```
= model(data...) # instantiate model on the data
m = vi(m, vi_alg) # perform VI on `m` using the VI method `vi_alg`, which returns a `VariationalPosterior` q
```

# Variational inference (VI) in Turing.jl

In this post we’ll have a look at what’s know as **variational inference (VI)**, a family of *approximate* Bayesian inference methods, and how to use it in Turing.jl as an alternative to other approaches such as MCMC. In particular, we will focus on one of the more standard VI methods called **Automatic Differentation Variational Inference (ADVI)**.

Here we will focus on how to use VI in Turing and not much on the theory underlying VI. If you are interested in understanding the mathematics you can checkout our write-up or any other resource online (there a lot of great ones).

Using VI in Turing.jl is very straight forward. If `model`

denotes a definition of a `Turing.Model`

, performing VI is as simple as

Thus it’s no more work than standard MCMC sampling in Turing.

To get a bit more into what we can do with `vi`

, we’ll first have a look at a simple example and then we’ll reproduce the tutorial on Bayesian linear regression using VI instead of MCMC. Finally we’ll look at some of the different parameters of `vi`

and how you for example can use your own custom variational family.

We first import the packages to be used:

```
using Random
using Turing
using Turing: Variational
using StatsPlots, Measures
Random.seed!(42);
```

## Simple example: Normal-Gamma conjugate model

The Normal-(Inverse)Gamma conjugate model is defined by the following generative process

\[\begin{align} s &\sim \mathrm{InverseGamma}(2, 3) \\ m &\sim \mathcal{N}(0, s) \\ x_i &\overset{\text{i.i.d.}}{=} \mathcal{N}(m, s), \quad i = 1, \dots, n \end{align}\]

Recall that *conjugate* refers to the fact that we can obtain a closed-form expression for the posterior. Of course one wouldn’t use something like variational inference for a conjugate model, but it’s useful as a simple demonstration as we can compare the result to the true posterior.

First we generate some synthetic data, define the `Turing.Model`

and instantiate the model on the data:

```
# generate data
= randn(2000); x
```

```
@model function model(x)
~ InverseGamma(2, 3)
s ~ Normal(0.0, sqrt(s))
m for i in 1:length(x)
~ Normal(m, sqrt(s))
x[i] end
end;
```

```
# Instantiate model
= model(x); m
```

Now we’ll produce some samples from the posterior using a MCMC method, which in constrast to VI is guaranteed to converge to the *exact* posterior (as the number of samples go to infinity).

We’ll produce 10 000 samples with 200 steps used for adaptation and a target acceptance rate of 0.65

If you don’t understand what “adaptation” or “target acceptance rate” refers to, all you really need to know is that `NUTS`

is known to be one of the most accurate and efficient samplers (when applicable) while requiring little to no hand-tuning to work well.

`setprogress!(false)`

`= sample(m, NUTS(), 10_000); samples_nuts `

```
┌ Info: Found initial step size
└ ϵ = 0.025
```

Now let’s try VI. The most important function you need to now about to do VI in Turing is `vi`

:

`@doc(Variational.vi)`

```
vi(model, alg::VariationalInference)
vi(model, alg::VariationalInference, q::VariationalPosterior)
vi(model, alg::VariationalInference, getq::Function, θ::AbstractArray)
```

Constructs the variational posterior from the `model`

and performs the optimization following the configuration of the given `VariationalInference`

instance.

# Arguments

`model`

:`Turing.Model`

or`Function`

z ↦ log p(x, z) where`x`

denotes the observations`alg`

: the VI algorithm used`q`

: a`VariationalPosterior`

for which it is assumed a specialized implementation of the variational objective used exists.`getq`

: function taking parameters`θ`

as input and returns a`VariationalPosterior`

`θ`

: only required if`getq`

is used, in which case it is the initial parameters for the variational posterior

Additionally, you can pass

- an initial variational posterior
`q`

, for which we assume there exists a implementation of`update(::typeof(q), θ::AbstractVector)`

returning an updated posterior`q`

with parameters`θ`

. - a function mapping \(\theta \mapsto q_{\theta}\) (denoted above
`getq`

) together with initial parameters`θ`

. This provides more flexibility in the types of variational families that we can use, and can sometimes be slightly more convenient for quick and rough work.

