Bayesian/causal networks: just a fast check

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In summary, the conversation discusses a DAG for a Bayesian network with discrete variables and known priors and conditional probabilities. The question is asked whether the DAG satisfies the Markov Assumption, to which the answer is no due to V being dependent on a non-descendant variable S. The conversation also clarifies some parts of the diagram and addresses potential misunderstandings.
  • #1
carllacan
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Hi!

I just want to check that I'm getting this right.

Suppose we have a DAG for a Bayesian network:

V S
\ /
\ /
C

Each variable is discrete and has two possible values, named v1 and v2 and similar for the others.

We know the priors P(v) and P(s) and also the conditional on C, P(c1,v1,s1), P(c1,v1,s2)...
Then [itex]P(v1|c1)=\frac{P(c1,v1,s1)+P(c1,v1,s2)}{P(c1,v1,s1)+P(c1,v1,s2)+P(c1,v2,s1)+P(c1,v2,s2)}[/itex]
And [itex]P(v1|c1,s1)= P(c1,vs,s1)[/itex]

Is that right?

Also, is the fact that P(v1|c1,s1) and P(v1|c1,s2) are different mean that the conditional probabilities of V are not independent of its non-descendants?

And therefore that is what makes this DAG not satisfy the Markov Assumption?

Thank you for your time.
 
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  • #2
Sorry, I am not knowledgeable in this area. Its been a while since you posted this, so I thought I'd respond even though I quite possibly won't be very helpful.
Your "diagram"
V S
\ /
\ /
C
is unclear -- did you mean V→C←S ? (you can't use spaces to format a diagram in many forums)
I also don't know what P(c1,vs,s1) is. What does vs mean??
Finally, when you say that P(v1|c1, s1) is different from P(v1|c1,s2) do you mean logically (in the general case) or that you know P(s1) ≠ P(s2) [≠ 0.5 ] ? (yes, generally its different, but its actual value is indeterminate here.)
The other problem that I have is that I don't understand your diagram (again, its probably my own ignorance).
Lets say I am correct in assuming your diagram is V→C←S... (I hope we agree that a Markov Assumption is about future states.) Does this mean that V(t=i) influences C(t=i+1)? or that V(t=i) influences C(t=i) ? (where t is sequential time intervals) (I assume here that V and S are independent variables with some unknown T/F probability). Obviously, only if the past state(s) don't influence the future state(s) do we have a Markov Process.
I hope I haven't totally wasted your time. If your diagram is timeless, then it says nothing about Markov Processes. If it is describing flow of states, then by definition any X→Y means Y is dependent on what X was, and hence isn't Markovian... I think?
 
  • #3
Thank you for your answer.

Yes, I meant V→C←S, I thought it was going to look as intended.

With P(c1,vs,s1) I meant P(c1,v1,s1), its a typo (I can't edit it). I meant the joint probability of C = c1, V = v1 and S = s1.

My diagram doesn't involve time its a causal network. The Markov Condition requires, according to my textbook (Neapolitan's Learning Bayesian Networks)
Each variable is conditionally independent of the set of all its nondescendants given the set of all its parents.

In my example V depends on S, which is a non-descendant, and I just want to check if this is why the Markov Condition is not true here.
 

What is a Bayesian/causal network?

A Bayesian/causal network is a graphical model that represents the probabilistic relationships between variables in a system. It uses Bayesian inference to update the probabilities of these variables based on new evidence or data. Causal networks also show the causal relationships between variables, indicating which variables directly influence others.

What are the advantages of using Bayesian/causal networks?

Bayesian/causal networks allow for probabilistic reasoning, which is useful in situations where there is uncertainty or incomplete information. They also provide a visual representation of complex systems, making it easier to understand the relationships between variables. Additionally, these networks can be updated with new data, allowing for continuous learning and improvement.

How are Bayesian/causal networks constructed?

Bayesian/causal networks are constructed by identifying the variables in a system and determining their causal relationships. This information is then represented graphically using nodes and directed edges. The network is then populated with probabilities, which can be based on prior knowledge, expert opinions, or data.

What is the difference between Bayesian and causal networks?

Bayesian networks are a type of probabilistic graphical model that uses Bayesian inference to update probabilities based on new evidence. Causal networks, on the other hand, focus on showing the causal relationships between variables. This means that while Bayesian networks can represent any type of probabilistic relationship, causal networks specifically show cause-and-effect relationships between variables.

How are Bayesian/causal networks used in real-world applications?

Bayesian/causal networks have a wide range of applications in various fields such as medicine, finance, and engineering. They can be used for predictive modeling, decision making, risk analysis, and more. For example, in healthcare, these networks can be used to diagnose diseases and determine the effectiveness of different treatments.

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