Getting started
RxInfer.jl
is a Julia package for Bayesian Inference on Factor Graphs by Message Passing. It supports both exact and variational inference algorithms.
RxInfer.jl
package forms an ecosystem around three main packages: ReactiveMP.jl
exports a reactive message passing based Bayesian inference engine, Rocket.jl
is the core library that enables reactivity and GraphPPL.jl
library simplifies model and constraints specification. ReactiveMP.jl
engine is a successor of the ForneyLab
package. It follows the same ideas and concepts for message-passing based inference, but uses new reactive and efficient message passing implementation under the hood. The API between two packages is different due to a better flexibility, performance and new reactive approach for solving inference problems.
This page provides the necessary information you need to get started with Rxinfer
. We will show the general approach to solving inference problems with RxInfer
by means of a running example: inferring the bias of a coin.
Installation
Install RxInfer
through the Julia package manager:
] add RxInfer
Importing RxInfer
To add RxInfer
package (and all associated packages) into a running Julia session simply run:
using RxInfer
Read more about about using
in the Using methods from RxInfer section of the documentation.
Example: Inferring the bias of a coin
The RxInfer
approach to solving inference problems consists of three phases:
- Model specification:
RxInfer
usesGraphPPL
package for model specification part. It offers a domain-specific language to specify your probabilistic model. - Inference specification:
RxInfer
inference API usesReactiveMP
inference engine under the hood and has been designed to be as flexible as possible. It is compatible both with asynchronous infinite data streams and with static datasets. For most of the use cases it consists of the same simple building blocks. In this example we will show one of the many possible ways to infer your quantities of interest. - Inference execution: Given model specification and inference procedure it is pretty straightforward to use reactive API from
Rocket
to pass data to the inference backend and to run actual inference.
Coin flip simulation
Let's start by creating some dataset. One approach could be flipping a coin N times and recording each outcome. For simplicity in this example we will use static pre-generated dataset. Each sample can be thought of as the outcome of single flip which is either heads or tails (1 or 0). We will assume that our virtual coin is biased, and lands heads up on 75% of the trials (on average).
First let's setup our environment by importing all needed packages:
using RxInfer, Distributions, Random
Next, let's define our dataset:
rng = MersenneTwister(42)
n = 10
p = 0.75
distribution = Bernoulli(p)
dataset = float.(rand(rng, Bernoulli(p), n))
10-element Vector{Float64}:
1.0
1.0
1.0
1.0
0.0
0.0
1.0
1.0
0.0
1.0
Model specification
In a Bayesian setting, the next step is to specify our probabilistic model. This amounts to specifying the joint probability of the random variables of the system.
Likelihood
We will assume that the outcome of each coin flip is governed by the Bernoulli distribution, i.e.
\[y_i \sim \mathrm{Bernoulli}(\theta),\]
where $y_i = 1$ represents "heads", $y_i = 0$ represents "tails". The underlying probability of the coin landing heads up for a single coin flip is $\theta \in [0,1]$.
Prior
We will choose the conjugate prior of the Bernoulli likelihood function defined above, namely the beta distribution, i.e.
\[\theta \sim Beta(a, b),\]
where $a$ and $b$ are the hyperparameters that encode our prior beliefs about the possible values of $\theta$. We will assign values to the hyperparameters in a later step.
Joint probability
The joint probability is given by the multiplication of the likelihood and the prior, i.e.
\[P(y_{1:N}, θ) = P(θ) \prod_{i=1}^N P(y_i | θ).\]
Now let's see how to specify this model using GraphPPL's package syntax.
# GraphPPL.jl export `@model` macro for model specification
# It accepts a regular Julia function and builds an FFG under the hood
@model function coin_model(n)
# `datavar` creates data 'inputs' in our model
# We will pass data later on to these inputs
# In this example we create a sequence of inputs that accepts Float64
y = datavar(Float64, n)
# We endow θ parameter of our model with some prior
θ ~ Beta(2.0, 7.0)
# or, in this particular case, the `Uniform(0.0, 1.0)` prior also works:
# θ ~ Uniform(0.0, 1.0)
# We assume that outcome of each coin flip is governed by the Bernoulli distribution
for i in 1:n
y[i] ~ Bernoulli(θ)
end
# We return references to our data inputs and θ parameter
# We will use these references later on during inference step
return y, θ
end
As you can see, RxInfer
offers a model specification syntax that resembles closely to the mathematical equations defined above. We use datavar
function to create "clamped" variables that take specific values at a later date. θ ~ Beta(2.0, 7.0)
expression creates random variable θ
and assigns it as an output of Beta
node in the corresponding FFG.