By default, i.e. when calling `vi(m, advi)`

, Turing use a *mean-field* approximation with a multivariate normal as the base-distribution. Mean-field refers to the fact that we assume all the latent variables to be *independent*. This the “standard” ADVI approach; see Automatic Differentiation Variational Inference (2016) for more. In Turing, one can obtain such a mean-field approximation by calling `Variational.meanfield(model)`

for which there exists an internal implementation for `update`

:

`@doc(Variational.meanfield)`

`meanfield([rng, ]model::Model)`

Creates a mean-field approximation with multivariate normal as underlying distribution.

Currently the only implementation of `VariationalInference`

available is `ADVI`

, which is very convenient and applicable as long as your `Model`

is differentiable with respect to the *variational parameters*, that is, the parameters of your variational distribution, e.g. mean and variance in the mean-field approximation.

`@doc(Variational.ADVI)`

`struct ADVI{AD} <: AdvancedVI.VariationalInference{AD}`

Automatic Differentiation Variational Inference (ADVI) with automatic differentiation backend `AD`

.

# Fields

`samples_per_step::Int64`

: Number of samples used to estimate the ELBO in each optimization step.`max_iters::Int64`

: Maximum number of gradient steps.`adtype::Any`

: AD backend used for automatic differentiation.

To perform VI on the model `m`

using 10 samples for gradient estimation and taking 1000 gradient steps is then as simple as:

```
# ADVI
= ADVI(10, 1000)
advi = vi(m, advi); q
```

```
┌ Info: [ADVI] Should only be seen once: optimizer created for θ
└ objectid(θ) = 0xe71f71459f6ae878
```

Unfortunately, for such a small problem Turing’s new `NUTS`

sampler is *so* efficient now that it’s not that much more efficient to use ADVI. So, so very unfortunate…

With that being said, this is not the case in general. For very complex models we’ll later find that `ADVI`

produces very reasonable results in a much shorter time than `NUTS`

.

And one significant advantage of using `vi`

is that we can sample from the resulting `q`

with ease. In fact, the result of the `vi`

call is a `TransformedDistribution`

from Bijectors.jl, and it implements the Distributions.jl interface for a `Distribution`

:

` q isa MultivariateDistribution`

`true`

This means that we can call `rand`

to sample from the variational posterior `q`

`histogram(rand(q, 1_000)[1, :])`

and `logpdf`

to compute the log-probability

`logpdf(q, rand(q))`

`4.974678862345768`

Let’s check the first and second moments of the data to see how our approximation compares to the point-estimates form the data:

`var(x), mean(x)`

`(0.9853179971937871, 0.014079819023689953)`

`mean(rand(q, 1000); dims=2)...,) (`

`(0.9857616152084558, 0.003756099419659516)`

AssertionError: Mean of m (VI posterior, 1000 samples): 0.0038075909149039334 Stacktrace: [1] top-level scope @ ~/work/TuringTutorials/TuringTutorials/tutorials/09-variational-inference/index.qmd:160

That’s pretty close! But we’re Bayesian so we’re not interested in *just* matching the mean. Let’s instead look the actual density `q`

.

For that we need samples:

```
= rand(q, 10000);
samples size(samples)
```

`(2, 10000)`

```
= histogram(
p1 1, :]; bins=100, normed=true, alpha=0.2, color=:blue, label="", ylabel="density"
samples[
)density!(samples[1, :]; label="s (ADVI)", color=:blue, linewidth=2)
density!(samples_nuts, :s; label="s (NUTS)", color=:green, linewidth=2)
vline!([var(x)]; label="s (data)", color=:black)
vline!([mean(samples[1, :])]; color=:blue, label="")
= histogram(
p2 2, :]; bins=100, normed=true, alpha=0.2, color=:blue, label="", ylabel="density"
samples[
)density!(samples[2, :]; label="m (ADVI)", color=:blue, linewidth=2)
density!(samples_nuts, :m; label="m (NUTS)", color=:green, linewidth=2)
vline!([mean(x)]; color=:black, label="m (data)")
vline!([mean(samples[2, :])]; color=:blue, label="")
plot(p1, p2; layout=(2, 1), size=(900, 500), legend=true)
```

For this particular `Model`

, we can in fact obtain the posterior of the latent variables in closed form. This allows us to compare both `NUTS`

and `ADVI`

to the true posterior \(p(s, m \mid x_1, \ldots, x_n)\).

