# AdaptiveRejectionSampling

This package is useful for efficientlysampling from log-concave univariate density functions.

```
using AdaptiveRejectionSampling
using Plots
```

## Sampling from a shifted normal distribution

```
# Define function to be sampled
μ, σ = 1.0, 2.0
f(x) = exp(-0.5(x - μ)^2 / σ^2) / sqrt(2pi * σ^2)
support = (-Inf, Inf)
# Build the sampler and simulate 10,000 samples
sampler = RejectionSampler(f, support, max_segments = 5)
@time sim = run_sampler!(sampler, 10000);
```

```
0.010434 seconds (192.15 k allocations: 3.173 MiB)
```

Let's verify the result

```
# Plot the results and compare to target distribution
x = range(-10.0, 10.0, length=100)
envelop = [eval_envelop(sampler.envelop, xi) for xi in x]
target = [f(xi) for xi in x]
histogram(sim, normalize = true, label = "Histogram")
plot!(x, [target envelop], width = 2, label = ["Normal(μ, σ)" "Envelop"])
```

## Let's try a Gamma

```
α, β = 5.0, 2.0
f(x) = β^α * x^(α-1) * exp(-β*x) / gamma(α)
support = (0.0, Inf)
# Build the sampler and simulate 10,000 samples
sampler = RejectionSampler(f, support)
@time sim = run_sampler!(sampler, 10000)
# Plot the results and compare to target distribution
x = range(0.0, 5.0, length=100)
envelop = [eval_envelop(sampler.envelop, xi) for xi in x]
target = [f(xi) for xi in x]
histogram(sim, normalize = true, label = "Histogram")
plot!(x, [target envelop], width = 2, label = ["Gamma(α, β)" "Envelop"])
```

```
0.007299 seconds (182.00 k allocations: 3.027 MiB)
```

## Truncated distributions and unknown normalisation constant

We don't to provide an exact density--it will sample up to proportionality--and we can do truncated distributions

```
α, β = 5.0, 2.0
f(x) = β^α * x^(α-1) * exp(-β*x) / gamma(α)
support = (1.0, 3.5)
# Build the sampler and simulate 10,000 samples
sampler = RejectionSampler(f, support)
@time sim = run_sampler!(sampler, 10000)
# Plot the results and compare to target distribution
x = range(0.01, 8.0, length=100)
envelop = [eval_envelop(sampler.envelop, xi) for xi in x]
target = [f(xi) for xi in x]
histogram(sim, normalize = true, label = "histogram")
plot!(x, [target envelop], width = 2, label = ["target density" "envelop"])
```

```
0.007766 seconds (181.82 k allocations: 3.024 MiB)
```

## Elastic Net distribution

The following example arises from elastic net regression and smoothing problems. In these cases, the integration constants are not available analytically.

```
# Define function to be sampled
function f(x, μ, λ1, λ2)
δ = x - μ
nl = λ1 * abs(δ) + λ2 * δ^2
return exp(-nl)
end
support = (-Inf, Inf)
# Build the sampler and simulate 10,000 samples
μ, λ1, λ2 = 0.0, 2.0, 1.0
sampler = RejectionSampler(x -> f(x, μ, λ1, λ2), support, max_segments = 5)
@time sim = run_sampler!(sampler, 10000);
# Plot the results and compare to target distribution
x = range(-2.3, 2.3, length=100)
envelop = [eval_envelop(sampler.envelop, xi) for xi in x]
target = [f(xi, μ, λ1, λ2) for xi in x]
histogram(sim, normalize = true, label = "histogram")
plot!(x, [target envelop], width = 2, label = ["target density" "envelop"])
```

To cite please use

```
@manual{tec2018ars,
title = {AdaptiveRejectionSampling.jl},
author = {Mauricio Tec},
year = {2018},
url = {https://github.com/mauriciogtec/AdaptiveRejectionSampling.jl}
}
```