To quickly check the list of all available factor nodes that can be used in the model specification language call ?make_node
or Base.doc(make_node)
.
Inference specification
Automatic inference specification
Once we have defined our model, the next step is to use RxInfer
API to infer quantities of interests. To do this we can use a generic inference
function that supports static datasets.
result = inference(
model = coin_model(length(dataset)),
data = (y = dataset, )
)
Inference results:
Posteriors | available for (θ)
θestimated = result.posteriors[:θ]
Beta{Float64}(α=9.0, β=10.0)
println("mean: ", mean(θestimated))
println("std: ", std(θestimated))
mean: 0.47368421052631576
std: 0.11164843913471803
Read more information about the inference
function in the Static Inference documentation section.
Manual inference specification
There is a way to manually specify an inference procedure for advanced use-cases. RxInfer
API is flexible in terms of inference specification and is compatible both with real-time inference processing and with static datasets. In most of the cases for static datasets, as in our example, it consists of same basic building blocks:
- Return variables of interests from model specification
- Subscribe on variables of interests posterior marginal updates
- Pass data to the model
- Unsubscribe
Here is an example of inference procedure:
function custom_inference(data)
n = length(data)
# `coin_model` function from `@model` macro returns a reference to the model generator object
# we need to use the `create_model` function to get actual model object
model, (y, θ) = create_model(coin_model(n))
# Reference for future posterior marginal
mθ = nothing
# `getmarginal` function returns an observable of future posterior marginal updates
# We use `Rocket.jl` API to subscribe on this observable
# As soon as posterior marginal update is available we just save it in `mθ`
subscription = subscribe!(getmarginal(θ), (m) -> mθ = m)
# `update!` function passes data to our data inputs
update!(y, data)
# It is always a good practice to unsubscribe and to
# free computer resources held by the subscription
unsubscribe!(subscription)
# Here we return our resulting posterior marginal
return mθ
end
custom_inference (generic function with 1 method)
Inference execution
Here after everything is ready we just call our inference
function to get a posterior marginal distribution over θ
parameter in the model.
θestimated = custom_inference(dataset)
Marginal(Beta{Float64}(α=9.0, β=10.0))
println("mean: ", mean(θestimated))
println("std: ", std(θestimated))
mean: 0.47368421052631576
std: 0.11164843913471803
using Plots
rθ = range(0, 1, length = 1000)
p1 = plot(rθ, (x) -> pdf(Beta(2.0, 7.0), x), title="Prior", fillalpha=0.3, fillrange = 0, label="P(θ)", c=1,)
p2 = plot(rθ, (x) -> pdf(θestimated, x), title="Posterior", fillalpha=0.3, fillrange = 0, label="P(θ|y)", c=3)
plot(p1, p2, layout = @layout([ a; b ]))
In our dataset we used 10 coin flips to estimate the bias of a coin. It resulted in a vague posterior distribution, however RxInfer
scales very well for large models and factor graphs. We may use more coin flips in our dataset for better posterior distribution estimates:
dataset_100 = float.(rand(rng, Bernoulli(p), 100))
dataset_1000 = float.(rand(rng, Bernoulli(p), 1000))
dataset_10000 = float.(rand(rng, Bernoulli(p), 10000))
θestimated_100 = custom_inference(dataset_100)
θestimated_1000 = custom_inference(dataset_1000)
θestimated_10000 = custom_inference(dataset_10000)
p3 = plot(title = "Posterior", legend = :topleft)
p3 = plot!(p3, rθ, (x) -> pdf(θestimated_100, x), fillalpha = 0.3, fillrange = 0, label = "P(θ|y_100)", c = 4)
p3 = plot!(p3, rθ, (x) -> pdf(θestimated_1000, x), fillalpha = 0.3, fillrange = 0, label = "P(θ|y_1000)", c = 5)
p3 = plot!(p3, rθ, (x) -> pdf(θestimated_10000, x), fillalpha = 0.3, fillrange = 0, label = "P(θ|y_10000)", c = 6)
plot(p1, p3, layout = @layout([ a; b ]))
With larger dataset our posterior marginal estimate becomes more and more accurate and represents real value of the bias of a coin.
println("mean: ", mean(θestimated_10000))
println("std: ", std(θestimated_10000))
mean: 0.7529223698671196
std: 0.004310967761308469
Where to go next?
There are a set of examples available in RxInfer
repository that demonstrate the more advanced features of the package and also Examples section in the documentation. Alternatively, you can head to the Model specification which provides more detailed information of how to use RxInfer
to specify probabilistic models. Inference execution section provides a documentation about RxInfer
API for running reactive Bayesian inference.