Someone made an annotated list of contexts where we perceive chance:
https://www.stat.berkeley.edu/~aldous/R ... d/100.html
Implicit in the list is that "we" usually think like Bayesians (even if the explicit terminology of Bayesian probability can be difficult or possibly sometimes disavowed).
Of interest here are 44, 45, 46, and 48.
The discussion of 51, 52, and 53 is illuminating.
The author was "deliberately trying to exclude mathematical methodology from this list," so it represents many modes of thinking that are organic, so to speak, and intuitive to many.
https://www.stat.berkeley.edu/~aldous/R ... d/100.html
Implicit in the list is that "we" usually think like Bayesians (even if the explicit terminology of Bayesian probability can be difficult or possibly sometimes disavowed).
Of interest here are 44, 45, 46, and 48.
(44) Probability and the law
illustrated by the the famous "beyond reasonable doubt" criterion for criminal conviction, deliberately never quantified as a numerical probability. Is it true in fact (and should it be true in principle) that "reasoning under uncertainty" is done differently in legal contexts than in other contexts? Technical material can be found in the journal Law, Probability & Risk. Recall that DNA profiling is one setting where numerical probabilities are brought into court.
To what extent uncertainty about knowledge should be perceived in terms of chance is one of those contentious philosophical questions. Let me use the commonsense rule: does it feel natural to use the word "likely" in a given context? Via this rule we should include
(45) Uncertain validity of scientific theories
exemplified by
Most of the observed increase in global average temperatures since the mid-20th century is very likely due to the observed increase in anthropogenic GHG concentrations.
Similarly we should include
(46) Uncertainty about historical events
such as
It is likely that [during 1909-1913] Hitler experienced and possible that he shared the general antisemitism common among middle-class German nationalists.
...
(48) Uncertainty about easily-checkable facts
If you want to call a friend and think you probably remember the correct number, then you have to decide whether to call that number or to look up the correct number. So this is a context where there is a direct link between probability assessments and actual actions.
illustrated by the the famous "beyond reasonable doubt" criterion for criminal conviction, deliberately never quantified as a numerical probability. Is it true in fact (and should it be true in principle) that "reasoning under uncertainty" is done differently in legal contexts than in other contexts? Technical material can be found in the journal Law, Probability & Risk. Recall that DNA profiling is one setting where numerical probabilities are brought into court.
To what extent uncertainty about knowledge should be perceived in terms of chance is one of those contentious philosophical questions. Let me use the commonsense rule: does it feel natural to use the word "likely" in a given context? Via this rule we should include
(45) Uncertain validity of scientific theories
exemplified by
Most of the observed increase in global average temperatures since the mid-20th century is very likely due to the observed increase in anthropogenic GHG concentrations.
Similarly we should include
(46) Uncertainty about historical events
such as
It is likely that [during 1909-1913] Hitler experienced and possible that he shared the general antisemitism common among middle-class German nationalists.
...
(48) Uncertainty about easily-checkable facts
If you want to call a friend and think you probably remember the correct number, then you have to decide whether to call that number or to look up the correct number. So this is a context where there is a direct link between probability assessments and actual actions.
The discussion of 51, 52, and 53 is illuminating.
The distinction between probability and statistics is somewhat arbitrary. Some previous examples feature explicitly in statistics textbooks, and in many "everyday life" examples the only way to guess future probabilities is from implicitly "statistical" knowledge of the past. Here are some more explicitly statistical examples.
(51) (Mis)use of population statistics as probabilities for an individual.
When conceiving a child one can use the population statistics (about 50% of children are girls) to say that the chance your child will be a girl is about 50%. But this is a rather uncommon instance. Consider instead death by lightning; the U.S. population statistics are about 1 per 5 million people per year. However, pointing to a given individual and saying "your chance of death by lightning in the next year is about 1 in 5 million" is just wrong; the chance varies enormously between different individuals, according to their propensity to be outdoors during thunderstorms. Perhaps most instance are intermediate between these extremes. U.S. '"life expectancy" tables put the chance of a 21-year old living past 70 at about 77%. How this applies to a typical individual is a matter of judgement; someone with no visible health problems presumably has a slightly higher survival probability.
(52) Statistical estimates of probabilities for an individual, based on data for that individual and population statistics concerning relationships between factors
This is a huge topic, exemplified by credit scores which seek to predict your likelihood of defaulting on a loan using data about you and the past record of similar people; or the algorithms by which amazon.com offers you suggestions based on your past purchases and browsing; cf. the Netflix Prize . Originally based on classical statistical techniques like multiple regression, it is nowadays often regarded more as part of the theory of algorithms under names such as machine learning .
(53) Repeatable chance experiments.
Here I am cheating by stating a conceptual category rather than specific real-world setting. "Dropping a thumbtack, and seeing if it ends point-up or point-down" is the iconic classroom example of a repeatable chance experiment, generalizing (yyy - games of chance based on artifacts with physical symmetry) because we do not have an a priori knowledge of the probabilities. Items (yyy - Random sampling for representativeness) and (yyy - Randomized controlled trials) fit the category, as does the classical topic of measurement error. The category is conceptually prominent for two reasons. Dogmatic frequentists assert this is the only context in which numerical probabilities make sense. Second, much classic mathematical probability concerns conclusions (law of large numbers, central limit theorem) of the hypothesis that observations are independent identically distributed random variables, the mathematical formalization of "repeatable chance experiment".
(51) (Mis)use of population statistics as probabilities for an individual.
When conceiving a child one can use the population statistics (about 50% of children are girls) to say that the chance your child will be a girl is about 50%. But this is a rather uncommon instance. Consider instead death by lightning; the U.S. population statistics are about 1 per 5 million people per year. However, pointing to a given individual and saying "your chance of death by lightning in the next year is about 1 in 5 million" is just wrong; the chance varies enormously between different individuals, according to their propensity to be outdoors during thunderstorms. Perhaps most instance are intermediate between these extremes. U.S. '"life expectancy" tables put the chance of a 21-year old living past 70 at about 77%. How this applies to a typical individual is a matter of judgement; someone with no visible health problems presumably has a slightly higher survival probability.
(52) Statistical estimates of probabilities for an individual, based on data for that individual and population statistics concerning relationships between factors
This is a huge topic, exemplified by credit scores which seek to predict your likelihood of defaulting on a loan using data about you and the past record of similar people; or the algorithms by which amazon.com offers you suggestions based on your past purchases and browsing; cf. the Netflix Prize . Originally based on classical statistical techniques like multiple regression, it is nowadays often regarded more as part of the theory of algorithms under names such as machine learning .
(53) Repeatable chance experiments.
Here I am cheating by stating a conceptual category rather than specific real-world setting. "Dropping a thumbtack, and seeing if it ends point-up or point-down" is the iconic classroom example of a repeatable chance experiment, generalizing (yyy - games of chance based on artifacts with physical symmetry) because we do not have an a priori knowledge of the probabilities. Items (yyy - Random sampling for representativeness) and (yyy - Randomized controlled trials) fit the category, as does the classical topic of measurement error. The category is conceptually prominent for two reasons. Dogmatic frequentists assert this is the only context in which numerical probabilities make sense. Second, much classic mathematical probability concerns conclusions (law of large numbers, central limit theorem) of the hypothesis that observations are independent identically distributed random variables, the mathematical formalization of "repeatable chance experiment".
The author was "deliberately trying to exclude mathematical methodology from this list," so it represents many modes of thinking that are organic, so to speak, and intuitive to many.
Statistics: Posted by Peter Kirby — Tue Jan 14, 2025 1:01 pm