# Bayesian Networks: P(a,b,c,d) Calculation

• prashantgolu
In summary, the first line of the equation is correct based on the assumptions of conditional independence in the two Bayesian networks. However, the simplifications in the second line may not be valid without further clarification on how the variables are related.
prashantgolu
suppose i have 2 bayesian networks...
a->b and c->d ie b depends on a and d depends on b
now P(a,b,c,d)=P(a)*P(b|a)*P(c|a,b)*P(d|a,b,c)
=P(a)*P(b|a)*P(c)*P(d|c)

Am i right...?

prashantgolu said:
suppose i have 2 bayesian networks...
a->b and c->d ie b depends on a and d depends on b
now P(a,b,c,d)=P(a)*P(b|a)*P(c|a,b)*P(d|a,b,c)
=P(a)*P(b|a)*P(c)*P(d|c)

Am i right...?

It's not clear what you are trying to say since the term on the left is not a conditional probability.

Note: $$P(A|B,C,D)= \frac{A\cap (B\cup C\cup D)}{B\cup C\cup D}$$

Last edited:
By ',' i mean intersection
P(x,y)=p(x|y)*p(y)
like this...

prashantgolu said:
suppose i have 2 bayesian networks...
a->b and c->d ie b depends on a and d depends on b
now P(a,b,c,d)=P(a)*P(b|a)*P(c|a,b)*P(d|a,b,c)
=P(a)*P(b|a)*P(c)*P(d|c)

Am i right...?

OK, then the first line looks correct. However I don't know how the assumptions you're making regarding conditional independence allow you to get your simplification on the second line. If the two networks (a,b), (c,d) are conditionally independent, then I believe your simplifications are correct if they are taken individually. However, I'm not sure why you would combine them since they have no common variables.

EDIT: In other words, am I supposed to know that P(c|a,b)=P(c)? How does the fact that d depends on c tell me that? (I assume that you made a mistake when you stated d depends on b since your formula indicates d depends on c.)

Last edited:

Yes, you are correct in your calculation for P(a,b,c,d) in this specific scenario. However, it is important to note that Bayesian networks can have more complex structures and dependencies, so this formula may not always apply. It is always best to carefully analyze the structure of the network and the conditional probabilities before making any calculations. Additionally, it is important to verify that all the probabilities used in the calculation are accurate and consistent.

## 1. What is a Bayesian network?

A Bayesian network is a probabilistic graphical model used to represent the relationships between variables and their dependencies. It consists of nodes representing variables and directed edges representing the causal relationships between them.

## 2. What is the purpose of calculating P(a,b,c,d) in a Bayesian network?

Calculating P(a,b,c,d) in a Bayesian network allows us to determine the joint probability of a set of variables, given the evidence of other variables. This can be useful in making predictions or decisions based on the relationships between variables.

## 3. How is P(a,b,c,d) calculated in a Bayesian network?

P(a,b,c,d) is calculated using the Bayes' rule, which involves multiplying the conditional probabilities of each node in the network based on the given evidence. This calculation can be done manually or with the help of specialized software.

## 4. What are the benefits of using Bayesian networks in data analysis?

Bayesian networks offer several advantages in data analysis, such as handling uncertainty and incomplete data, modeling complex relationships between variables, and incorporating prior knowledge into the analysis. They can also provide a graphical representation of the relationships between variables, making it easier to interpret and communicate the results.

## 5. Are there any limitations of using Bayesian networks?

One limitation of Bayesian networks is that they assume all variables are independent of each other, which may not always be the case in real-world situations. They also require a large amount of data to accurately model complex relationships between variables. Additionally, the accuracy of the results depends heavily on the quality of the data and the assumptions made in the model.

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