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Lecture 15: Markov Chains Summary

* Introduction

* Markov Chains

Introduction

* Stochastic Processes = Processes that evolve over time in a probabilistic manner

* Most processes are stochastic

? If not, the future is fully determined

* We make the assumption that stochastic processes are fully described by two sets of information:

* The current state

* Transition probabilities

* Stochastic processes with such assumptions are Markov Processes

* The simplifying assumption is that history does not matter

* Markov Chain = Markov Process in discrete time (i.e. weeks or generations) Example of modelling a Markov Chain We can use a Markov Chain to model the future class structure of a society

*

* Current state = current class structure (what proportion of individuals are in each class)

* Transition probabilities = i.e. the probability that the son will be each class given that his father was upper class etc.

* [?] future class structure depends on current and we do not need to know about the past

* Another assumption that is commonly made:

* Stationary transition probabilities = That the transition probabilities stay the same

* With this we can model the long run equilibrium state

* These determine the long run outcomes

? Not the current distribution

* Good at modelling market share in the short run

Markov Chains

* Examples include:

* Brand switching in consumer purchases

* Changes in social class over generations

* Changes in staff employed at different levels in firms

* Progress of a disease in populations

* Models movement between different states over time

* In any time period (stage) a unit will be in one and only one state

* Between states there can be a transition to any of a number of other states

Transition Probabilities

* Transition probability from state to :In Markov chains, it depends on and and not on how state was reached Course Notes Page 37

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