*The code below is just work to get the marginals \(p(s \mid x_1, \ldots, x_n)\) and \(p(m \mid x_1, \ldots, x_n)\). Feel free to skip it.*

```
# closed form computation of the Normal-inverse-gamma posterior
# based on "Conjugate Bayesian analysis of the Gaussian distribution" by Murphy
function posterior(μ₀::Real, κ₀::Real, α₀::Real, β₀::Real, x::AbstractVector{<:Real})
# Compute summary statistics
= length(x)
n = mean(x)
x̄ = sum(xi -> (xi - x̄)^2, x)
sum_of_squares
# Compute parameters of the posterior
= κ₀ + n
κₙ = (κ₀ * μ₀ + n * x̄) / κₙ
μₙ = α₀ + n / 2
αₙ = β₀ + (sum_of_squares + n * κ₀ / κₙ * (x̄ - μ₀)^2) / 2
βₙ
return μₙ, κₙ, αₙ, βₙ
end
= posterior(0.0, 1.0, 2.0, 3.0, x)
μₙ, κₙ, αₙ, βₙ
# marginal distribution of σ²
# cf. Eq. (90) in "Conjugate Bayesian analysis of the Gaussian distribution" by Murphy
= InverseGamma(αₙ, βₙ)
p_σ² = z -> pdf(p_σ², z)
p_σ²_pdf
# marginal of μ
# Eq. (91) in "Conjugate Bayesian analysis of the Gaussian distribution" by Murphy
= μₙ + sqrt(βₙ / (αₙ * κₙ)) * TDist(2 * αₙ)
p_μ = z -> pdf(p_μ, z)
p_μ_pdf
# posterior plots
= plot()
p1 histogram!(samples[1, :]; bins=100, normed=true, alpha=0.2, color=:blue, label="")
density!(samples[1, :]; label="s (ADVI)", color=:blue)
density!(samples_nuts, :s; label="s (NUTS)", color=:green)
vline!([mean(samples[1, :])]; linewidth=1.5, color=:blue, label="")
plot!(range(0.75, 1.35; length=1_001), p_σ²_pdf; label="s (posterior)", color=:red)
vline!([var(x)]; label="s (data)", linewidth=1.5, color=:black, alpha=0.7)
xlims!(0.75, 1.35)
= plot()
p2 histogram!(samples[2, :]; bins=100, normed=true, alpha=0.2, color=:blue, label="")
density!(samples[2, :]; label="m (ADVI)", color=:blue)
density!(samples_nuts, :m; label="m (NUTS)", color=:green)
vline!([mean(samples[2, :])]; linewidth=1.5, color=:blue, label="")
plot!(range(-0.25, 0.25; length=1_001), p_μ_pdf; label="m (posterior)", color=:red)
vline!([mean(x)]; label="m (data)", linewidth=1.5, color=:black, alpha=0.7)
xlims!(-0.25, 0.25)
plot(p1, p2; layout=(2, 1), size=(900, 500))
```

## Bayesian linear regression example using ADVI

This is simply a duplication of the tutorial 5. Linear regression but now with the addition of an approximate posterior obtained using `ADVI`

.

As we’ll see, there is really no additional work required to apply variational inference to a more complex `Model`

.

This section is basically copy-pasting the code from the linear regression tutorial.

`Random.seed!(1);`

```
using FillArrays
using RDatasets
using LinearAlgebra
```

```
# Import the "Default" dataset.
= RDatasets.dataset("datasets", "mtcars");
data
# Show the first six rows of the dataset.
first(data, 6)
```

Row | Model | MPG | Cyl | Disp | HP | DRat | WT | QSec | VS | AM | Gear | Carb |
---|---|---|---|---|---|---|---|---|---|---|---|---|

String31 | Float64 | Int64 | Float64 | Int64 | Float64 | Float64 | Float64 | Int64 | Int64 | Int64 | Int64 | |

1 | Mazda RX4 | 21.0 | 6 | 160.0 | 110 | 3.9 | 2.62 | 16.46 | 0 | 1 | 4 | 4 |

2 | Mazda RX4 Wag | 21.0 | 6 | 160.0 | 110 | 3.9 | 2.875 | 17.02 | 0 | 1 | 4 | 4 |

3 | Datsun 710 | 22.8 | 4 | 108.0 | 93 | 3.85 | 2.32 | 18.61 | 1 | 1 | 4 | 1 |

4 | Hornet 4 Drive | 21.4 | 6 | 258.0 | 110 | 3.08 | 3.215 | 19.44 | 1 | 0 | 3 | 1 |

