Cohen Tervaert, G.D. (2009) Decay of memory on Bayesian networks. Bachelor's Thesis, Mathematics.
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Abstract
This thesis deals with systems of random variables attached to the nodes of a graph, depending on each other according to the directed edges in the graph. The theory of "random systems with interaction" is an area of probability theory which is growing rapidly. The goal of this theory is understanding cooperative effects in large random systems. Applications are primarily in statistical physics but also other fields such as biology or medicine. In statistical physics and thermodynamics, when one tries to explain a physical phenomenon like ferromagnetism, the concept phase transition is useful. A phase transition is a sudden shift of the state, for example matter going from a liquid state to a gas. Phase transitions can only occur in systems with an infinite number of random variables. However, only systems with a finite number of random variables are considered in this thesis to approximate the infinite case. A specific type of probability distribution will be used, namely Bayesian Networks. In chapter 1 a definition of Bayesian Networks will be given, this definition will be followed by an auxiliary lemma with a proof in chapter 2, excluding the in uence of more remote generations under certain circumstances.In chapter 3, an Ising-type model with ferromagnetic interaction from parent-spins to the spin of the children is introduced. Then the central theorem of this thesis about decay of memory on Bayesian Networks will be formulated and proven in chapter 4. Chapter 5 will be about Dobrushin's uniqueness condition and an explanation of why it doesn't cover the above theorem. The case where vertices have more than 2 parents which is not covered by the theorem will be discussed in chapter 6. The conclusion of this thesis will be given in chapter 7. Finally, in chapter 8 some ideas for future investigation are suggested.
| Item Type: | Thesis (Bachelor's Thesis) |
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| Supervisor name: | Kulske, C. |
| Degree programme: | Mathematics |
| Thesis type: | Bachelor's Thesis |
| Language: | English |
| Date Deposited: | 15 Feb 2018 07:28 |
| Last Modified: | 17 Apr 2019 12:39 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/8606 |
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