Sampling from power posterior

Specific Domains Probabilistic programming
1

Hey,
I was wondering if someone could guide me (or direct me to the relevant code) on how to sample from the power posterior given a Turing model. By power posterior, I mean the posterior to modified joint : p(x|y; b) \propto p(y|x)^b p(x), where b \in [0,1].

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2

I don’t know about Turing, but it should be relatively straightforward to build a logdensity function like this for Soss.

I like the way the power posterior connects the prior and posterior, but I don’t know any applications for sampling from it. Is this something that comes up a lot in some area?

3

It appears in particular for Thermodynamic Integration for which I created a small package : GitHub - theogf/ThermodynamicIntegration.jl: Thermodynamic Integration for Turing models and more.
I wanted to create some nice wrapper where the user gives a model and it automatically spits out the evidence. I would be happy to create a wrapper for Soss models as well!

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4

Nice! In the log-density this just comes in as a multiplicative constant, so I don’t think there will be any measurable overhead (the rest of the computation is so much more). So I may just add a switch for this to logdensity.

What’s a good descriptive name for the exponent b as a keyword argument?

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5

In physics they call it the coupling parameter cf : Thermodynamic integration - Wikipedia

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6

I’m seeing it called the temperature in some places:

Is that consistent with other uses of “temperature”?

7

Also

8

I think the issue with temperature it that it could be confused with tempered posteriors, which are quite different: for : p(x|y) \propto \exp(-V(x,y)),
the tempered posterior is given by p(x|y;T)\propto \exp(-\frac{V(x,y)}{T}) where T is the temperature

9

The MiniBatchContext in DynamicPPL and Turing can help you with that. The loglike_scalar in DynamicPPL.jl/contexts.jl at master · TuringLang/DynamicPPL.jl · GitHub is b.

10

Oh that does sound perfect, it obviously has a completely different target but it should work!

11

The “Probabilistic Integration” paper calls it the inverse temperature. Both b and 1/T appear as an exponent, so maybe the same?

12

Yeah, there are various names for the same, e.g. power posterior or cold posterior. I have seen the term power posterior mostly been used by statisticians.

13

Fair enough, I would personally disagree with the use of the term (a temperature is not restricted to [0,1] :slight_smile: ) and there is still my argument of how confusing it can be with tempered posterior. But I think as long everything is properly described all terms are fine.

14

But these two are different :smiley: For power posterior you only act on the likelihood, while for cold posteriors you act on the whole joint.
Cold posteriors are used in simulated annealing for instance.

15

Hm, I think there is a paper on BNNs which defines a cold posterior just as a power posterior.

16

But maybe I remember it wrong. It’s a while ago that I look at this paper.

17

If you mean this one : How Good is the Bayes Posterior in Deep Neural Networks Really? | Florian Wenzel they use the same definition that I gave you 2021-02-09_15-54

I got pretty confused at first as well, that’s why I am so confident now :laughing:

Cold posteriors are tempered posteriors where T<1

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18

Ah, yes. See I remembered it wrong. :sweat_smile:
But at least I remembered correctly that it was the same as some existing concept. Haha.

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19

So I tried the following :

function power_logjoint(model, β)
    ctx = DynamicPPL.MiniBatchContext(DynamicPPL.DefaultContext(), β)
    spl = DynamicPPL.SampleFromPrior()
    return function f(z)
        vi = DynamicPPL.VarInfo(model)
        varinfo = DynamicPPL.VarInfo(vi, ctx, z)
        model(varinfo, spl, ctx)
        return DynamicPPL.getlogp(varinfo)
    end
end

But I get the error

ERROR: LoadError: MethodError: no method matching getspace(::DynamicPPL.MiniBatchContext{DynamicPPL.DefaultContext,Float64})
Closest candidates are:
  getspace(::Union{DynamicPPL.SampleFromPrior, DynamicPPL.SampleFromUniform}) at /home/theo/.julia/packages/DynamicPPL/wf0dU/src/sampler.jl:9
  getspace(::GibbsConditional{S,C} where C) where S at /home/theo/.julia/packages/Turing/a9ANC/src/inference/gibbs_conditional.jl:60
  getspace(::SMC{space,R} where R) where space at /home/theo/.julia/packages/Turing/a9ANC/src/inference/Inference.jl:425
  ...
Stacktrace:
 [1] DynamicPPL.VarInfo(::DynamicPPL.VarInfo{NamedTuple{(:x,),Tuple{DynamicPPL.Metadata{Dict{DynamicPPL.VarName{:x,Tuple{}},Int64},Array{MvNormal{Float64,PDMats.PDiagMat{Float64,Array{Float64,1}},FillArrays.Zeros{Float64,1,Tuple{Base.OneTo{Int64}}}},1},Array{DynamicPPL.VarName{:x,Tuple{}},1},Array{Float64,1},Array{Set{DynamicPPL.Selector},1}}}},Float64}, ::DynamicPPL.MiniBatchContext{DynamicPPL.DefaultContext,Float64}, ::Array{ForwardDiff.Dual{ForwardDiff.Tag{ThermodynamicIntegration.var"#f#33"{DynamicPPL.Model{var"#7#8",(:y,),(),(),Tuple{Array{Float64,1}},Tuple{}},DynamicPPL.MiniBatchContext{DynamicPPL.DefaultContext,Float64},DynamicPPL.SampleFromPrior},Float64},Float64,5},1}) at /home/theo/.julia/packages/DynamicPPL/wf0dU/src/varinfo.jl:115
20

Full stacktrace please.