# Conjugate inference

**URL:** <https://discourse.julialang.org/t/conjugate-inference/84536>\
**Category:** Probabilistic Programming\
**Tags:** conjugate-priors\
**Created:** [July 20, 2022, 1:11pm UTC](https://discourse.julialang.org/t/conjugate-inference/84536 "2022-07-20T13:11:11Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![filchristou](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/filchristou/32/26760_2.png) [@filchristou](https://discourse.julialang.org/u/filchristou)\
**Post date:** [July 20, 2022, 1:11pm UTC](https://discourse.julialang.org/t/conjugate-inference/84536/1 "2022-07-20T13:11:11Z")

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Is there a package that analytically and automatically computers posteriors based on conjugate priors?

I found [GitHub - JuliaStats/ConjugatePriors.jl: A Julia package to support conjugate prior distributions.](https://github.com/JuliaStats/ConjugatePriors.jl) and the name looks promising.  
I saw from a Turing.jl tutorial that you can do the following:

```julia
pri = NormalGamma(μ₀, κ₀, α₀, β₀)
post = posterior(pri, Normal, x)

```

I tried doing the same for a Bernoulli-Beta conjugate pair but didn’t work out:

```julia-repl
julia> using Distributions, ConjugatePriors
julia> prior = Beta(10,10);
julia> poste = posterior(prior, Bernoulli, [0.2, 0.3]);
suffstats is not implemented for (Distributions.Bernoulli, Vector{Float64}).

    1. error(::String)@error.jl:33
    2. suffstats(::Type{Distributions.Bernoulli}, ::Vector{Float64})@genericfit.jl:5
    3. posterior_canon(::Distributions.Beta{Float64}, ::Type{Distributions.Bernoulli}, ::Vector{Float64})@fallbacks.jl:3
    4. posterior(::Distributions.Beta{Float64}, ::Type{Distributions.Bernoulli}, ::Vector{Float64})@fallbacks.jl:7
    5. top-level scope@Local: 1[inlined]

```

Either the package is WIP or I miss something because I would consider Beta-Bernoulli to be very basic functionality.  
Sadly there is no documentation in ConjugatePriors.jl which is a little bit frustrating.

---

<div class="post-metadata">

**Author:** ![gdalle](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/gdalle/32/27854_2.png) [@gdalle](https://discourse.julialang.org/u/gdalle)\
**Post date:** [July 20, 2022, 1:43pm UTC](https://discourse.julialang.org/t/conjugate-inference/84536/2 "2022-07-20T13:43:42Z")

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It’s true that this package is a bit old and needs some love. IIRC @mschauer has started a PR to modernize it. As for me I hoped I would be able to contribute, but I have no time right now. In any case, you’re welcome to lend a hand 😉

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**Author:** ![opera\_malenky](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/opera_malenky/32/8213_2.png) [@opera\_malenky](https://discourse.julialang.org/u/opera_malenky)\
**Post date:** [July 20, 2022, 3:09pm UTC](https://discourse.julialang.org/t/conjugate-inference/84536/3 "2022-07-20T15:09:30Z")

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I haven’t used it myself, but I believe [ReactiveMP](https://github.com/biaslab/ReactiveMP.jl) includes a number of conjugate priors to speed things up.

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**Author:** ![cscherrer](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/cscherrer/32/7631_2.png) [@cscherrer](https://discourse.julialang.org/u/cscherrer)\
**Post date:** [July 20, 2022, 3:18pm UTC](https://discourse.julialang.org/t/conjugate-inference/84536/4 "2022-07-20T15:18:05Z")

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I can confirm that ReactiveMP has very strong support for accelerating computations when conjugate priors are available. Tagging @bvdmitri in case he has more to say about this.

[MeasureTheory.jl](https://github.com/cscherrer/MeasureTheory.jl) also has the infrastructure to support this, but currently that’s the extent of it - such accelerations are not yet implemented.

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**Author:** ![bvdmitri](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bvdmitri/32/22935_2.png) [@bvdmitri](https://discourse.julialang.org/u/bvdmitri)\
**Post date:** [July 20, 2022, 5:14pm UTC](https://discourse.julialang.org/t/conjugate-inference/84536/5 "2022-07-20T17:14:22Z")

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@filchristou ReactiveMP.jl supports infrastructure to run very efficient Bayesian inference in models with conjugate pairs. A large table of rules has been implemented for many conjugate pairs (but not for all…). You can check-out this example on Beta-Bernoulli distributions: [Getting Started · ReactiveMP.jl](https://biaslab.github.io/ReactiveMP.jl/stable/man/getting-started/#Joint-probability). On my laptop ReactiveMP.jl computes **exact** posterior with less than 1 seconds from dataset with `100_000` points:

```julia
julia> p
0.75

julia> dataset = float.(rand(rng, Bernoulli(p), n))

julia> length(dataset)
100000

julia> @btime inference(
           model = Model($coin_model, length($dataset)),
           data = (y = $dataset, )
       )
  693.036 ms (10499774 allocations: 465.60 MiB)

julia> result.posteriors[:θ][end]
Beta{Float64}(α=74987.0, β=25022.0)

julia> mean_var(result.posteriors[:θ][end])
(0.7498025177734003, 1.8757994411965801e-6)

```

If you are interested we have a list of examples available in the [documentation](https://biaslab.github.io/ReactiveMP.jl/stable/examples/overview/) and in the [repository](https://github.com/biaslab/ReactiveMP.jl/tree/master/demo). Worth noting though that ReactiveMP supports not only conjugate pairs.