5 | Hornet Sportabout | 18.7 | 8 | 360.0 | 175 | 3.15 | 3.44 | 17.02 | 0 | 0 | 3 | 2 |

6 | Valiant | 18.1 | 6 | 225.0 | 105 | 2.76 | 3.46 | 20.22 | 1 | 0 | 3 | 1 |

```
# Function to split samples.
function split_data(df, at=0.70)
= size(df, 1)
r = Int(round(r * at))
index = df[1:index, :]
train = df[(index + 1):end, :]
test return train, test
end
# A handy helper function to rescale our dataset.
function standardize(x)
return (x .- mean(x; dims=1)) ./ std(x; dims=1)
end
function standardize(x, orig)
return (x .- mean(orig; dims=1)) ./ std(orig; dims=1)
end
# Another helper function to unstandardize our datasets.
function unstandardize(x, orig)
return x .* std(orig; dims=1) .+ mean(orig; dims=1)
end
function unstandardize(x, mean_train, std_train)
return x .* std_train .+ mean_train
end
```

`unstandardize (generic function with 2 methods)`

```
# Remove the model column.
select!(data, Not(:Model))
# Split our dataset 70%/30% into training/test sets.
= split_data(data, 0.7)
train, test = copy(train)
train_unstandardized
# Standardize both datasets.
= standardize(Matrix(train))
std_train = standardize(Matrix(test), Matrix(train))
std_test
# Save dataframe versions of our dataset.
= DataFrame(std_train, names(data))
train_cut = DataFrame(std_test, names(data))
test_cut
# Create our labels. These are the values we are trying to predict.
= train_cut[:, :MPG]
train_label = test_cut[:, :MPG]
test_label
# Get the list of columns to keep.
= filter(x -> !in(x, ["MPG"]), names(data))
remove_names
# Filter the test and train sets.
= Matrix(train_cut[:, remove_names]);
train = Matrix(test_cut[:, remove_names]); test
```

```
# Bayesian linear regression.
@model function linear_regression(x, y, n_obs, n_vars, ::Type{T}=Vector{Float64}) where {T}
# Set variance prior.
~ truncated(Normal(0, 100), 0, Inf)
σ²
# Set intercept prior.
~ Normal(0, 3)
intercept
# Set the priors on our coefficients.
~ MvNormal(Zeros(n_vars), 10.0 * I)
coefficients
# Calculate all the mu terms.
= intercept .+ x * coefficients
mu return y ~ MvNormal(mu, σ² * I)
end;
```

```
= size(train)
n_obs, n_vars = linear_regression(train, train_label, n_obs, n_vars); m
```

## Performing VI

First we define the initial variational distribution, or, equivalently, the family of distributions to consider. We’re going to use the same mean-field approximation as Turing will use by default when we call `vi(m, advi)`

, which we obtain by calling `Variational.meanfield`

. This returns a `TransformedDistribution`

with a `TuringDiagMvNormal`

as the underlying distribution and the transformation mapping from the reals to the domain of the latent variables.

```
= Variational.meanfield(m)
q0 typeof(q0)
```

MultivariateTransformed{TuringDiagMvNormal{Vector{Float64}, Vector{Float64}}, Stacked{Vector{Any}, Vector{UnitRange{Int64}}}} (alias for Bijectors.TransformedDistribution{DistributionsAD.TuringDiagMvNormal{Array{Float64, 1}, Array{Float64, 1}}, Stacked{Array{Any, 1}, Array{UnitRange{Int64}, 1}}, ArrayLikeVariate{1}})

`= ADVI(10, 10_000) advi `

`ADVI{AutoForwardDiff{nothing, Nothing}}(10, 10000, AutoForwardDiff{nothing, Nothing}(nothing))`

Turing also provides a couple of different optimizers:

`TruncatedADAGrad`

(default)`DecayedADAGrad`

as these are well-suited for problems with high-variance stochastic objectives, which is usually what the ELBO ends up being at different times in our optimization process.

With that being said, thanks to Requires.jl, if we add a `using Flux`

prior to `using Turing`

we can also make use of all the optimizers in `Flux`

, e.g. `ADAM`

, without any additional changes to your code! For example:

```
using Flux, Turing
using Turing.Variational
vi(m, advi; optimizer=Flux.ADAM())
```

just works.

