howto custom Likelihood

Author: bgctw

docs/make.jl

@@ -24,6 +24,7 @@ makedocs(;
         ],
         "How to" => [
             ".. use GPU" => "tutorials/lux_gpu.md",
+            ".. specify log-Likelihood" => "tutorials/logden_user.md",
             ".. model independent parameters" => "tutorials/blocks_corr.md",
             ".. model site-global corr" => "tutorials/corr_site_global.md",
         ],

docs/src/tutorials/logden_user.md

@@ -0,0 +1,225 @@
+# How to specify a custom LogLikelihood of the observations
+
+
+``` @meta
+CurrentModule = HybridVariationalInference  
+```
+
+This guide shows how the user can specify a customized log-density function.
+
+## Motivation
+
+The loglikelihood function assigns a cost to the mismatch between predictions and
+observations. This often needs to be customized to the specific inversion.
+
+This guide walks through he specification of such a function and inspects
+differences among two log-likelihood functions.
+Specifically, it will assume observation errors to be independently distributed
+according to a LogNormal distribution with a specified fixed relative error,
+compared to an inversion assuming observation error to be distributed independently normal.
+
+First load necessary packages.
+
+``` julia
+using HybridVariationalInference
+using ComponentArrays: ComponentArrays as CA
+using Bijectors
+using SimpleChains
+using MLUtils
+using JLD2
+using Random
+using CairoMakie
+using PairPlots   # scatterplot matrices
+```
+
+This tutorial reuses and modifies the fitted object saved at the end of the
+[Basic workflow without GPU](@ref) tutorial, that used a log-Likelihood
+function assuming observation error to be distributed independently normal.
+
+``` julia
+fname = "intermediate/basic_cpu_results.jld2"
+print(abspath(fname))
+prob = probo_normal = load(fname, "probo");
+```
+
+## Write the LogLikelihood Function
+
+The function signature corresponds to the one of [`neg_logden_indep_normal`](@ref).
+of signature
+
+`neg_log_den_user(y_pred, y_obs, y_unc; kwargs...)`
+
+It takes inputs of predictions, `y_pred`, observations, `y_obs`,
+and uncertainties parameters, `y_unc` and returns the logarithm of the
+likelihhood up to a constant.
+
+All of the arguments are vectors of the same length specifying predictions and
+observations for one site.
+If `y_pred`, `y_obs` are given as a matrix of several column-vectors, their summed
+Likelihood is computed.
+
+The density of a LogNormal distribution is
+
+$$
+\frac{ 1 }{ x  \sqrt{2 \pi \sigma^2} } \exp\left( -\frac{ (\ln x-\mu)^2 }{2 \sigma^2} \right)$$
+
+where x is the observation, μ is the log of the prediction, and σ is the scale
+parameter that is related to the relative error, $c_v$ by $\sigma = \sqrt{ln(c^2_v + 1)}$.
+
+Taking the log:
+
+$$
+ -ln x -\frac{1}{2} ln \sigma^2 -\frac{1}{2} ln (2 \pi) -\frac{ (\ln x-\mu)^2 }{2 \sigma^2}$$
+
+Negating and dropping the constants $-\frac{1}{2} ln (2 \pi)$ and $-\frac{1}{2} ln \sigma^2$
+
+$$
+ ln x + \frac{1}{2} \left(\frac{ (\ln x-\mu)^2 }{\sigma^2} \right)$$
+
+``` julia
+function neg_logden_lognormalep_lognormal(y_pred, y_obs::AbstractArray{ET}, y_unc; 
+  σ2 = log(abs2(ET(0.02)) + ET(1))) where ET
+    lnx = log.(CA.getdata(y_obs))
+    μ = log.(CA.getdata(y_pred))
+    nlogL = sum(lnx .+ abs2.(lnx .- μ) ./ (ET(2) .* σ2))  
+    #nlogL = sum(lnx + (log(σ2) .+ abs2.(lnx .- μ) ./ σ2) ./ ET(2)) # nonconstant σ2
+    return (nlogL)
+end
+```
+
+If information on the different relative error by observation was available,
+we could pass that information using the DataLoader with `y_unc`, rather than
+assuming a constant relative error across observations.
+
+## Update the problem and redo the inversion
+
+HybridProblem has keyword argument `py` to specify the function of negative Log-Likelihood.
+
+``` julia
+prob_lognormal = HybridProblem(prob; py = neg_logden_lognormalep_lognormal)
+
+using OptimizationOptimisers
+import Zygote
+
+solver = HybridPosteriorSolver(; alg=Adam(0.02), n_MC=3)
+
+(; probo) = solve(prob_lognormal, solver; 
+    callback = callback_loss(100), # output during fitting
+    epochs = 20,
+); probo_lognormal = probo;
+```
+
+## Compare results between assumptions of observation error
+
+First, draw a sample form the inversion assumping normal and a sample from
