I have some trouble understanding the benefits of Bayesian networks.
Am I correct that the key benefit of the network is that one does not need to use the chain rule of probability in order to calculate joint distributions?
So, using the chain rule:
$$ P(A_1, \dots, A_n) = \prod_{i=1}^n (A_i \mid \cap_{j=1}^{i-1} A_j) $$
leads to the same result as the following (assuming the nodes are structured by a Bayesian network)?
$$ P(A_1, \dots, A_n) = \prod_{i=1}^n P(A_i \mid \text{parents}(A_i)) $$
Yes, you are correct that one of the key benefits of Bayesian networks is that they allow you to calculate joint probability distributions without directly using the chain rule of probability. Bayesian networks provide a more efficient way to represent and compute joint probabilities by exploiting the conditional independence properties of the variables in the network.
In Bayesian networks, you represent the joint probability distribution by decomposing it into a product of conditional probabilities. This representation takes advantage of the conditional independence assumptions implied by the network structure, which often leads to more efficient computations and a more compact representation of the joint distribution.
While this is one of the key benefits, it is not the only benefit. Others include:
Modularity and interpretability: Bayesian networks represent complex relationships among variables in a graphical form, making it easy to visualize and understand the dependencies and independencies between variables. This modularity aids in knowledge representation, making it easy to update and modify the network structure as new information becomes available.
Incorporation of domain knowledge: Bayesian networks can incorporate expert knowledge in the form of prior probabilities and network structure. This allows for the inclusion of valuable information that might not be readily available in the data itself.
Inference and reasoning: Bayesian networks facilitate probabilistic reasoning under uncertainty, allowing you to perform various types of inferences such as causal reasoning, diagnostic reasoning, and predictive reasoning. They can be used to answer queries about the probabilities of events given new evidence or observations.
Handling missing data: Bayesian networks can naturally deal with incomplete or missing data. They can be used to predict the values of missing variables based on the observed data and the underlying probabilistic relationships between variables.
