Bayesian econometrics is a statistical and mathematical method of problem solving that relies on an investigator's convictions as to an expected outcome, instead of just relying on evidence provided by the data available. This is based on the premise of Baye's Theorem, which is a mathematical formula that is used to prove any hypothesis where preexisting ideas are supported by evidence. It is a form of subjective reasoning that places emphasis on a researcher's initial degree of belief, and uses evidence to shape conclusions based on that initial belief.
One of the foundational elements of Bayesian econometrics is that Bayesian principles are based on conditional probability. That is, the likelihood of an event occurring is looked at first based on the condition that a prior event took place to set the stage for it. The formula for this is that a probability for both of these events occurring must be divided by the probability or condition that the first event did, in fact, take place.
Conditional probability as a feature of Bayesian econometrics is an attempt to more closely model the real world when calculating the likely occurrence of future events. It relies on probability distributions, which are varying levels of uncertainty instead of just pure randomness, on which to base future outcome calculations. This means that Bayesian econometrics takes a more evidential support approach as a premise, by attempting to quantify the degree of belief or confidence individuals have in an outcome as an input to predicting the actual outcome. This has relevance in economics fields such as consumer confidence, where group expectations have a tremendous impact on what becomes reality.
Insufficient data is often a problem in weighted statistical calculations that attempt to yield meaningful results, and Bayesian regression analysis offers a solution to this. It allows for estimates of prior information as input into the calculations. This approach of using prior density functions to arrive at posterior density functions has the potential to yield much more useful solutions to problems.
Bayesian methods are not often used, however, for several reasons. It is difficult to formally account for the subjective beliefs of a population and form them into a meaningful mathematical distribution. Calculating the proper outcome to the posterior distribution is also open to interpretation, and any results obtained only have value if you agree with the beliefs and assumptions that were used to start with. Economists also state that Bayesian econometrics is focused too much on theory and technique, and not enough on developing this theory towards current economic models that attempt to predict real world events and trends.