Text fixes

Author: penelopeysm

tutorials/gaussian-mixture-models/index.qmd

@@ -160,8 +160,10 @@ We consider the samples of the location parameters $\mu_1$ and $\mu_2$ for the t
 plot(chains[["μ[1]", "μ[2]"]]; legend=true)
 ```
 
-In the course of sampling, it is possible that the values of the parameters $\mu_1$ and $\mu_2$ will be swapped with each other.
-This is because either model parameter $\mu_k$ to be assigned to either of the corresponding true means, and since the resulting mixture distribution is still the same even if we swap the parameters $\mu_1$ and $\mu_2$, the model does not distinguish between the two assignments.
+From the plots above, we can see that the chains have converged to seemingly different values for the parameters $\mu_1$ and $\mu_2$.
+However, these actually represent the same solution: it does not matter whether we assign $\mu_1$ to the first cluster and $\mu_2$ to the second, or vice versa, since the resulting sum is the same.
+In other words, the posterior distribution is fundamentally multimodal.
+(In principle it is also possible for the parameters to swap places _within_ a single chain, although this does not happen in this example.)
 For more information see the [Stan documentation](https://mc-stan.org/users/documentation/case-studies/identifying_mixture_models.html), or Bishop's book, where the concept of _identifiability_ is discussed.
 
 Having $\mu_1$ and $\mu_2$ swap can complicate the interpretation of the results, especially when different chains converge to different assignments.