A. select/refine models
For uniform parameter-priors the most-likely parameters have maximum likelihood pmax[data|params+model] as well, and for log-normally distributed measurement errors this is nothing other than the least squares fit. A more challenging inverse problem is that of choosing models (or idea-sets) from a set of competing alternatives. The emergent cross-disciplinary strategy here is to choose the model whose probability p[data|model] marginalized over all possible parameter values is largest.
This amounts to choosing the model which is least surprised by the data i.e. for which cross-entropy Sp/r ≈ ln[1/p[data|model]] in nats is smallest. The marginalization process in effect adds an Occam-factor penalty[1] for fit-parameters to the minimized log-likelihood ln[1/pmax[data|params+model]], which can either be estimated by detailed analysis of parameter prior-probabilities or via information criteria like AIC (Akaike Information Criterion adds 1 nat/fit-parameter) now popular in the behavioral sciences[7] .
using data to refine & select models
Models often help one calculate the probability of something, depending on the value of certain model-parameters. One's choice or estimation of parameter-values often depends on data from prior observations.
This is a classic "inverse problem", for which Bayes' product-rule plays a key role. The steps in general are:
come up with a likelihood expression i.e. a probability of the data given a model,
assign prior probabilities to parameter-values based on information available separately,
integrate the product of these to get a global-likelihood normalization constant, so that
the normalized product becomes the posterior-probability of parameter-values given the data.
related references
Phil C. Gregory (2005) Bayesian logical data analysis for the physical sciences: A comparative approach with Mathematica support (Cambridge U. Press, Cambridge UK) preview.
Burnham, K. P. and Anderson D. R. (2002) Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, Second Edition (Springer Science, New York) ISBN 978-0-387-95364-9.