The previous examples on parameter estimation and model comparison have demonstrated the benefits of Bayesian probability theory for quantitative inference based on prior knowledge and measured data. However, Bayesian probability theory is not a magic black box capable of compensating for badly designed experiments. Information absent in the data cannot be revealed by any kind of data analysis. This immediately raises the question of how the information provided by a measurement can be quantified and, in a next step, how to optimize experiments to maximize the information gain. Here one of the very recent areas of applied Bayesian data analysis is entered: Bayesian experimental design is an increasingly important topic driven by progress in computer power and algorithmic improvements [132, 214]. So far it has been implicitly assumed that there is little choice in the actual execution of the experiment, in other words, the data to be analysed were assumed to be given. While this is the most widespread use of data analysis, the active selection of data holds great promise to improve the measurement process. There are several scenarios in which an active selection of the data to be collected or evaluated is obviously very advantageous, for example:

• Expensive and/or time-consuming measurements, thus one wants to know where to look next to learn as much as possible – or when to stop performing further experiments.

Stochastic processes

If a classical particle moves in an external force field F(x) its motion is deterministic and it could be called a ‘deterministic process’. If the particle interacts with randomly distributed obstacles its motion is still deterministic, but the trajectory depends on the random characteristics of the obstacles, and maybe also on the random choice of the initial conditions. Such a motion is an example of a ‘stochastic process’, or rather a ‘random process’.

In a more abstract generalized definition, a stochastic process is a random variable Xξ that depends on an additional (deterministic) independent variable ξ, which can be discrete or continuous. In most cases it stands for an index ξ ∈ N or time t, space x or energy E. An almost trivial ubiquitous stochastic process is given by additive noise η(t) on a time-dependent signal s(t), i.e. Xt = s(t) + η(t). As a consequence, Xt is no longer continuous. The most apparent applications of stochastic processes are time series of any kind that depend on some random impact. A broad field of applications are time series occurring, for example, in business, finance, engineering, medical applications and of course in physics. Beyond time series analysis, stochastic processes are at the heart of diffusion Langevin dynamics, Feynman's path integrals [43], as well as Klauder's stochastic quantization [62], which represents an unconventional approach to quantum mechanics. Here we will give a concise introduction and present a few pedagogical examples.