integrated RL, spell check

bayesian-intro.md

@@ -1,7 +1,7 @@
 Introduction to Posterior Inference
 =======================
 
-We have to keep in mind that for all measurements, models, and discretizations we have uncertainties. In the former, this typically appears in the form of measurements errors, model equations usually encompass only parts of a system we're interested in, and for numerical simulations we inherently introduce discretization errors. So a very important question to ask here is how sure we can be sure that an answer we obtain is the correct one. From a statistics viewpoint, we'd like to know the probability distribution for the posterior, i.e., the outcomes.
+We should keep in mind that for all measurements, models, and discretizations we have uncertainties. For the former, this typically appears in the form of measurements errors, while model equations usually encompass only parts of a system we're interested in, and for numerical simulations we inherently introduce discretization errors. So a very important question to ask here is how sure we can be sure that an answer we obtain is the correct one. From a statistics viewpoint, we'd like to know the probability distribution for the posterior, i.e., the different outcomes that are possible.
 
 This admittedly becomes even more difficult in the context of machine learning:
 we're typically facing the task of approximating complex and unknown functions.
@@ -10,22 +10,36 @@ yields a _maximum likelihood estimation_ (MLE) for the parameters of the network
 However, this MLE viewpoint does not take any of the uncertainties mentioned above into account:
 for DL training, we likewise have a numerical optimization, and hence an inherent
 approximation error and uncertainty regarding the learned representation.
-Ideally, we could change our learning problem such that we could do _posterior inference_,
+Ideally, we should reformulate our learning problem such that it enables _posterior inference_,
 i.e. learn to produce the full output distribution. However, this turns out to be an
 extremely difficult task.
 
 This where so called _Bayesian neural network_ (BNN) approaches come into play. They 
-make posterior inference possible by making assumptions about the probability 
-distributions of individual parameters of the network. Nonetheless, the task
+make a form of posterior inference possible by making assumptions about the probability 
+distributions of individual parameters of the network. With a distribution for the
+parameters we can evaluate the network multiple times to obtain different versions
+of the output, and in this way sample the distribution of the output.
+
+Nonetheless, the task
 remains very challenging. Training a BNN is typically significantly more difficult
 than training a regular NN. However, this should come as no surprise, as we're trying to 
-learn something fundamentally different in this case: a full probability distribution 
-instead of a point estimate.
+learn something fundamentally different here: a full probability distribution 
+instead of a point estimate. (All previous chapters "just" dealt with
+learning such point estimates.)
 
 ![Divider](resources/divider5.jpg)
 
+## Introduction to Bayesian Neural Networks
+
+
+**TODO, integrate Maximilians intro section here**
+...
+
+
 ## A practical example
 
-first example here with airfoils, extension from {doc}`supervised-airfoils`
-
+As a first real example for posterior inference with BNNs, let's revisit the
+case of turbulent flows around airfoils, from {doc}`supervised-airfoils`. However,
+in contrast to the point estimate learned in this section, we'll now aim for
+learning the full posterior.