Make GMM tutorial a bit less demanding

Author: penelopeysm

tutorials/gaussian-mixture-models/index.qmd

@@ -38,11 +38,11 @@ Random.seed!(3)
 
 # Define Gaussian mixture model.
 w = [0.5, 0.5]
-μ = [-3.5, 0.5]
-mixturemodel = MixtureModel([MvNormal(Fill(μₖ, 2), I) for μₖ in μ], w)
+μ = [-2.0, 2.0]
+mixturemodel = MixtureModel([MvNormal(Fill(μₖ, 2), 0.2 * I) for μₖ in μ], w)
 
 # We draw the data points.
-N = 60
+N = 30
 x = rand(mixturemodel, N);
 ```
 
@@ -112,7 +112,7 @@ model = gaussian_mixture_model(x);
 ```
 
 We run a MCMC simulation to obtain an approximation of the posterior distribution of the parameters $\mu$ and $w$ and assignments $k$.
-We use a `Gibbs` sampler that combines a [particle Gibbs](https://www.stats.ox.ac.uk/%7Edoucet/andrieu_doucet_holenstein_PMCMC.pdf) sampler for the discrete parameters (assignments $k$) and a Hamiltonian Monte Carlo sampler for the continuous parameters ($\mu$ and $w$).
+We use a `Gibbs` sampler that combines a [particle Gibbs](https://www.stats.ox.ac.uk/%8Edoucet/andrieu_doucet_holenstein_PMCMC.pdf) sampler for the discrete parameters (assignments $k$) and a Hamiltonian Monte Carlo sampler for the continuous parameters ($\mu$ and $w$).
 We generate multiple chains in parallel using multi-threading.
 
 ```{julia}
@@ -145,7 +145,7 @@ let
         # μ[1] and μ[2] can switch places, so we sort the values first.
         chain = Array(chains[:, ["μ[1]", "μ[2]"], i])
         μ_mean = vec(mean(chain; dims=1))
-        @assert isapprox(sort(μ_mean), μ; rtol=0.1) "Difference between estimated mean of μ ($(sort(μ_mean))) and data-generating μ ($μ) unexpectedly large!"
+        @assert isapprox(sort(μ_mean), μ; atol=0.5) "Difference between estimated mean of μ ($(sort(μ_mean))) and data-generating μ ($μ) unexpectedly large!"
     end
 end
 ```
@@ -212,7 +212,7 @@ let
         # μ[1] and μ[2] can no longer switch places. Check that they've found the mean
         chain = Array(chains[:, ["μ[1]", "μ[2]"], i])
         μ_mean = vec(mean(chain; dims=1))
-        @assert isapprox(sort(μ_mean), μ; rtol=0.4) "Difference between estimated mean of μ ($(sort(μ_mean))) and data-generating μ ($μ) unexpectedly large!"
+        @assert isapprox(sort(μ_mean), μ; atol=0.5) "Difference between estimated mean of μ ($(sort(μ_mean))) and data-generating μ ($μ) unexpectedly large!"
     end
 end
 ```
@@ -350,7 +350,7 @@ let
         # μ[1] and μ[2] can no longer switch places. Check that they've found the mean
         chain = Array(chains[:, ["μ[1]", "μ[2]"], i])
         μ_mean = vec(mean(chain; dims=1))
-        @assert isapprox(sort(μ_mean), μ; rtol=0.4) "Difference between estimated mean of μ ($(sort(μ_mean))) and data-generating μ ($μ) unexpectedly large!"
+        @assert isapprox(sort(μ_mean), μ; atol=0.5) "Difference between estimated mean of μ ($(sort(μ_mean))) and data-generating μ ($μ) unexpectedly large!"
     end
 end
 ```