For this problem we’ll use the `DecayedADAGrad`

from Turing:

`= Variational.DecayedADAGrad(1e-2, 1.1, 0.9) opt `

`AdvancedVI.DecayedADAGrad(0.01, 1.1, 0.9, IdDict{Any, Any}())`

```
= vi(m, advi, q0; optimizer=opt)
q typeof(q)
```

MultivariateTransformed{TuringDiagMvNormal{Vector{Float64}, Vector{Float64}}, Stacked{Vector{Any}, Vector{UnitRange{Int64}}}} (alias for Bijectors.TransformedDistribution{DistributionsAD.TuringDiagMvNormal{Array{Float64, 1}, Array{Float64, 1}}, Stacked{Array{Any, 1}, Array{UnitRange{Int64}, 1}}, ArrayLikeVariate{1}})

*Note: as mentioned before, we internally define a update(q::TransformedDistribution{<:TuringDiagMvNormal}, θ::AbstractVector) method which takes in the current variational approximation q together with new parameters z and returns the new variational approximation. This is required so that we can actually update the Distribution object after each optimization step.*

*Alternatively, we can instead provide the mapping \(\theta \mapsto q_{\theta}\) directly together with initial parameters using the signature vi(m, advi, getq, θ_init) as mentioned earlier. We’ll see an explicit example of this later on!*

To compute statistics for our approximation we need samples:

`= rand(q, 10_000); z `

Now we can for example look at the average

`= vec(mean(z; dims=2)) avg `

```
12-element Vector{Float64}:
0.21624785317661832
0.0031558668201944542
0.37231221668710746
-0.11018007422716512
-0.08101937336233356
0.615155109186401
-0.0017316410253455064
0.07866223388683276
-0.06573931922747253
0.12502799799059774
0.18738303423173688
-0.6202772215781214
```

The vector has the same ordering as the model, e.g. in this case `σ²`

has index `1`

, `intercept`

has index `2`

and `coefficients`

has indices `3:12`

. If you forget or you might want to do something programmatically with the result, you can obtain the `sym → indices`

mapping as follows:

```
= bijector(m, Val(true));
_, sym2range sym2range
```

`(intercept = UnitRange{Int64}[2:2], σ² = UnitRange{Int64}[1:1], coefficients = UnitRange{Int64}[3:12])`

For example, we can check the sample distribution and mean value of `σ²`

:

```
histogram(z[1, :])
union(sym2range[:σ²]...)] avg[
```

```
1-element Vector{Float64}:
0.21624785317661832
```

`union(sym2range[:intercept]...)] avg[`

```
1-element Vector{Float64}:
0.0031558668201944542
```

`union(sym2range[:coefficients]...)] avg[`

```
10-element Vector{Float64}:
0.37231221668710746
-0.11018007422716512
-0.08101937336233356
0.615155109186401
-0.0017316410253455064
0.07866223388683276
-0.06573931922747253
0.12502799799059774
0.18738303423173688
-0.6202772215781214
```

*Note: as you can see, this is slightly awkward to work with at the moment. We’ll soon add a better way of dealing with this.*

With a bit of work (this will be much easier in the future), we can also visualize the approximate marginals of the different variables, similar to `plot(chain)`

:

```
function plot_variational_marginals(z, sym2range)
= []
ps
for (i, sym) in enumerate(keys(sym2range))
= union(sym2range[sym]...) # <= array of ranges
indices if sum(length.(indices)) > 1
= 1
offset for r in indices
= density(
p :];
z[r, ="$(sym)[$offset]",
title=10,
titlefontsize="",
label="Density",
ylabel=1.5mm,
margin
)push!(ps, p)
+= 1
offset end
else
= density(
p first(indices), :];
z[="$(sym)",
title=10,
titlefontsize="",
label="Density",
ylabel=1.5mm,
margin
)push!(ps, p)
end
end
return plot(ps...; layout=(length(ps), 1), size=(500, 2000), margin=4.0mm)
end
```

`plot_variational_marginals (generic function with 1 method)`

`plot_variational_marginals(z, sym2range)`

And let’s compare this to using the `NUTS`

sampler:

`= sample(m, NUTS(), 10_000); chain `

```
┌ Info: Found initial step size
└ ϵ = 0.025
```

`plot(chain; margin=12.00mm)`