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exponential.jl
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### A Pluto.jl notebook ###
# v0.17.4
using Markdown
using InteractiveUtils
# ╔═╡ b136d1ee-eeaa-4d80-bcb3-d97c4520171f
let
using PlutoUI
TableOfContents()
end
# ╔═╡ d38f21e1-11cf-4d80-bdaf-a542b782ae0d
begin
using NestedSamplers
using ForwardDiff
using FiniteDifferences
using Distributions
using LinearAlgebra
using StatsBase
using MCMCChains
using StatsPlots
using NLsolve
using BayesianOptimization
using GaussianProcesses
using Gaston
set(term="svg")
set(showable="svg")
set(termopts="size 550,325 font 'JuliaMono,12'")
end
# ╔═╡ afb580b8-6517-4b98-87ed-9584bae6055e
md"""
# *MEANS*: Maximum entropy as nested sampling
A quick proof of principle to show that it is in fact doable!
This notebook focuses on automatically differentiating $\log Z$ wrt. to the Lagrange multiplier $\lambda$.
!!! note "Abstract"
MEANS is about:
- Differentiating the log partition function $\log Z$ directly using *automatic differentiation*. This allows one to solve for the Lagrange multipliers in maximum entropy (or miniumum Kullback-Leibler distance) problems and **evaluate** all kinds of statistics directly.
- And, in the same line of reasoning, dynamically **sample** from the new maximum entropy (or miniumum Kullback-Leibler distance) distribution by casting it as a posterior in a standard nested sampling problem.
!!! terminology "Single constraints"
Nested sampling can simulate any temperature (even negative ones), so if there is only one constraint -- which is the vast majority of use cases -- **you only need to do the nested sampling run** for $\lambda = 1$ to "map out" the structure of the ME problem. Then varying the Lagrange multipliers can be done after the run is finished, which modulates the a posteriori sampling weights. We don't implement the sample reweighting procedure in this notebook, however.
"""
# ╔═╡ 0c0c8fbb-018f-4971-96cf-893f292618dd
md"""
## Notes
Start with explaining maximum entropy (ME) with `Exp(λ)` as an example. Derive from $q(x) = Uniform(0, L)$ with $L > 0$ and take $L \rightarrow \infty$. Emphasize that ME is about *updating* distributions with constraints.
### AD software
Working:
- `ForwardDiff` + `NestedSamplers` (Julia)
AD software that did not immediately work out of the box:
- `jaxns` (Python): forward differentiation gives back nans. Reverse mode differentiation chokes on a while loop (needs `lax.scan`)
- `Zygote` + `NestedSamplers` (Julia): chokes on a try/except block
"""
# ╔═╡ 4c10e97d-ba75-476c-8679-5e41a5b4fd66
md"""
## `Exponential(x|λ)`
"""
# ╔═╡ 42170bbe-6e56-4199-bf20-378b6fa681d4
begin
const L = 10.
function logz(λ)
logl(X) = -λ*sum(X)
prior(X) = L*sum(X)
bounds = Bounds.MultiEllipsoid
prop = Proposals.Slice(slices=10)
model = NestedModel(logl, prior)
sampler = Nested(1, 10; bounds=bounds, proposal=prop)
chain, state = sample(model, sampler; dlogz=0.1)
return state.logz
end
dlogz(λ) = ForwardDiff.derivative(logz, λ);
end
# ╔═╡ 0b1ff5d3-db2f-4a64-83c0-0614552f5f0e
begin
λ = collect(0:0.01:10)
z_analytical(λ) = (1 - exp(-L*λ))/(L*λ)
logz_analytical(λ) = log(z_analytical(λ))
dlogz_analytical(λ) = L/(exp(L*λ) - 1) - 1/λ
lz = logz.(λ)
dlz = dlogz.(λ);
end
# ╔═╡ 7b044833-bbfb-4802-964f-59b2394b5d0c
md"""
### `logz`
The noisy estimate seems to be quite sufficient for our goals. Its accuracy can simply be increased by increasing the number of live points in the nested sampling call.
"""
# ╔═╡ 6773cbd3-85fc-46c6-bf98-c996f4b40ae4
let
Gaston.plot(
λ, lz, leg="'nested sampling'",
Axes(xlabel = :λ, ylabel = "'log Z(λ)'")
)
Gaston.plot!(λ, logz_analytical.(λ), leg="'analytical'")
end
# ╔═╡ a1b378eb-421e-423b-b22e-aad02a03a5ab
md"""
### `dlogz`
The estimate of the gradient `dlogz(λ)` is less noisier than the estimate of `logz(λ)` itself. This is remarkable because differentiation is high-pass. This points toward positive correlations being exploited (a difference of positively correlated variables can be estimated more accurately than their sum), but where, how and why exactly?
"""
# ╔═╡ 08ce46e3-19ff-4b62-ba16-b0c92c748a38
let
Gaston.plot(
λ, dlz, leg="'nested sampling + automatic differentiation'",
Axes(xlabel = :λ, ylabel = "'dlog[Z(λ)]/dλ'", key="bottom")
)
Gaston.plot!(λ, dlogz_analytical.(λ), leg="'analytical'")
end
# ╔═╡ 0144cbd1-2146-4548-b054-2008c49afb82
md"""
#### Finite differences
Finally, the finite difference methods all diverge for any order `p`. It is also much slower.
