Waïss Azizian

OnlineSampling.jl is a Julia package for online Bayesian inference on reactive models, i.e., streaming probabilistic models.

Online sampling provides 1) a small macro based domain specific language to describe reactive models and 2) a semi-symbolic inference algorithm which combines exact solutions using Belief Propagation for trees of Gaussian random variables, and approximate solutions using Particle Filtering.

[OnlineSampling](https://github.com/wazizian/OnlineSampling.jl) is a probabilistic programming language that focuses on reactive models, i.e., streaming probabilistic models based on the synchronous model of execution.
Programs execute synchronously in lockstep on a global discrete logical clock.
Inputs and outputs are data streams, programs are stream processors.
For such models, inference is a reactive process that returns the distribution of parameters at the current time step given the observations so far.

## Synchronous Reactive Programming

We use Julia's macro system to program reactive models in a style reminiscent of synchronous dataflow programming languages.

A stream function is introduced by the macro `@node`.
Inside a `node`, the macro `@init` can be used to initialize a variable.
Another macro `@prev` can then be used to access the value of a variable at the previous time step.

Then, the macro `@nodeiter` turns a node into a Julia iterator which unfolds the execution of a node and returns the current value at each step.

For examples, the following function `cpt` implements a simple counter incremented at each step, and prints its value

```julia
@node function cpt() 
    @init x = 0 
    x = @prev(x) + 1 
    return x
end

for x in @nodeiter T = 10 cpt() 
    println(x)
end
```


## Reactive Probabilistic Programming

Reactive constructs `@init` and `@prev` can be mixed with probabilistic constructs to program reactive probabilistic models.

Following recent probabilistic languages (e.g., Turing.jl), probabilistic constructs are the following:
- `x = rand(D)` introduces a random variable `x` with the prior distribution `D`.
- `@observe(x, v)` conditions the models assuming the random variable `x` takes the value `v`.

For example, the following example is a HMM where we try to estimate the position of a moving agent from noisy observations.

```julia
speed = 1.0
noise = 0.5
    
@node function model()
    @init x = rand(MvNormal([0.0], ScalMat(1, 1000.0))) # x_0 ~ N(0, 1000)
    x = rand(MvNormal(@prev(x), ScalMat(1, speed)))     # x_t ~ N(x_{t-1}, speed)
    y = rand(MvNormal(x, ScalMat(1, noise)))            # y_t ~ N(x_t, noise)
    return x, y
end
@node function hmm(obs)
    x, y = @nodecall model()
    @observe(y, obs) # assume y_t is observed with value obs_t 
    return x
end

steps = 100
obs = rand(steps, 1)
cloud = @nodeiter particles = 1000 hmm(eachrow(obs)) # launch the inference with 1000 particles (return an iterator)

for (x, o) in zip(cloud, obs)                            
    samples = rand(x, 1000)                                    # sample the 1000 values from the posterior     
    println("Estimated: ", mean(samples), " Observation: ", o)
end
```


## Semi-symbolic algorithm

The inference method is a Rao-Blackwellised particle filter, a semi-symbolic algorithm which tries to analytically compute closed-form solutions, and falls back to a particle filter when symbolic computations fail.
For Gaussian random variables with linear relations, we implemented belief propagation if the factor graph is a tree. 
As a result, in the previous HMM example, belief propagation is able to recover the equation of a Kalman filter and compute the exact solution and only one particle is necessary as shown below.

```julia
cloud = @noderun particles = 1 algo = belief_propagation hmm(eachrow(obs)) # launch the inference with 1 particles for all observations
d = dist(cloud.particles[1])                                               # distribution for the last state
```

## Internals
This package relies on Julia's metaprogramming capabilities. 
Under the hood, the macro `@node` generates a stateful stream processor which closely mimic the `Iterator` interface of Julia. The state correspond to the memory used to store all the variables accessed via `@prev`.

The heavy lifting to create these functions is done by a Julia macro which acts on the Abstract Syntax Tree. The transformations at this level include, for `t > 0`, adding the code to retrieve the previous internal state, update it and return it.

However, some transformations are best done at a later stage of the Julia pipeline. 
One of them is the handling of calls to `@prev` during the initial step `t = 0`.
To seamlessly handle the various constructs of the Julia language, these calls are invalidated at the level of Intermediate Representation (IR) thanks to the package `IRTools`.

Another operation at the IR level is the automatic realization of a symbolic variable undergoing an unsupported transform: when a function is applied to a random variable and there is no method matching the variable type, this variable is automatically sampled. 

We also provide a "pointer-minimal" implementation of belief propagation: during execution when a random variables is not referenced anymore by the program, it can be freed by the garbage collector (GC).

Waïss Azizian

Talks:

OnlineSampling : online inference on reactive models

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