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

## Examples

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 normalization 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"])


### Tips for numerical stability, use of logdensity

Here are some tips:

• Make sure the logdensity is numerically stable in the domain and avoid logdensity values > 25 (since the evaluation of the envelop requires exponentials);
• Use log densities instead of densities using the keyword logdensity=true;
• Specify a min_slope and max_slope to find better initial points. The default is 1e-6 and 1e6, respectively. The min_slope is the minimum slope of the logdensity in the initial points of the envelop in absolute value. In general, it is a good idea to leave min_slope with the default and try max_slope=10.0 or a smaller number.
• Try setting δ to a smaller value in the search_grid. The default is 0.5.

⚠️ Warning ⚠️: Using logdensity=true will be the default in v1.0.

Here is an example

import StatsFuns: logsumexp
n = 50
k = 10
alpha = 0.5
tau = 0.5
theta = 1.0

# a complicated logdensity
logf(v) = n * v - (n - k * alpha) * logsumexp([v, log(tau)]) - theta / alpha * ( (tau + exp(v) )^alpha )

# run sampler
δ = 0.1
support = (-Inf, Inf)
search = (0.0, 10.0)
sampler = RejectionSampler(logf, support, δ, max_segments=10, logdensity=true, search_range=search, max_slope=10.0)
@time sim = run_sampler!(sampler, 10000)

[ Info: initial points found at 1.08, 5.43 with grads 9.94522619043481, -9.98968199019509
0.016296 seconds (371.21 k allocations: 6.850 MiB)

x = range(0, 10, length=200)
normconst = sum(f.(x)) * (x[2] - x[1])
envelop = [eval_envelop(sampler.envelop, xi) for xi in x] ./ normconst
target = [f(xi) for xi in x] ./ normconst

# make two plots of logf and f
p1 = plot(logf, -20, 20, label = "logf")
p2 = histogram(sim, normalize=true, label="histogram")
plot!(p2, x, [target envelop], width=2, label=["target density" "envelop"])

plot(p1, p2, layout = (1, 2))


## Citation

@manual{tec2018ars,