+the inversion assuming loglornally distributed observation errors.
+
+``` julia
+n_sample_pred = 400
+(y_normal, θsP_normal, θsMs_normal) = (; y, θsP, θsMs) = predict_hvi(
+  Random.default_rng(), probo_normal; n_sample_pred)
+(y_lognormal, θsP_lognormal, θsMs_lognormal) = (; y, θsP, θsMs) = predict_hvi(
+  Random.default_rng(), probo_lognormal; n_sample_pred)
+```
+
+Get the original observations from the DataLoader of the problem, and
+compute the residuals.
+
+``` julia
+train_loader = get_hybridproblem_train_dataloader(probo_normal; scenario=())
+y_o = train_loader.data[3]
+resid_normal = y_o .- y_normal
+resid_lognormal = y_o .- y_lognormal
+```
+
+And compare plots of some of the results.
+
+``` julia
+i_out = 4
+i_site = 1
+fig = Figure(); ax = Axis(fig[1,1], xlabel="observations error (y_obs - y_pred)",ylabel="probability density")
+#hist!(ax, resid_normal[i_out,i_site,:], label="normal", normalization=:pdf)
+density!(ax, resid_normal[i_out,i_site,:], alpha = 0.8, label="normal")
+density!(ax, resid_lognormal[i_out,i_site,:], alpha = 0.8, label="lognormal")
+axislegend(ax, unique=true)
+fig
+```
+
+![](logden_user_files/figure-commonmark/cell-8-output-1.png)
+
+The density plot of the observation residuals does not show the lognormal shape.
+The used synthetic observations were actually noramally
+distributed around predictions with true parameters.
+
+How does the wrong assumption of observation error influence the parameter
+posterior?
+
+``` julia
+i_site = 1
+fig = Figure(); ax = Axis(fig[1,1], xlabel="global parameter K2",ylabel="probability density")
+#hist!(ax, resid_normal[i_out,i_site,:], label="normal", normalization=:pdf)
+density!(ax, θsP_normal[:K2,:], alpha = 0.8, label="normal")
+density!(ax, θsP_lognormal[:K2,:], alpha = 0.8, label="lognormal")
+axislegend(ax, unique=true)
+fig
+```
+
+![](logden_user_files/figure-commonmark/cell-9-output-1.png)
+
+The marginal posterior of the global parameters is also similar, with a small
+trend of lower values.
+
+``` julia
+i_site = 1
+θln = vcat(θsP_lognormal, θsMs_lognormal[i_site,:,:])
+θln_nt = NamedTuple(Symbol("$(k)_lognormal") => CA.getdata(θln[k,:]) for k in keys(θln[:,1])) # 
+#θn = vcat(θsP_normal, θsMs_normal[i_site,:,:])
+#θn_nt = NamedTuple(Symbol("$(k)_normal") => CA.getdata(θn[k,:]) for k in keys(θn[:,1])) # 
+# ntc = (;θn_nt..., θln_nt...)
+plt = pairplot(θln_nt)
+```
+
+![](logden_user_files/figure-commonmark/cell-10-output-1.png)
+
+The corner plot of the independent-parameters estimate  
+looks similar and shows correlations between site parameters, $r_1$ and $K_1$.
+
+``` julia
+i_out = 4
+fig = Figure(); ax = Axis(fig[1,1], xlabel="mean(y)",ylabel="sd(y)")
+ymean_normal = [mean(y_normal[i_out,s,:]) for s in axes(y_normal, 2)]
+ysd_normal = [std(y_normal[i_out,s,:]) for s in axes(y_normal, 2)]
+scatter!(ax, ymean_normal, ysd_normal, label="normal") 
+ymean_lognormal = [mean(y_lognormal[i_out,s,:]) for s in axes(y_lognormal, 2)]
+ysd_lognormal = [std(y_lognormal[i_out,s,:]) for s in axes(y_lognormal, 2)]
+scatter!(ax, ymean_lognormal, ysd_lognormal, label="lognormal") 
+axislegend(ax, unique=true)
+fig
+```
+
+![](logden_user_files/figure-commonmark/cell-11-output-1.png)
+
+The predicted magnitude of error in predictions for the fourth observation across sites
+is of the same magnitude,
+and still shows (although weaker) pattern of decreasing uncertainty with
+increasing value.
+
+``` julia
+plot_sd_vs_mean = (par) -> begin
+  fig = Figure(); ax = Axis(fig[1,1], xlabel="mean($par)",ylabel="sd($par)")
+  θmean_normal = [mean(θsMs_normal[s,par,:]) for s in axes(θsMs_normal, 1)]
+  θsd_normal = [std(θsMs_normal[s,par,:]) for s in axes(θsMs_normal, 1)]
+  scatter!(ax, θmean_normal, θsd_normal, label="correlated") 
+  θmean_lognormal = [mean(θsMs_lognormal[s,par,:]) for s in axes(θsMs_lognormal, 1)]
+  θsd_lognormal = [std(θsMs_lognormal[s,par,:]) for s in axes(θsMs_lognormal, 1)]
+  scatter!(ax, θmean_lognormal, θsd_lognormal, label="independent") 
+  axislegend(ax, unique=true)
+  fig
+end
+plot_sd_vs_mean(:K1)
+```
+
+![](logden_user_files/figure-commonmark/cell-12-output-1.png)
+
+For the assumed fixed relative error,the uncertainty in the model
+parameters, $K_1$, across sites is similar to the uncertainty with nornal log-likelihood.