UKI comments

docs/src/learning_rate_scheduler.md

@@ -50,10 +50,12 @@ Right: the error and spread of the different timesteppers at their final iterati
 
 Finding the Posterior (terminating at ``T=1``):
  - DMC with termination (purple), closely mimics a small-fixed timestep (green) that finishes stepping at ``T=1``. Both retain more spread than other approaches, and DMC is far more efficient, typically terminating after around 10-20 steps, where fixed-stepping takes 50. We see that (for this experiment) this is a conservative estimate, as continuing to solve (e.g. brown) until later times often leads to a better error while still retaining similar "error vs spread" curves (before inevitable collapse). This is consistent with the concept of approximating the posterior, over seeking an optimizer.
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+- The behavior observed in UKI is similar to EKI
+
 Optimizing the objective function (continuing ``T \to \infty``):
 - Large fixed step (orange). This is very efficient, but can get stuck when drawn too large, (perhaps unintuitive from a gradient-descent perspective). It typically also collapses the ensemble. On average it gives lower error to the true parameters than DMC. 
  - Both EKSStable and DMC with continuation schedulers, perform very similarly. Both retain good ensemble spread during convergence, and collapse after finding a local optimum. This optimum on average has the best error to the true parameters in this experiment. They appear to consistently find the same optimum as ``T\to\infty`` but DMC finds this in fewer iterations.
+- The UKI behavior is largely similar to EKI here, except that ensemble spread is retained in the ``T\to\infty`` limit in all cases, from inflation of the parameter covariance (``\Sigma_\omega``) within our implementation. 
 
 ### DMC as a default in future?