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# Bayesian Linear Regression in Julia

This is a simple package that does one thing, Bayesian Linear Regression, in around 100 lines of code.

It is actively maintained, but it might appear inactive as it's one of those packages which requires very little maintenence because it's very simple.

## Intended Use and Functionality

The interface sits at roughly the same level as that of Distributions.jl. This means that while you won't find a scikit-learn-style `fit` function, you will find all of the primitives that you need to construct such a function to suit your particular problem. In particular, one can:

• Construct a `BayesianLinearRegressor` (BLR) object by providing a mean-vector and precision matrix for the weights of said regressor. This object represents a distribution over (linear) functions.
• A `BayesianLinearRegressor` is a `AbstractGP`, and implements the primary AbstractGP API.
• Think of an instance of `BayesianLinearRegressor` as a very restricted GP, where the time complexity of inference scales linearly in the number of observations `N`.

## Conventions

A `BayesianLinearRegressor` in `D` dimensions works with data where:

• inputs `X` should be a `D x N` matrix of `Real`s where each column is from one data point.
• outputs `y` should be an `N`-vector of `Real`s, where each element is from one data point.

## Example Usage

``````# Install the packages if you don't already have them installed
] add AbstractGPs, BayesianLinearRegressors LinearAlgebra Random Optim Plots Zygote
using AbstractGPs, BayesianLinearRegressors, LinearAlgebra, Random, Optim, Plots, Zygote

# Fix seed for re-producibility.
rng = MersenneTwister(123456)

# Construct a BayesianLinearRegressor prior over linear functions of `X`.
mw, Λw = zeros(2), Diagonal(ones(2))
f = BayesianLinearRegressor(mw, Λw)

# Index into the regressor and assume heterscedastic observation noise `Σ_noise`.
N = 10
X = collect(hcat(collect(range(-5.0, 5.0, length=N)), ones(N))')
Σ_noise = Diagonal(exp.(randn(N)))
fX = f(X, Σ_noise)

# Generate some toy data by sampling from the prior.
y = rand(rng, fX)

# Compute the adjoint of `rand` w.r.t. everything given random sensitivities of y′.
_, back_rand = Zygote.pullback(
(X, Σ_noise, mw, Λw)->rand(rng, BayesianLinearRegressor(mw, Λw)(X, Σ_noise), 5),
X, Σ_noise, mw, Λw,
)
back_rand(randn(N, 5))

# Compute the `logpdf`. Read as `the log probability of observing `y` at `X` under `f`, and
# Gaussian observation noise with zero-mean and covariance `Σ_noise`.
logpdf(fX, y)

# Compute the gradient of the `logpdf` w.r.t. everything.
(X, Σ_noise, y, mw, Λw)->logpdf(BayesianLinearRegressor(mw, Λw)(X, Σ_noise), y),
X, Σ_noise, y, mw, Λw,
)

# Perform posterior inference. Note that `f′` has the same type as `f`.
f′ = posterior(fX, y)

# Compute `logpdf` of the observations under the posterior predictive.
logpdf(f′(X, Σ_noise), y)

# Sample from the posterior predictive distribution.
N_plt = 1000
X_plt = hcat(collect(range(-6.0, 6.0, length=N_plt)), ones(N_plt))'
f′X_plt = rand(rng, f′(X_plt, eps()), 100) # Samples with machine-epsilon noise for stability

# Compute some posterior marginal statisics.
normals = marginals(f′(X_plt, eps()))
m′X_plt = mean.(normals)
σ′X_plt = std.(normals)

# Plot the posterior. This uses the default AbstractGPs plotting recipes.
posterior_plot = plot();
plot!(posterior_plot, X_plt[1, :], f′(X_plt, eps()); color=:blue, ribbon_scale=3);
sampleplot!(posterior_plot, X_plt[1, :], f′(X_plt, eps()); color=:blue, samples=10);
scatter!(posterior_plot, X[1, :], y; # Observations.
markercolor=:red,
markershape=:circle,
markerstrokewidth=0.0,
markersize=4,
markeralpha=0.7,
label="",
);
display(posterior_plot);
```

## Up For Grabs

- Scikit-learn style interface: it wouldn't be too hard to implement a scikit-learn - style interface to handle basic regression tasks, so please feel free to make a PR that implements this.
- Monte Carlo VI (MCVI): i.e. variational inference using the reparametrisation trick. This could be very useful when working with large data sets and applying big non-linear transformations, such as neural networks, to the inputs as it would enable mini-batching. I would envise at least supporting both a dense approximate posterior covariance and diagonal (i.e. mean-field), where the latter is for small-moderate dimensionalities and the latter for very high-dimensional problems.

## Bugs, Issues, and PRs

Please do report and bugs you find by raising an issue. Please also feel free to raise PRs, especially if for one of the above `Up For Grabs` items. Raise an issue to discuss the extension in detail before opening a PR if you prefer though.

## Related Work

[BayesianLinearRegression.jl](https://github.com/cscherrer/BayesianLinearRegression.jl) is closely related, but appears to be a WIP and hasn't been touched in around a year or so (as of 27-03-2019).
``````

03/20/2019

16 days ago

51 commits