# Interpretation of concentration of posteriors in the limit of infinitely many independent versus dependent random variables

Disclaimer: the setup and specific example may not be a minimal example to illustrate the point, but I am not well-versed in these topics enough to construct a smaller example without accidentally omitting important details.

# The setup

Consider a statistical model --Markov random field-- defined with respect to a graph $$G=(V,E)$$, where each node in the graph represents a random variable, and undirected edges represent dependencies. For example: $$A,B,D$$ are dependent on each other, but $$A|D$$ is independent of $$E$$.

Now to define the model in question, supposed each variable, which we'll denote as $$X_i$$s for convenience, can take on values in $$\mathcal{X}=\{1,2,3\}$$.associate an edge a strictly positive $$w_{s,t} \in \mathbb{R}_{>0}$$ for $$(s,t)\in E$$. Given $$G$$, its joint mass function (assume discrete space for the random variables) is $$P(X) \propto \exp\{ \langle \boldsymbol \beta, \boldsymbol \phi(X) \rangle \}$$ Where sufficient statistics $$\phi(X) = (\phi_{k})_{k \in \mathcal{X}\times\mathcal{X}}$$, $$\phi_k(X) = \sum_{e} w_e \boldsymbol 1_k(X_e)$$, where $$\boldsymbol{1}_k(X_e)$$ is exchangeable in its argument. The parameters $$\boldsymbol \beta$$ are constrained to a simplex, i.e., $$\sum_{k\in\mathcal{X}\times\mathcal{X}}\beta_k=1, \forall k\in\mathcal{X}\times\mathcal{X}, \beta_k > 0$$. Hopefully this setup is clear, please let me know if there is anything I should clarify.

# A specific example

Consider two scenarios:

1. I observe $$10$$ realizations $$X^{(1)}, X^{(2)},\dots, X^{(10)}$$ independently drawn from a model as defined above with respect to a graph $$G = (V, E)$$ where $$|V| = 10$$.
2. I have one realization $$X'^{(1)}$$ drawn from a model as defined above with respect to a graph $$G' = (V',E')$$ where $$|V'|=100$$.

By writing out the sufficient statistics of the above two scenarios, we get a sum of the original sufficient statistics $$\phi_{10}(X^{(1)}, X^{(2)},\dots, X^{(10)}) = \sum_{j=1}^{10} \phi(X^{(i)})$$ for 1, and $$\phi'(X')$$ for 2.

# My question

RE earlier disclaimer: I like to think the parallel to this setup is something along the lines of $$n$$ iid univariate Gaussian random variables versus one $$n$$-dimensional Gaussian random variable.

When I perform inference, posterior inference in my case, I would expect both models to have posteriors concentrate around their respective "truths". However, I am having trouble understanding how they differ as $$|V'| = \sum_j |V^{(j)}| = n$$ tends to infinity. On one hand, I think this is comparing apples to oranges in the sense that in one scenario (scenario 1), the interpretation of our parameter $$\boldsymbol \beta$$ pertains to a fixed graph, while scenario 2's parameter is interpretted as a parameter that dictates the dependencies of an increasing large graph...

Perhaps my question is a bit ill-formed, but I hope I communicated enough to express my concerns/confusion, and any discussion or comments will be really appreciated.