Can someone explain why Bayesian networks are called "Bayesian"

Question

I have been reading Jensen's book on Bayesian Networks and Decision Graphs as well as the Deep Learning book by Bengio, et. al. I am trying to understand why undirected graphs are referred to as Bayesian? I am familiar with Bayesian models from statistics and have written my share of Bayesian regression models, etc., usually with uninformed priors. So I understand a Bayesian network is not a regression model. In some cases a Bayesnet is a classification model while in other cases it seems to be used to encode some causal relationships like a structural equation model.

I was hoping someone could explain why Bayesian networks or undirected graphical models are called "Bayesian"? When I think of Bayesian models, I think of prior and posterior distributions on the parameters of interest--usually informed priors. But when I look at these graphical models, they don't really seem to focus on prior distributions? Are they called Bayesian because they make assumptions about the ability to factor the joint distribution into some product of a limited set of conditional distributions? Is that modelling assumption the reason that these are called bayesian?

My sense is that by encoding the Graph, the modeller makes some specific assumptions about $X$ being a parent of some outcome $Y$, and that this is considered a prior distribution. But I am not sure if I am interpreting that correctly.

I did read the following post, but it did not clear up the questions posted above because it does not address these notions of prior and posterior distributions, which is usually the basis for Bayesian model.

Why Bayesian Networks? Why Markov Networks?

Why are directed graphical models called Bayesian?

I just want to be clear about the language. There are many models that encode certain assumptions about the causal relationships in the model but are not called Bayesian. For example, time series models like AR1 and ARIMA and others make very specific assumptions about the nature of a shock and how its effect diminishes over time. But we don't call those time series models Bayesian.

Answer 1

I thought Bayesian is called Bayesian because of this Bayes formula:

$$P(\theta|X)=\frac{P(X|\theta)P(\theta)}{P(X)}$$

This means two things: 1) every parameter has a distribution; 2) a joint distribution between model parameters and data exists.

For the second point, I'd like to quote this quote:

If you use Bayes' rule, you assume that a joint distribution between model parameters and data exists. This, however, only exists if your data and your parameters both exist, in the same σ-algebra. You can't have it both ways. You have to think your model really exists somewhere.

The above two points don't hold in frequentist because it only optimizes the likelihood $P(X|\theta)$, assuming that $P(\theta|X)=P(X|\theta)$.

We can see that frequentist doesn't utilize the Bayes formula to connect the parameters and data.

Answer 2

Bayesian networks are always directed. They're represented through Directed Acyclic Graphs, so no, you shouldn't see someone referring to an undirected graph as a bayesian network. They're probably referring to a Markov Random Field (MRF), which is undirected and therefore can represent some relationships that a Bayesian Network, being directed and acyclic, can't. On the other hand, there are relationships that can be represented through a Bayesian Network that can not through a Markov Random Field.

Both structure learning of a Bayesian network and inference tasks using a bayesian network are performed with bayesian statistics. People could have named it something else, but in the end they decided to call it bayesian networks and it stuck :-)