# BayesianNonparametrics.jl

*BayesianNonparametrics* is a Julia package implementing state-of-the-art Bayesian nonparametric models for medium-sized unsupervised problems. The software package brings Bayesian nonparametrics to non-specialists allowing the widespread use of Bayesian nonparametric models. Emphasis is put on consistency, performance and ease of use allowing easy access to Bayesian nonparametric models inside Julia.

*BayesianNonparametrics* allows you to

- explain discrete or continous data using: Dirichlet Process Mixtures or Hierarchical Dirichlet Process Mixtures
- analyse variable dependencies using: Variable Clustering Model
- fit multivariate or univariate distributions for discrete or continous data with conjugate priors
- compute point estimtates of Dirichlet Process Mixtures posterior samples

#### News

*BayesianNonparametrics* is Julia 0.7 / 1.0 compatible

## Installation

You can install the package into your running Julia installation using Julia's package manager, i.e.

```
pkg> add BayesianNonparametrics
```

## Documentation

Documentation is available in Markdown: documentation

## Example

The following example illustrates the use of *BayesianNonparametrics* for clustering of continuous observations using a Dirichlet Process Mixture of Gaussians.

After loading the package:

```
using BayesianNonparametrics
```

we can generate a 2D synthetic dataset (or use a multivariate continuous dataset of interest)

```
(X, Y) = bloobs(randomize = false)
```

and construct the parameters of our base distribution:

```
μ0 = vec(mean(X, dims = 1))
κ0 = 5.0
ν0 = 9.0
Σ0 = cov(X)
H = WishartGaussian(μ0, κ0, ν0, Σ0)
```

After defining the base distribution we can specify the model:

```
model = DPM(H)
```

which is in this case a Dirichlet Process Mixture. Each model has to be initialised, one possible initialisation approach for Dirichlet Process Mixtures is a k-Means initialisation:

```
modelBuffer = init(X, model, KMeansInitialisation(k = 10))
```

The resulting buffer object can now be used to apply posterior inference on the model given `X`

. In the following we apply Gibbs sampling for 500 iterations without burn in or thining:

```
models = train(modelBuffer, DPMHyperparam(), Gibbs(maxiter = 500))
```

You shoud see the progress of the sampling process in the command line. After applying Gibbs sampling, it is possible explore the posterior based on their posterior densities,

```
densities = map(m -> m.energy, models)
```

number of active components

```
activeComponents = map(m -> sum(m.weights .> 0), models)
```

or the groupings of the observations:

```
assignments = map(m -> m.assignments, models)
```

The following animation illustrates posterior samples obtained by a Dirichlet Process Mixture:

Alternatively, one can compute a point estimate based on the posterior similarity matrix:

```
A = reduce(hcat, assignments)
(N, D) = size(X)
PSM = ones(N, N)
M = size(A, 2)
for i in 1:N
for j in 1:i-1
PSM[i, j] = sum(A[i,:] .== A[j,:]) / M
PSM[j, i] = PSM[i, j]
end
end
```

and find the optimal partition which minimizes the lower bound of the variation of information:

```
mink = minimum(length(m.weights) for m in models)
maxk = maximum(length(m.weights) for m in models)
(peassignments, _) = pointestimate(PSM, method = :average, mink = mink, maxk = maxk)
```

The grouping wich minimizes the lower bound of the variation of information is illustrated in the following image: