A light-weight package to factor out interfaces and associated APIs for modelling languages for probabilistic programming. High level goals are:

• Definition of an interface of few abstract types and a small set of functions that should be supported by all probabilistic programs and trace types.
• Provision of some commonly used functionality and data structures, e.g., for managing variable names and traces.

This should facilitate reuse of functions in modelling languages, to allow end users to handle models in a consistent way, and to simplify interaction between different languages and sampler implementations, from very rich, dynamic languages like Turing.jl to highly constrained or simplified models such as GPs, GLMs, or plain log-density problems.

A more short term goal is to start a process of cleanly refactoring and justifying parts of DynamicPPL.jl’s design, and hopefully to get on closer terms with Soss.jl.

There are at least two somewhat incompatible conventions used for the term “model”. None of this is particularly exact, but:

• In Turing.jl, if you write down a @model function and call it on arguments, you get a model object paired with (a possibly empty set of) observations. This can be treated as instantiated “conditioned” object with fixed values for parameters and observations.
• In Soss.jl, “model” is used for a symbolic “generative” object from which concrete functions, such as densities and sampling functions, can be derived, and which you can later condition on (and in turn get a conditional density etc.).

Relevant discussions: 1, 2.

There are three interrelating aspects that this interface intends to standardize:

• Density calculation
• Sampling
• “Conversions” between different conditionings of models

Therefore, the interface consists of an AbstractProbabilisticProgram supertype, together with functions

• condition(::Model, ::Trace) -> ConditionedModel
• decondition(::ConditionedModel) -> GenerativeModel
• sample(::Model, ::Sampler = Exact(), [Int]) (from AbstractMCMC.sample)
• logdensityof(::Model, ::Trace) and densityof(::Model, ::Trace) (from DensityInterface.jl)

First, an infrastructural requirement which we will need below to write things out.

The kinds of models we consider are, at least in a theoretical sense, distributions over traces – types which carry collections of values together with their names. Existing realizations of these are VarInfo in Turing.jl, choice maps in Gen.jl, and the usage of named tuples in Soss.jl.

Traces solve the problem of having to name random variables in function calls, and in samples from models. In essence, every concrete trace type will just be a fancy kind of dictionary from variable names (ideally, VarNames) to values.

Since we have to use this kind of mapping a lot in the specification of the interface, let’s for now just choose some arbitrary macro-like syntax like the following:

@T(Y[1] = …, Z = …)

Some more ideas for this kind of object can be found at the end.

The purpose of this part is to provide common names for how we want a model instance to be understood. As we have seen, in some modelling languages, model instances are primarily generative, with some parameters fixed, while other instance types pair model instances conditioned on observations. What I call “conversions” here is just an interface to transform between these two views and unify the involved objects under one language.

Let’s start from a generative model with parameter μ:

# (hypothetical) generative spec a la Soss
@generative_model function foo_gen(μ)
    X ~ Normal(0, μ)
    Y[1] ~ Normal(X)
    Y[2] ~ Normal(X + 1)
end

Applying the “constructor” foo_gen now means to fix the parameter, and should return a concrete object of the generative type:

g = foo_gen(μ=…)::SomeGenerativeModel

With this kind of object, we should be able to sample and calculate joint log-densities from, i.e., over the combined trace space of X, Y[1], and Y[2] – either directly, or by deriving the respective functions (e.g., by converting form a symbolic representation).

For model types that contain enough structural information, it should then be possible to condition on observed values and obtain a conditioned model:

condition(g, @T(Y = …))::SomeConditionedModel

For this operation, there will probably exist syntactic sugar in the form of

g | @T(Y = …)

Now, if we start from a Turing.jl-like model instead, with the “observation part” already specified, we have a situation like this, with the observations Y fixed in the instantiation:

# conditioned spec a la DPPL
@model function foo(Y, μ)
    X ~ Normal(0, μ)
    Y[1] ~ Normal(X)
    Y[2] ~ Normal(X + 1)
end

m = foo(Y=…, μ=…)::SomeConditionedModel

From this we can, if supported, go back to the generative form via decondition, and back via condition:

decondition(m) == g::SomeGenerativeModel
m == condition(g, @T(Y = …))

(with equality in distribution).

In the case of Turing.jl, the object m would at the same time contain the information about the generative and posterior distribution condition and decondition can simply return different kinds of “tagged” model types which put the model specification into a certain context.

Soss.jl pretty much already works like the examples above, with one model object being either a JointModel or a ConditionedModel, and the | syntax just being sugar for the latter.

A hypothetical DensityModel, or something like the types from LogDensityProblems.jl, would be a case for a model type that does not support the structural operations condition and decondition.

The invariances between these operations should follow normal rules of probability theory. Not all methods or directions need to be supported for every modelling language; in this case, a MethodError or some other runtime error should be raised.

There is no strict requirement for generative models and conditioned models to have different types or be tagged with variable names etc. This is a choice to be made by the concrete implementation.

Decomposing models into prior and observation distributions is not yet specified; the former is rather easy, since it is only a marginal of the generative distribution, while the latter requires more structural information. Perhaps both can be generalized under the query function I discuss at the end.

Sampling in this case refers to producing values from the distribution specified in a model instance, either following the distribution exactly, or approximating it through a Monte Carlo algorithm.

All sampleable model instances are assumed to implement the AbstractMCMC interface – i.e., at least step, and accordingly sample, steps, Samples. The most important aspect is sample, though, which plays the role of rand for distributions.

The results of sample generalize rand – while rand(d, N) is assumed to give you iid samples, sample(m, sampler, N) returns a sample from a sequence (known as chain in the case of MCMC) of length N approximating m’s distribution by a specific sampling algorithm (which of course subsumes the case that m can be sampled from exactly, in which case the “chain” actually is iid).

Depending on which kind of sampling is supported, several methods may be supported. In the case of a (posterior) conditioned model with no known sampling procedure, we just have what is given through AbstractMCMC:

sample([rng], m, N, sampler; [args…]) # chain of length N using `sampler`

In the case of a generative model, or a posterior model with exact solution, we can have some more methods without the need to specify a sampler:

sample([rng], m; [args…])    # one random sample
sample([rng], m, N; [args…]) # N iid samples; equivalent to `rand` in certain cases

It should be possible to implement this by a special sampler, say, Exact (name still to be discussed), that can then also be reused for generative sampling:

step(g, spl = Exact(), state = nothing) # IID sample from exact distribution with trivial state
sample(g, Exact(), [N]) 

with dispatch failing for models types for which exact sampling is not possible (or not implemented).

This could even be useful for Monte Carlo methods not being based on Markov Chains, e.g., particle-based sampling using a return type with weights, or rejection sampling.

Not all variants need to be supported – for example, a posterior model might not support sample(m) when exact sampling is not possible, only sample(m, N, alg) for Markov chains.

rand is then just a special case when “trivial” exact sampling works for a model, e.g. a joint model.

Since the different “versions” of how a model is to be understood as generative or conditioned are to be expressed in the type or dispatch they support, there should be no need for separate functions logjoint, loglikelihood, etc., which force these semantic distinctions on the implementor; we therefore adapt the interface of DensityInterface.jl. Its main function logdensityof should suffice for variants, with the distinction being made by the capabilities of the concrete model instance.

DensityInterface.jl also requires the trait function DensityKind, which is set to HasDensity() for the AbstractProbabilisticProgram type. Additional functions

DensityInterface.densityof(d, x) = exp(logdensityof(d, x))
DensityInterface.logdensityof(d) = Base.Fix1(logdensityof, d)
DensityInterface.densityof(d) = Base.Fix1(densityof, d)

are provided automatically (repeated here for clarity).

Note that logdensityof strictly generalizes logpdf, since the posterior density will of course in general be unnormalized and hence not a probability density.

The evaluation will usually work with the internal, concrete trace type, like VarInfo in Turing.jl:

logdensityof(m, vi)

But the user will more likely work on the interface using probability expressions:

logdensityof(m, @T(X = …))

(Note that this would replace the current prob string macro in Turing.jl.)

Densities need (and usually, will) not be normalized.

It should be able to make this fall back on the internal method with the right definition and implementation of maketrace:

logdensityof(m, t::ProbabilityExpression) = logdensityof(m, maketrace(m, t))

There is one open question – should normalized and unnormalized densities be able to be distinguished? This could be done by dispatch as well, e.g., if the caller wants to make sure normalization:

logdensityof(g, @T(X = …, Y = …, Z = …); normalized=Val{true})

Although there is proably a better way through traits; maybe like for arrays, with NormalizationStyle(g, t) = IsNormalized()?

Note that this needs to be a macro, if written this way, since the keys may themselves be more complex than just symbols (e.g., indexed variables.) (Don’t hang yourselves up on that @T name though, this is just a working draft.)

The idea here is to standardize the construction (and manipulation) of abstract probability expressions, plus the interface for turning them into concrete traces for a specific model – like @formula and apply_schema from StatsModels.jl are doing.

Maybe the following would suffice to do that:

maketrace(m, t)::tracetype(m, t)

where maketrace produces a concrete trace corresponding to t for the model m, and tracetype is the corresponding eltype–like function giving you the concrete trace type for a certain model and probability expression combination.

Possible extensions of this idea:

• Pearl-style do-notation: @T(Y = y | do(X = x))
• Allowing free variables, to specify model transformations: query(m, @T(X | Y))
• “Graph queries”: @T(X | Parents(X)), @T(Y | Not(X)) (a nice way to express Gibbs conditionals!)
• Predicate style for “measure queries”: @T(X < Y + Z)

The latter applications are the reason I originally liked the idea of the macro being called @P (or even @𝓅 or @ℙ), since then it would look like a “Bayesian probability expression”: @P(X < Y + Z). But this would not be so meaningful in the case of representing a trace instance.

Perhaps both @T and @P can coexist, and both produce different kinds of ProbabilityExpression objects?

NB: the exact details of this kind of “schema application”, and what results from it, will need to be specified in the interface of AbstractModelTrace, aka “the new VarInfo”.

This part is even draftier than the above – we’ll try out things in DynamicPPL.jl first

As I have said before:

There are many aspects that make VarInfo a very complex data structure.

Currently, there is an insane amount of complexity and implementation details in DynamicPPL.jl’s varinfo.jl, which has been rewritten multiple times with different concerns in mind – most times to improve concrete needs of Turing.jl, such as type stability, or requirements of specific samplers.

This unfortunately makes VarInfo extremely opaque: it is hard to refactor without breaking anything (nobody really dares touching it), and a lot of knowledge about Turing.jl/DynamicPPL.jl internals is needed in order to judge the effects of changes.

Recently, @torfjelde has shown that a much simpler implementation is feasible – basically, just a wrapped NamedTuple with a minimal interface.

The purpose of this proposal is twofold: first, to think about what a sufficient interface for AbstractModelTrace, the abstract supertype of VarInfo, should be, to allow multiple specialized variants and refactor the existing ones (typed/untyped and simple). Second, to view the problem as the design of an abstract data type: the specification of construction and modification mechanisms for a dictionary-like structure.

Related previous discussions:

• Discussion about VarName
• AbstractVarInfo representation

Additionally (but closely related), the second part tries to formalize the “subsumption” mechanism of VarNames, and its interaction with using VarNames as keys/indices.

Our discussions take place in what is a bit of a fuzzy zone between the part that is really “abstract”, and meant for the wider purpuse of AbstractPPL.jl – the implementation of probabilistic programming systems in general – and our concrete needs within DPPL. I hope to always stay abstract and reusable; and there are already a couple of candidates for APPL clients other than DPPL, which will hopefully keep us focused: simulation based calibration, SimplePPL (a BUGS-like frontend), and ParetoSmoothing.jl.

• For the end user of Turing.jl: nothing. You usually don’t use VarInfo, or the raw evaluator interface, anyways. (Although if the newer data structures are more user-friendly, they might occur in more places in the future?)
• For people having a look into code using VarInfo, or starting to hack on Turing.jl/DPPL.jl: a huge reduction in cognitive complexity. VarInfo implementations should be readable on their own, and the implemented functions layed out somewhere. Its usages should look like for any other nice, normal data structure.
• For core DPPL.jl implementors: same as the previous, plus: a standard against which to improve and test VarInfo, and a clearly defined design space for new data structures.
• For AbstractPPL.jl clients/PPL implementors: an interface to program against (as with the rest of APPL), and an existing set of well-specified, flexible trace data types with different characteristics.

And in terms of implementation work in DPPL.jl: once the interface is fixed (or even during fixing it), varinfo.jl will undergo a heavy refactoring – which should make it simpler! (No three different getter functions with slightly different semantics, etc…).

The basic idea is for all VarInfos to behave like ordered dictionaries with VarName keys – all common operations should just work. There are two things that make them more special, though:

1. “Fancy indexing”: since VarNames are structured themselves, the VarInfo should be have a bit like a trie, in the sense that all prefixes of stored keys should be retrievable. Also, subsumption of VarNames should be respected (see end of this document):

   vi[@varname(x.a)] = [1,2,3]
   vi[@varname(x.b)] = [4,5,6]
   vi[@varname(x.a[2])] == 2
   vi[@varname(x)] == (; a = [1,2,3], b = [4,5,6])

   Generalizations that go beyond simple cases (those that you can imagine by storing individual setfield!s in a tree) need not be implemented in the beginning; e.g.,

   vi[@varname(x[1])] = 1
   vi[@varname(x[2])] = 2
   keys(vi) == [x[1], x[2]]
   
   vi[@varname(x)] = [1,2]
   keys(vi) == [x]

2. (This has to be discussed further.) Information other than the sampled values, such as flags, metadata, pointwise likelihoods, etc., can in principle be stored in multiple of these “VarInfo dicts” with parallel structure. For efficiency, it is thinkable to devise a design such that multiple fields can be stored under the same indexing structure.

   vi[@varname(x[1])] == 1
   vi[@varname(x[1])].meta["bla"] == false

   or something in that direction.

   (This is logically equivalent to a dictionary with named tuple values. Maybe we can do what DictTable does?)

   The old order field, indicating at which position in the evaluator function a variable has been added (essentially a counter of insertions) can actually be left out completely, since the dictionary is specified to be ordered by insertion.

   The important question here is: should the “joint data structure” behave like a dictionary of NamedTuples (eltype(vi) == @NamedTuple{value::T, ℓ::Float64, meta}), or like a struct of dicts with shared keys (eltype(vi.value) <: T, eltype(vi.ℓ) <: Float64, …)?

The required dictionary functions are about the following:

• Pure functions:
  • iterate, yielding pairs of VarName and the stored value
  • IteratorEltype == HasEltype(), IteratorSize = HasLength()
  • keys, values, pairs, length consistent with iterate
  • eltype, keytype, valuetype
  • get, getindex, haskey for indexing by VarName
  • merge to join two VarInfos
• Mutating functions:
  • insert!!, set!!
  • merge!! to add and join elements (TODO: think about merge)
  • setindex!!
  • empty!!, delete!!, unset!! (Are these really used anywhere? Not having them makes persistent implementations much easier!)

I believe that adopting the interface of Dictionaries.jl, not Base.AbstractDict, would be ideal, since their approach make key sharing and certain operations naturally easy (particularly “broadcast-style”, i.e., transformations on the values, but not the keys).

Other Base functions, like enumerate, should follow from the above.

length might appear weird – but it should definitely be consistent with the iterator.

It would be really cool if merge supported the combination of distinct types of implementations, e.g., a dynamic and a tuple-based part.

To support both mutable and immutable/persistent implementations, let’s require consistent BangBang.jl style mutators throughout.

Transformations should ideally be handled explicitely and from outside: automatically by the compiler macro, or at the places required by samplers.

Implementation-wise, they can probably be expressed as folds?

map(v -> link(v.dist, v.value), vi)

There are multiple possible approaches to handle this:

1. As a special case of conversion: Vector(vi)
2. copy!(vals_array, vi).
3. As a fold: mapreduce(v -> vec(v.value), append!, vi, init=Float64[])

Also here, I think that the best implementation would be through a fold. Variants (1) or (2) might additionally be provided as syntactic sugar.

What follows is mostly an attempt to formalize subsumption.

First, remember that in Turing.jl we can always work with concretized VarNames: begin/end, :, and boolean indexing are all turned into some form of concrete cartesian or array indexing (assuming this suggestion being implemented). This makes all index comparisons static.

Now, VarNames have a compositional structure: they can be built by composing a root variable with more and more lenses (VarName{v}() starts off with an IdentityLens):

julia> vn = VarName{:x}() ∘ Setfield.IndexLens((1:10, 1) ∘ Setfield.IndexLens((2, )))
x[1:10,1][2]

(Note that the composition function, ∘, is really in wrong order; but this is a heritage of Setfield.jl.)

By “subsumption”, we mean the notion of a VarName expressing a more nested path than another one:

subsumes(@varname(x.a), @varname(x.a[1]))
@varname(x.a) ⊒ @varname(x.a[1]) # \sqsupseteq
@varname(x.a) ⋢ @varname(x.a[1]) # \nsqsubseteq

Thus, we have the following axioms for VarNames (“variables” are VarName{n}()):

1. x ⊑ x for all variables x
2. x ≍ y for x ≠ y (i.e., distinct variables are incomparable; x ⋢ y and y ⋢ x) (≍ is \asymp)
3. x ∘ ℓ ⊑ x for all variables x and lenses ℓ
4. x ∘ ℓ₁ ⊑ x ∘ ℓ₂ ⇔ ℓ₁ ⊑ ℓ₂

For the last axiom to work, we also have to define subsumption of individual, non-composed lenses:

1. PropertyLens(a) == PropertyLens(b) ⇔ a == b, for all symbols a, b
2. FunctionLens(f) == FunctionLens(g) ⇔ f == g (under extensional equality; I’m only mentioning this in case we ever generalize to Bijector-ed variables like @varname(log(x)))
3. IndexLens(ι₁) ⊑ IndexLens(ι₂) if the index tuple ι₂ covers all indices in ι₁; for example, _[1, 2:10] ⊑ _[1:10, 1:20]. (This is a bit fuzzy and not all corner cases have been considered yet!)
4. IdentityLens() == IdentityLens()
5. ℓ₁ ≍ ℓ₂, otherwise

Together, this should make VarNames under subsumption a reflexive poset.

The fundamental requirement for VarInfos is then:

vi[x ∘ ℓ] == get(vi[x], ℓ)

So we always want the following to work, automatically:

vi = insert!!(vi, vn, x)
vi[vn] == x

(the trivial case), and

x = set!!(x, ℓ₁, a)
x = set!!(x, ℓ₂, b)
vi = insert!!(vi, vn, x)
vi[vn ∘ ℓ₁] == a
vi[vn ∘ ℓ₂] == b

since vn subsumes both vn ∘ ℓ₁ and vn ∘ ℓ₂.

Whether the opposite case is supported may depend on the implementation. The most complicated part is “unification”:

vi = insert!!(vi, vn ∘ ℓ₁, a)
vi = insert!!(vi, vn ∘ ℓ₂, b)
get(vi[vn], ℓ₁) == a
get(vi[vn], ℓ₂) == b

where vn ∘ ℓ₁ and vn ∘ ℓ₂ need to be recognized as “children” of a common parent vn.