# Issue with the Metropolis adjusted Langevin sampler from AdvancedMH.jl

**URL:** <https://discourse.julialang.org/t/issue-with-the-metropolis-adjusted-langevin-sampler-from-advancedmh-jl/110155>\
**Category:** Statistics\
**Tags:** question, statistics, turing, monte-carlo, mcmc\
**Created:** [February 13, 2024, 2:46pm UTC](https://discourse.julialang.org/t/issue-with-the-metropolis-adjusted-langevin-sampler-from-advancedmh-jl/110155 "2024-02-13T14:46:52Z")\
**Posts on this page:** 2\
**Page:** 1

<div class="post-metadata">

**Author:** ![e3c6](https://avatars.discourse-cdn.com/v4/letter/e/e79b87/32.png) [@e3c6](https://discourse.julialang.org/u/e3c6)\
**Post date:** [February 13, 2024, 2:46pm UTC](https://discourse.julialang.org/t/issue-with-the-metropolis-adjusted-langevin-sampler-from-advancedmh-jl/110155/1 "2024-02-13T14:46:52Z")

</div>

I am trying to use a Metropolis adjusted Langevin sampler, from `AdvancedMH.MALA`. I did a simple test to see if things are working correctly, which is to sample from a multivariate normal distribution. Here is the code:

```julia
import AdvancedMH, LogDensityProblems, Distributions

struct TheNormalLogDensity{M}
    A::M
end

# can do gradient
LogDensityProblems.capabilities(::Type{<:TheNormalLogDensity}) = LogDensityProblems.LogDensityOrder{1}()

LogDensityProblems.dimension(d::TheNormalLogDensity) = size(d.A, 1)
LogDensityProblems.logdensity(d::TheNormalLogDensity, x) = -x' * d.A * x / 2

function LogDensityProblems.logdensity_and_gradient(d::TheNormalLogDensity, x)
    return -x' * d.A * x / 2, -d.A * x
end

Σ = [1.5 0.35; 0.35 1.0]
σ² = 0.01
spl = AdvancedMH.MALA(g -> Distributions.MvNormal((σ² / 2) .* g, σ² * I))

chain = AdvancedMH.sample(TheNormalLogDensity(inv(Σ)), spl, 500000; initial_params=ones(2))
data = stack([c.params for c = chain])

```

I can then compute an empirical covariance matrix from the sampled data, which should be close to `Σ` because we have a lot of samples here.

```julia
# this should be close to Σ
C = data * data' / size(data, 2)

```

But actually, I get an empirical covariance that seems very different from `Σ`:

```julia
> data * data' / size(data, 2)
2×2 Matrix{Float64}:
 0.507211 0.124855
 0.124855 0.341631

```

Am I doing something wrong here?

---

<div class="post-metadata">

**Author:** ![e3c6](https://avatars.discourse-cdn.com/v4/letter/e/e79b87/32.png) [@e3c6](https://discourse.julialang.org/u/e3c6)\
**Post date:** [February 13, 2024, 10:13pm UTC](https://discourse.julialang.org/t/issue-with-the-metropolis-adjusted-langevin-sampler-from-advancedmh-jl/110155/2 "2024-02-13T22:13:39Z")

</div>

This seems to be a bug: [MALA implementation results in the wrong covariance · Issue #95 · TuringLang/AdvancedMH.jl · GitHub](https://github.com/TuringLang/AdvancedMH.jl/issues/95)