On Individual Risk

Presentation by Sarah Dean for AIPP 8/7/23

Example: Weather

A weather forecaster appears on television every night and issues a statement of the form “The probability of precipitation tomorrow is 30 %” (where the quoted probability will of course vary from day to day). Different forecasters issue different probabilities.

Example: Risk Assessment

• Actuarial Risk Assessment Instruments (ARAIs): statistical procedures for assessing “the risk” of an individual becoming violent
• Classification of Violence Risk (COVR) software outputs e.g. “The likelihood that XXX will commit a violent act toward another person in the next several months is estimated to be between 20 and 32 %, with a best estimate of 26 %.”

Example: BRCA

Writing in the New York Times (14 May 2013) about her decision to have a preventive double mastectomy, the actress Angelina Jolie said: “I carry a faulty gene, BRCA1, which sharply increases my risk of developing breast cancer and ovarian cancer. My doctors estimated that I had an 87 percent risk of breast cancer and a 50 percent risk of ovarian cancer, although the risk is different in the case of each woman.”

Interpretations of probability

Theories of probability regard it as fundamentally an attribute of groups  (groupist) or individuals (individualist).

1. Classical probability
2. Enumerative probability
3. Frequency probability
4. Formal probability
5. Personal probability
6. Propensity and chance
7. Logical probability

• Classical: focus on equally likely elementary outcomes (e.g. rolling die, drawing cards, flipping coins)
• Enumerative: (also "finite frequentism") probabilities are the proportions of a finite population exhibiting an attribute
  • Different scope of application, e.g. probability of a new-born child being assigned female is classically 0.5 but 0.49 under enumerative interpretation
• Frequency: enumerative but over infinite set of individuals (e.g. proportion observed from repeated idential trials in the limit)
• Formal: a statistical model of random variables and their joint probability distribution. Fine grained individualist structure, though claims of groupist interpretation. E.g. Bernoulli model and independent trials converges by Law of Large Numbers.
  • Metaphysical probability: groupist w.r.t. multiverse

• Personal: How much would you bet? "The degree of confidence of an individual, at a given instant and with a given set of information, regarding the occurrence of an event."
• Propensity and chance: under specified circumstances, there is a certain (objective) propensity towards certain outcomes. Little guidance on how to assess this. Championed by Popper.
• Logical: a probability value  \(P(A|B)\) quantifies the degree to which a rational being should believe in \(A\) given information \(B\). Associated with Keynes.

Example: Weather

"There is a 30 % chance of rain tomorrow" survey

•  “three out of ten meteorologists believe it will rain” (enumerative)
• “if you look up to the sky and see 100 clouds, then 30 of them are black” (enumerative)
• “not about time, the amount of rain that will fall”
• “If we had 100 lives, it would rain in 30 of these tomorrow” (metaphysical)
• “A probability is only about whether or not there is rain, but does not say anything about the time and region [or amount]” (individualist)

Official: when the weather conditions are “like today,” in three out of ten such cases there will be rain the next day. (Frequentist, but what about appearance of personal probabilities by forecaster?)

Example: Risk Assessment

ARAI probabilities based on statistical analysis of data on groups, using a model incorporating formal probabilities. However, they are intended as predictions for individual cases, not as descriptions of group behaviour. Furthermore, the confidence intervals quantify uncertainty arising from the use of finite data to estimate a risk value, with is relevant to the group not to the individual.

Example: BRCA

Sounds like individualist risk (perhaps a personal probability of her doctor), but appears to be an enumerative probability. (More recent numbers are lower: 55-65% and 40%)

I carry a faulty gene, BRCA1, which sharply increases my risk of developing breast cancer and ovarian cancer. My doctors estimated that I had an 87 percent risk of breast cancer and a 50 percent risk of ovarian cancer, although the risk is different in the case of each woman.”

Group to individual inference

How likely is a coin to land on heads?

• Frequentist: observed frequency after lengthy experimentation is justified as a individual probability assignment as long as all tosses are repeated trials of the same phenomenon under identical conditions
• Personal: assessment depends on observed tosses so far and boils down to empirical frequency as long as you believe the tosses are exchangeable
• Exchangability is hard to justify in more complex scenarios (e.g. assessing an individual student's risk of failing a statistics exam given information about their performance on other exams and the performance of other students)

Problem of the Reference Class

How much information about an individual to include? E.g. mortality risk by age, gender, location, occupation, health issues, etc...  plagues both frequentist and personal probability interpretations

Individual to group inference

• Start with individual risks (probability forecast) and then ensure consistency with observed group frequencies (validity)
• Probability forecast for an individual and the available background information
• Validity is determined by comparing probabilities to the true outcomes to ensure calibration
  • Overall calibration, probability calibration, subset calibration, and information calibration