Probability theory is nearly philosophical.

"Uncertainty concept in engineering" (and society) is not verified, esp. "small probability events", but is definitely a great topic for me to work on.

Orthodox concepts:

- probability: a measure of an event relative to a sample space
- likelihood: probability of an information set conditioned on an event
- confidence: pre-sampling probability

Alternative concepts:

- risk
- uncertainty
- possibility

**Probability triple**: (measure space, events, probability assignments)

**Three independent experiments**: (specimens, measurement, environment)

Priori distribution, or the whole concept of Bayesian methods, is not a falsifiable concept. This is where theory separates from reality.

Two popular views on Bayesian probability:

- objectivist view: The probability of a proposition corresponds to a reasonable belief everyone (even a "robot") sharing the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by requirements of rationality and consistency. {Jaynes, E.T., 1986}
- subjectivist view: Probability corresponds to a 'personal belief'; rationality and coherence constrain the probabilities a subject may have, but allow for substantial variation within those constraints. {de Finetti, 1974}

My own practice is to use Bayesian analysis in the presence of genuine prior information; to use empirical Bayes methods in the parallel cases situation; and otherwise to be cautious when invoking uninformative priors. In the last case, Bayesian calculations cannot be uncritically accepted and should be checked by other methods, which usually means frequentistically.[^1]

Bayesian methodsare easier to explain and understand than their frequentist counterparts.[^2]For frequentists the

priormust have a more objective foundation; ideally that is the relative frequency of events in repeatable, well-defined experiments.[^3]

If a problem can be solved by deductive reasoning, probability theory is not needed for it; thus our topic is the optimal processing of incomplete information.

Misuse of probability in disaster anticipation:

- Space Shuttle Challenger disaster
- earthquake forecast indicator(s)

Proper use of stochastic processes:

- Einstein's theory of Brownian motion

When we want to use probability theory for prediction (disasters) and decision making under uncertainty, we have to deal with the interpretation of probability.

- subjective vs. axiomatic
- Bayesian vs. frequentist
- combinatorial

**Probability assignment can be skewed, i.e. not homogeneous.**

For a certain universe $\Omega$ and sigma-algebra $\mathcal{F}$, design two probability assignments $P_1$ and $P_2$, such that under some random variable $X: ( \Omega, \mathcal{F} ) \rightarrow ( [0,1], \mathcal{B} )$, we get two CDFs $F_1 (x) = 1$; $F_2 (x) = x^n$.

Both probability assignments satisfy the axioms for probability measure, but $P_1$ assigns “homogenous” probability to the universe, under random variable $X$, $P_1$ assigns “skewed” probability to the universe, under the same random variable, with higher probability assigned to events with bigger realization.

Any two "admissible" probability assignments on a measurable space $( \Omega, \mathcal{F} )$ are "indistinguishable".

- "Admissible" probability assignment assigns non-degenerate probability density on the sigma-algebra.
- Sigma-algebra resembles topology; "admissible probability assignments" resemble homeomorphism; the generalized result resembles the principle of relativity.

随机理论的流行并不意味着人们开始重视不确定性，而是统计方法带动了理论研究。 统计方法适用面广，同时计算性能发展迅速，面对大规模数据有待处理，各学科都被引向统计方法（化学——HDMR，生物——DNA测序，实验设计——Response surface，计算机科学——Bayesian network）。 统计的理论基础则是 Kolmogorov 的主流概率论和随机过程理论。 至于大家是否真的理解不确定性的本质，则很值得怀疑。 [In this case, random variable is useless, while distribution (joint/marginal) is all that make a difference.]

As a result, the imaginary distinction between "probability theory" and "statistical inference" disappears. And the field achieves not only logical unity and simplicity, but far greater technical power and flexibility in applications.

[^1]: Bradley Efron, Bayes' Theorem in the 21st Century. Science, 7 June 2013.

[^2]: Bradley Efron, A Statistically Significant Future for Bayes' Rule—Response. Science, 26 July 2013.

[^3]: John Allen Paulos, The Mathematics of Changing Your Mind. The New York Times Books Review.