# markov-realization: Realizations of Markov chains.

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## Properties

Versions 0.1.0, 0.1.0, 0.2.1, 0.3.0, 0.3.1, 0.3.2, 0.3.3, 0.4 ChangeLog.md base (>=4.7 && <5), contravariant (>=1.5.1 && <1.6), discrimination (>=0.4 && <0.5), generic-deriving (>=1.12.4 && <1.13), MonadRandom (>=0.5.1.1 && <0.6) [details] BSD-3-Clause 2019 Alex Loomis Alex Loomis atloomis@math.arizona.edu Statistics https://github.com/alexloomis/markov https://github.com/alexloomis/markov/issues head: git clone git://github.com/alexloomis/markov.git by alexloomis at 2019-06-17T00:22:59Z

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# Markov Tutorial

Let Xn denote the nth state of a Markov chain with state space ℕ. For x ≠ 0 define transition probabilities

p(x,0) = q,

p(x,x) = r, and

p(x,x+1) = s.

When x = 0, let p(x,0) = q+r, p(x,x+1) = s. Let p(x,y) = 0 in all other cases. Suppose we wanted to find P[Xn = j ∩ d = k], where d denotes the number of transitions from a positive integer to zero. There are three values we need to track — extinctions, probability, and state. Extinctions add a value to a counter each time they happen and the counter takes integral values, so they can be represented by `Sum Int`. Probabilities are multiplied each step, and added when duplicate steps are combined. We want decimal probabilities, so we can represent this with `Product Rational`. We will make a new type for the state.

``````data Extinction = Extinction Int
deriving Generic
deriving newtype (Eq, Num, Show)
deriving anyclass Grouping
``````

All that remains is to make an instance of `Markov`.

``````instance Markov (Sum Int, Product Rational) Extinction where
transition x = case state x of
0 -> [ 0 >*< (q+r) >*< id
, 0 >*< s >*< (+1) ]
_ -> [ 1 >*< q >*< const 0
, 0 >*< r >*< id
, 0 >*< s >*< (+1) ]
where q = 0.1; r = 0.3; s = 0.6
``````

We can now easily see a list of states, deaths, and the probabilities.

```
> chain [pure 0 :: Sum Int :* Product Rational :* Extinction] !! 3

((0,8 % 125),0)
((0,111 % 500),1)
((1,51 % 500),0)
((0,9 % 25),2)
((1,9 % 250),1)
((0,27 % 125),3
```

This means that starting from a state of zero, after three time steps there is a 51/500 chance that the state is zero and there has been one death.