Bayesian data analysis

The goal of Bayesian or integrated data analysis (IDA) is to combine the information from a set of diagnostics providing complementary information in order to recover the best possible reconstruction of the actual state of the system subjected to measurement. Like Function parametrization (FP), this technique requires having a forward model to predict the measurement readings for any given state of the physical system; however

• instead of computing an estimate of the inverse of the forward model (as with FP), IDA finds the best model state corresponding to a specific measurement by a maximization procedure (maximization of the likelihood);
• the handling of error propagation is more sophisticated within IDA, allowing non-Gaussian error distributions and absolutely general and complex parameter interdependencies; and
• additionally, it provides a systematic way to include prior knowledge into the analysis.

The maximization process is CPU intensive, so that Bayesian analysis is not suited for real-time data analysis (unlike FP).

Bayes' Theorem

The method is based on Bayes' Theorem, expressed as follows:

$P({\vec {\alpha }}|{\vec {d}},{\vec {\sigma }},I)={\frac {{L}({\vec {d}}|{\vec {\alpha }},{\vec {\sigma }},I)\pi ({\vec {\alpha }}|I)}{\int {L}({\vec {d}}|{\vec {\alpha }},{\vec {\sigma }},I)\pi ({\vec {\alpha }}|I)d{\vec {\alpha }}}}$

Here, α are a set of model parameters, and P is the probability distribution of these parameters, given the experimental data d, their errors σ, and additional information I.

Bayes' Theorem expresses this probability distribution as a product of the likelihood L of obtaining the cited experimental data given some values of the model parameters α as well as σ and I, and the prior distribution π that expresses the knowledge concerning the model parameters preceding any measurement.

The likelihood L is computed using a forward model of the experiment, returning the value of simulated measurements while assuming a given physical state of the experimental system. It should be noted that this forward model (from system parameters to measurements) is often much easier to compute than the reverse mapping (from measurements to system parameters), as the latter is often the inverse of a projection, which is therefore typically ill-determined.

The normalization of the equation serves to maintain its character of a probability distribution, although it is not important for the determination of the best values of the parameters and their errors. The optimum reconstruction is determined by maximizing the posterior P, varying the parameters α.

See also

• Markov chain Monte Carlo