## Hidden Markov Models

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### Markov Models

A Markov model is a probabilistic process over a finite set, {S1, ..., Sk}, usually called its states. Each state-transition generates a character from the alphabet of the process. We are interested in matters such as the probability of a given state coming up next, pr(xt=Si), and this may depend on the prior history to t-1.

In computing, such processes, if they are reasonably complex and interesting, are usually called Probabilistic Finite State Automata (PFSA) or Probabilistic Finite State Machines (PFSM) because of their close links to deterministic and non-deterministic finite state automata as used in formal language theory.

Examples are (hidden) Markov Models of biased coins and dice, formal languages, the weather, etc.; Markov models and Hidden Markov Models (HMM) are used in Bioinformatics to model DNA and protein sequences.

### Order 0 Markov Models

An order 0 Markov model has no "memory": pr(xt = Si) = pr(xt' = Si), for all points t and t' in a sequence.

An order 0 Markov model is equivalent to a multinomial probability distribution. The issue of the accuracy with which the model's parameters should be stated, and hence the model's complexity, was investigated by Wallace and Boulton (1968, appendix).

### Order 1 Markov Models

An order 1 (first-order) Markov model has a memory of size 1. It is defined by a table of probabilities pr(xt=Si | xt-1=Sj), for i = 1..k & j = 1..k. You can think of this as k  order 0  Markov models, one for each Sj.

### Order m Markov Models

The order of a Markov model of fixed order, is the length of the history or context upon which the probabilities of the possible values of the next state depend. For example, the next state of an order 2 (second-order) Markov Model depends upon the two previous states.

### Example

#### Biased Coins

A large biased coin, `A', is described by the order 0 Markov model pr(H)=0.6, pr(T)=0.4.

A small biased coin, `B', pr(h)=0.3, pr(t)=0.7.

A third biased coin, 'C', pr(H)=0.8, pr(T)=0.2.

#### 3 Coins

Consider the following process:

1. Let X equal one of the coins `A' or `B' (above).
2. repeat
1. Throw the coin X and write down the result.
2. Toss coin `C'.
3. if `C' shows T then change X to the other one of {`A',`B'}, otherwise leave X unchanged.
3. until enough
We get the following first-order Markov model over {H,T,h,t}:
 xt-1 H T h t pr(xt=H) 0.6×0.8 0.4×0.8 0.3×0.2 0.7×0.2 pr(xt=T) 0.6×0.8 0.4×0.8 0.3×0.2 0.7×0.2 pr(xt=h) 0.6×0.2 0.4×0.2 0.3×0.8 0.7×0.8 pr(xt=t) 0.6×0.2 0.4×0.2 0.3×0.8 0.7×0.8

Below: The process can be drawn as a diagram of states (nodes) and transitions (edges), which makes it clear that it is equivalent to a finite state machine with probabilities associated with each transition: The probabilities on the arcs out of a given state must sum to 1.

e.g. output `HTHHTHHttthtttHHTHHHHtthtthttht...`

### Hidden Markov Models

A Hidden Markov Model (HMM) is simply a Markov Model in which the states are hidden. For example, suppose we only had the sequence of throws from the 3-coin example above, and that the upper-case v. lower-case information had been lost.

`HTHHTHHTTTHTTTHHTHHHHTTHTTHTTHT...`

We can never be absolutely sure which coin was used at a given point in the sequence but we can calculate the probability.

### Variations

The are a number of variations on HMM problems, e.g.

1. The number of states and transition probabilities are known
• Given data, find the optimum (most likely) position of the change points;
this is sometimes called the `segmentation problem' or the `cut-point problem'.
• How precisely should the points be stated?
• In some problems the run-lengths of the use of a component model (i.e. one of the coins `A' or `B' above) is generalised and modelled by some appropriate probability distribution.
2. The number of states is known, but the transition probabilities are not
• Estimate the transition probabilities.
• What accuracy is appropriate for the estimates?
3. The number of states, and the architecture, are unknown.
• Find a HMM / automaton which models the data "well".
• The simplest model has one state, the most complex model has one state per data value; almost certainly neither extreme is justified. Quantifying model complexity is therefore a crucial issue.
• NB. The cut-point positions are nuisance parameters for this problem.

### Automata (Variable Order) A probabilistic finite state automaton consists of states (drawn as nodes) and transitions (arcs, edges). Each transition causes a character to be output. The automaton moves from state to state outputing characters from its alphabet. Each state can be thought of as an order 0 Markov model, i.e. a multinomial probability distribution, over its outgoing transitions. On the other hand, two or more states may have outgoing transitions labelled with the same set of characters but having different probability distributions. The paths into such states can be thought of as contexts of varying lengths which govern the probability distribution of the character to be output next.

### Example

Above: The best probabilistic finite state automaton, in terms of model complexity and fit to the data, for the data  ``CAAAB/` `BBAAB/` `CAAB/` `BBAB/` `CAB/` `BBB/` `CB`'.  The character `/' is a delimiter, known a priori to cause a return to the start state. The problem was posed by Gaines (1976), the solution is from [TR32] (1983).

The examples start with `C` or `BB`, then there are a few `A`'s, and finally `B/`. The solution shows this. The probabilities of the transitions can be estimated from the observed frequencies.

### Inference

If the state transitions of the automaton are known, all that remains is to infer the probabilities of the transitions out of each state, which can be done using the observed transition frequencies (counting).

Sometimes the architecture of the model and the sequence of output characters are known, but not the sequence of state transitions.

In general, neither the architecture of the automaton nor the transitions are known - they are hidden. One must search for an automaton that explains the data well. It is necessary to weigh the trade-off between the complexity of the model and its fit to the data.

### Applications

MML and (hidden) finite state models have be used to: 