"""
# ╔═╡ e6c2236a-070d-4c84-9c14-15e77fd70455
begin
p = 5
finite_diff(fdm, λ) = fdm(p, 1)(logz, λ)
Gaston.plot(
λ, finite_diff.(backward_fdm, λ), leg="'Backward difference order $p'",
Axes(xlabel = :λ, ylabel = "'dlog[Z(λ)]/dλ'")
)
Gaston.plot!(λ, finite_diff.(central_fdm, λ), leg="'Central difference order $p'")
Gaston.plot!(λ, finite_diff.(forward_fdm, λ), leg="'Forward difference order $p'")
end
# ╔═╡ 18d76a9d-92ae-46d9-a2e4-3527137e1979
md"""
### `ddlogz`
The second order derivative $\frac{d^2[\log Z(\lambda)]}{d\lambda^2}$ is also possible and seems quite fast, too. At least in one dimension.
"""
# ╔═╡ 18db5688-783a-4aa5-a7ae-565dff41e67c
begin
ddlogz(λ) = ForwardDiff.derivative(dlogz, λ)
ddlogz_analytical(λ) = 1/λ^2 + L^2/(2 - 2*cosh(L*λ))
ddlz = ddlogz.(λ);
end
# ╔═╡ 6be03f4e-8f04-4fa2-ba02-1da97f745f91
begin
Gaston.plot(
λ, ddlz, leg="'nested sampling + automatic differentiation'",
Axes(xlabel = :λ, ylabel = "'d^2log[Z(λ)]/dλ^2'")
)
Gaston.plot!(λ, ddlogz_analytical.(λ), leg="'analytical'")
end
# ╔═╡ 5a6be9ec-4b60-4003-b1b9-2e847bdcd86f
md"""
### Solving for `λ` given `μ`
The problem to be solved is $\frac{L\lambda}{e^{L\lambda} - 1} + \mu\lambda - 1 = 0$. There is no analytical solution but this can be solved easily numerically.
In the general case, we try to find it automatically with Bayesian optimization. Can we use the second derivative?
"""
# ╔═╡ e10e0b04-2521-4661-a227-2cbe9171d474
begin
μ = 1.4
f(λ) = L*λ/(exp(L*λ) - 1) + μ*λ - 1
function residuals!(F, λ)
F[1] = f(λ[1])
end
res = nlsolve(residuals!, [1/μ], autodiff = :forward)
λ_optimal = res.zero[1]
@info "nlsolve result" res
@info "solution" λ_optimal
Gaston.plot(λ, f.(λ), leg="'Find λ such that f(λ)=0'", Axes(xlabel=:λ, ylabel="'f(λ)'"))
end
# ╔═╡ ebead40d-a95a-4c00-9833-8ea1e4f837bd
chain, state = let
function get_state(λ)
logl(X) = -λ*sum(X)
prior(X) = L*sum(X)
bounds = Bounds.MultiEllipsoid
prop = Proposals.Slice(slices=10)
model = NestedModel(logl, prior)
# We need to increase the number of live points to get good
# posterior p(x|λ)
sampler = Nested(1, 500; bounds=bounds, proposal=prop)
chain, state = sample(model, sampler; dlogz=0.2)
return chain, state
end
get_state(λ_optimal)
end;
# ╔═╡ 67385dc5-738d-4551-bc31-c0c9b489c79a
md"""
### Sampling from the ME distribution
Check the posterior samples of the maximum entropy (ME) distribution $p(x|λ_0)$ where $\lambda_0$ solves the given constraint $<x>_{p(x|λ_0)} = \mu$. And it works. Note that for good posterior samples we need to increase the number of live points for the nested sampling. Compare that to the fact that the number of live points for `dlogz` could be small to get already reasonable estimates.
"""
# ╔═╡ ef689287-6b2c-4f0f-831e-a5a430189946
begin
p_analytical(x, λ) = exp(λ*(L-x))*λ*(0<=x<=L)/(exp(L*λ) - 1)
# Need to give weights separately
StatsPlots.histogram(
chain.value.data[:,1,1], weights=chain.value.data[:,2,1],
normalize=:pdf, label="posterior samples from p(x|λ(μ))"
)
x = 0:0.01:L
StatsPlots.plot!(x, p_analytical.(x, λ_optimal), label="analytic")
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
BayesianOptimization = "4c6ed407-134f-591c-93fa-e0f7c164a0ec"
Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
FiniteDifferences = "26cc04aa-876d-5657-8c51-4c34ba976000"
ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
Gaston = "4b11ee91-296f-5714-9832-002c20994614"
GaussianProcesses = "891a1506-143c-57d2-908e-e1f8e92e6de9"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
MCMCChains = "c7f686f2-ff18-58e9-bc7b-31028e88f75d"
NLsolve = "2774e3e8-f4cf-5e23-947b-6d7e65073b56"
NestedSamplers = "41ceaf6f-1696-4a54-9b49-2e7a9ec3782e"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"
[compat]
BayesianOptimization = "~0.2.5"
Distributions = "~0.24.18"
FiniteDifferences = "~0.12.24"
ForwardDiff = "~0.10.27"
Gaston = "~1.0.5"
GaussianProcesses = "~0.12.4"
MCMCChains = "~4.14.1"
NLsolve = "~4.5.1"
NestedSamplers = "~0.8.1"
PlutoUI = "~0.7.38"
StatsBase = "~0.33.16"
StatsPlots = "~0.14.33"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised
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