# How to model a system with multiple probability distributions, each for a part of the system?

I need to build a complex probability model to describe some "real world" scenarios. The system consists of several types of objects, and the contraints upon these objects and their interactions are partially known. The goal is to determine the parameters of the model with the input of real world "training" data.

To this end, how can I put all these probability constraints all together to let them interect with each other in a good way?

For example, I want to model travelling behavior of a family:

1. Ed is college student, son of Tim and Barbara. They have a Ford, a VW and a bike.
2. Tim goes to work almost every working day.
3. Tim likes driving the VW to work, but sometimes also travel with buses.
4. Ed also likes driving the VW, but sometimes also ride to school.
5. Ed goes to school not very often(60%).
6. Barbara goes shopping regularly.
7. Barbara will drive only when 2 cars are both left at home, and she likes the Ford.
8. The cars will be more probable to malfunction if driven too often.
9. In raining days, nobody like the bike.

10.Tim sometimes drives the Ford, if he has driven the VW for a long period of time.

Assume I have a probabilit model for each of the 10. And I have the data of who has driven what in a whole year.

Now How can I put all the 10 constraints together into a dynamic system and find the proper parameters after input the one-year data?

What I want to depict is a system with many sub parts of different types. Each part has its own regularity and the parts interect with each other in time domain.

What is the computer-friendly way? better the state-of-the-art.

Thanks a lot!

Matt

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It depends a lot on the types of models you use to deal with subproblems. – dtldarek Jan 12 '13 at 16:36
You may want to look into Bayesian Networks (en.wikipedia.org/wiki/Bayesian_network). – Lopsy Jan 12 '13 at 17:02
thanks for naming Bayesian Networks, I think that's just what I was searching in my head. but is it capable to model dynamic systems? – Matt Jan 12 '13 at 17:43
@Matt: I'm not sure that what you're describing is a dynamical system. I don't meant to trivialize what you're trying to do here, but really it looks like your problem amounts to performing some linear algebra to determine the statistics governing a bunch of dependent Bernoulli random variables. I wouldn't even go so far as to call your system a Markov chain either, because it appears that the day to day decisions of which transport to use are independent (except for part 9, conceivably, if you decide that the car malfunctions and is taken out of consideration for a number of days). – A Blumenthal Jan 12 '13 at 20:50
@A Blumenthal: Thanks for your reply. The scenario described is just an simple example, because the real problem at hand is quite difficult to describe in short. – Matt Jan 12 '13 at 21:54