Probability Theory

Context:
- It can provide a Consistent Mathematical Framework for Uncertainty Quantification and Uncertainty Manipulating.
Example(s):
- Kolmogorov's Probability Theory.
- Bayes Theorem.
- Central Limit Theorem.
- Law of Large Numbers.
- …
Counter-Example(s):
- Decision Theory.
See: Law of Probability, Probability Function, Conditional Probability, Random Experiment, Statistical Inference, Statistical Mechanics, Entropy Measure.

References

(Wikipedia, 2013) ⇒ http://en.wikipedia.org/wiki/Probability_theory
- Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena.^[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion. If an individual coin toss or the roll of dice is considered to be a random event, then if repeated many times the sequence of random events will exhibit certain patterns, which can be studied and predicted. Two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem.
  As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics.

(Bishop, 2006) ⇒ Christopher M. Bishop. (2006). “Pattern Recognition and Machine Learning." Springer, Information Science and Statistics. ISBN:0387310738
- QUOTE: A key concept in the field of pattern recognition is that of uncertainty. It arises both through noise among measurements, as well as through the finite size of data sets. Probability theory provides a consistent framework for the quantification and manipulation of uncertainty and forms a one of the central foundsations for pattern recogtnition. When combined with decision theory, discussed in Section 1.5, it allows us to make optimal predictions given all the information available to us, even though that information may be incomplete or ambiguous. …
  … We have seen in Section 1.2 how probability theory provides us with a consistent mathematical framework for quantifying and manipulating uncertainty. Here we turn to the discussion of decision theory that, when combined with probability theory, allows us to make optimal decisions in situations involving uncertainty such as those encountered in pattern recognition.