trying to fix the mess that i made when i started writing docs

docs/src/tutorials/regression.md

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+
+
+``` @meta
+CurrentModule = LaplaceRedux
+```
+
+## Libraries
+
+Import the libraries required to run this example
+
+``` julia
+using Pkg; Pkg.activate("docs")
+# Import libraries
+using Flux, Plots, TaijaPlotting, Random, Statistics, LaplaceRedux
+theme(:wong)
+```
+
+## Data
+
+We first generate some synthetic data:
+
+``` julia
+using LaplaceRedux.Data
+n = 300       # number of observations
+σtrue = 0.30  # true observational noise
+x, y = Data.toy_data_regression(n;noise=σtrue)
+xs = [[x] for x in x]
+X = permutedims(x)
+```
+
+## MLP
+
+We set up a model and loss with weight regularization:
+
+``` julia
+data = zip(xs,y)
+n_hidden = 50
+D = size(X,1)
+nn = Chain(
+    Dense(D, n_hidden, tanh),
+    Dense(n_hidden, 1)
+)  
+loss(x, y) = Flux.Losses.mse(nn(x), y)
+```
+
+We train the model:
+
+``` julia
+using Flux.Optimise: update!, Adam
+opt = Adam(1e-3)
+epochs = 1000
+avg_loss(data) = mean(map(d -> loss(d[1],d[2]), data))
+show_every = epochs/10
+
+for epoch = 1:epochs
+  for d in data
+    gs = gradient(Flux.params(nn)) do
+      l = loss(d...)
+    end
+    update!(opt, Flux.params(nn), gs)
+  end
+  if epoch % show_every == 0
+    println("Epoch " * string(epoch))
+    @show avg_loss(data)
+  end
+end
+```
+
+## Laplace Approximation
+
+Laplace approximation can be implemented as follows:
+
+``` julia
+subset_w = :all
+la = Laplace(nn; likelihood=:regression, subset_of_weights=subset_w)
+fit!(la, data)
+plot(la, X, y; zoom=-5, size=(400,400))
+```
+
+![](regression_files/figure-commonmark/cell-6-output-1.svg)
+
+Next we optimize the prior precision $P_0$ and and observational noise $\sigma$ using Empirical Bayes:
+
+``` julia
+optimize_prior!(la; verbose=true)
+plot(la, X, y; zoom=-5, size=(400,400))
+```
+
+    loss(exp.(logP₀), exp.(logσ)) = 104.78561546028183
+    Log likelihood: -70.48742092717352
+    Log det ratio: 41.1390695290454
+    Scatter: 27.45731953717124
+    loss(exp.(logP₀), exp.(logσ)) = 104.9736282327825
+    Log likelihood: -74.85481357633174
+    Log det ratio: 46.59827618892447
+    Scatter: 13.639353123977058
+    loss(exp.(logP₀), exp.(logσ)) = 84.38222356291794
+    Log likelihood: -54.86985627702764
+    Log det ratio: 49.92347667032635
+    Scatter: 9.101257901454279
+
+    loss(exp.(logP₀), exp.(logσ)) = 84.53493863039972
+    Log likelihood: -55.013137224636
+    Log det ratio: 51.43622180356522
+    Scatter: 7.607381007962245
+    loss(exp.(logP₀), exp.(logσ)) = 83.95921598606084
+    Log likelihood: -54.41492266831395
+    Log det ratio: 51.794520967146354
+    Scatter: 7.294065668347427
+    loss(exp.(logP₀), exp.(logσ)) = 83.03505059021086
+    Log likelihood: -53.50540374805591
+    Log det ratio: 51.574749787874794
+    Scatter: 7.484543896435117
+
+    loss(exp.(logP₀), exp.(logσ)) = 82.97840036025443
+    Log likelihood: -53.468475394115416
+    Log det ratio: 51.17273666609066
+    Scatter: 7.847113266187348
+    loss(exp.(logP₀), exp.(logσ)) = 82.98550025321256
+    Log likelihood: -53.48508828283467
+    Log det ratio: 50.81442045868749
+    Scatter: 8.186403482068298
+    loss(exp.(logP₀), exp.(logσ)) = 82.9584040552644
+    Log likelihood: -53.45989630330948
+    Log det ratio: 50.59063282947659
+    Scatter: 8.406382674433235
+
+
+    loss(exp.(logP₀), exp.(logσ)) = 82.94465052328141
+    Log likelihood: -53.44600301956443
+    Log det ratio: 50.500079294094405
+    Scatter: 8.497215713339543
+
+![](regression_files/figure-commonmark/cell-7-output-5.svg)