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.

## Concepts

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)

### Bayesian probability

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 methods are easier to explain and understand than their frequentist counterparts.[^2]

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

## Use and Interpretations of Probability

### Use of Probability

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

### Interpretations of probability

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

## Notes

### On Probability Assignment

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.

### Relation with statistics

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.

## References

[^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.