# Implemented Distributions

POMDPTools contains several utility distributions to be used in the POMDPs `transition`

and `observation`

functions. These implement the appropriate methods of the functions in the distributions interface.

This package also supplies `showdistribution`

for pretty printing distributions as unicode bar graphs to the terminal.

## Sparse Categorical (`SparseCat`

)

`SparseCat`

is a sparse categorical distribution which is specified by simply providing a list of possible values (states or observations) and the probabilities corresponding to those particular objects.

Example: `SparseCat([1,2,3], [0.1,0.2,0.7])`

is a categorical distribution that assigns probability 0.1 to `1`

, 0.2 to `2`

, 0.7 to `3`

, and 0 to all other values.

`POMDPTools.POMDPDistributions.SparseCat`

— Type`SparseCat(values, probabilities)`

Create a sparse categorical distribution.

`values`

is an iterable object containing the possible values (can be of any type) in the distribution that have nonzero probability. `probabilities`

is an iterable object that contains the associated probabilities.

This is optimized for value iteration with a fast implementation of `weighted_iterator`

. Both `pdf`

and `rand`

are order n.

## Implicit

In situations where a distribution object is required, but the pdf is difficult to specify and only samples are required, `ImplicitDistribution`

provides a convenient way to package a sampling function.

`POMDPTools.POMDPDistributions.ImplicitDistribution`

— Type`ImplicitDistribution(sample_function, args...)`

Define a distribution that can only be sampled from using `rand`

, but has no explicit `pdf`

.

Each time `rand(rng, d::ImplicitDistribution)`

is called,

`sample_function(args..., rng)`

will be called to generate a new sample.

`ImplicitDistribution`

is designed to be used with anonymous functions or the `do`

syntax as follows:

**Examples**

`ImplicitDistribution(rng->rand(rng)^2)`

```
struct MyMDP <: MDP{Float64, Int} end
function POMDPs.transition(m::MyMDP, s, a)
ImplicitDistribution(s, a) do s, a, rng
return s + a + 0.001*randn(rng)
end
end
td = transition(MyMDP(), 1.0, 1)
rand(td) # will return a number near 2
```

## Bool Distribution

`POMDPTools.POMDPDistributions.BoolDistribution`

— Type`BoolDistribution(p_true)`

Create a distribution over Boolean values (`true`

or `false`

).

`p_true`

is the probability of the `true`

outcome; the probability of `false`

is 1-`p_true`

.

## Deterministic

`POMDPTools.POMDPDistributions.Deterministic`

— Type`Deterministic(value)`

Create a deterministic distribution over only one value.

This is intended to be used when a distribution is required, but the outcome is deterministic. It is equivalent to a Kronecker Delta distribution.

## Uniform

`POMDPTools.POMDPDistributions.Uniform`

— Type`Uniform(collection)`

Create a uniform categorical distribution over a collection of objects.

The objects in the collection must be unique (this is tested on construction), and will be stored in a `Set`

. To avoid this overhead, use `UnsafeUniform`

.

`POMDPTools.POMDPDistributions.UnsafeUniform`

— Type`UnsafeUniform(collection)`

Create a uniform categorical distribution over a collection of objects.

No checks are performed to ensure uniqueness or check whether an object is actually in the set when evaluating the pdf.

## Pretty Printing

`POMDPTools.POMDPDistributions.showdistribution`

— Function`showdistribution([io], [mime], d)`

Show a UnicodePlots.barplot representation of a distribution.

**Keyword Arguments**

`title::String=string(typeof(d))*" distribution"`

: title for the barplot.