provided that \emph{(i)} the number of confounders is at most the number of the observed variables, \emph{(ii)} the confounders do not have a Gaussian component, and \emph{(iii)} the causal structure of the SCM is known. If the third condition is relaxed, the SCM becomes finitely identifiable; more specifically, it belongs to a set of at most $n!$ linear SCMS, where $n$ is the number of observed variables. The confounders in all of these $n!$ SCMs share the same joint probability distribution function (PDF), which we obtain analytically. For the case where both the noise and confounders are Gaussian, we provide further insight into the existing counter-example-based unidentifiability result and demonstrate that every SCM with confounders can be represented as an SCM without confounders but with the same joint PDF.