Archive for category Probability and Risk
Women can’t do math. Hypatia of Alexandria and Émilie du Châtelet notwithstanding, this was asserted for thousands of years by men who controlled access to education. With men in charge it was a self-fulfilling prophecy. Women now represent the majority of college students and about 40% of math degrees. That’s progress.
Last week Marcio Rubio caught hell for taking Ilhan Omar’s statement about double standards and unfair terrorism risk assessment out of context. The quoted fragment was: “I would say our country should be more fearful of white men across our country because they are actually causing most of the deaths within this country…”
Most news coverage of the Rubio story (e.g. Vox) note that Omar did not mean that everyone should be afraid of white men as a group, but that, e.g., “violence by right-wing extremists, who are overwhelmingly white and male, really is a bigger problem in the United States today than jihadism.”
Let’s look at the numbers. Wikipedia, following the curious date-range choice of the US GAO, notes: “of the 85 violent extremist incidents that resulted in death since September 12, 2001, far-right politics violent extremist groups were responsible for 62 (73 percent) while radical Islamist violent extremists were responsible for 23 (27 percent).” Note that those are incident counts, not death counts. The fatality counts were 106 (47%) for white extremists and 119 (53%) for jihadists. Counting fatalities instead of incidents reverses the sense of the numbers.
Pushing the terminus post quem back one day adds the 2,977 9-11 fatalities to the category of deaths from jihadists. That makes 3% of fatalities from right wing extremists and 97% from radical Islamist extremists. Pushing the start date further back to 1/1/1990, again using Wikipedia numbers, would include the Oklahoma City bombing (white extremists, 168 dead), nine deaths from jihadists, and 14 other deaths from white wackos, including two radical Christian antisemites and professor Ted Kaczynski. So the numbers since 1990 show 92% of US terrorism deaths from jihadists and 8% from white extremists.
Barring any ridiculous adverse selection of date range (in the 3rd week of April, 1995, 100% of US terrorism deaths involved white extremists), Omar is very, very wrong in her data. The jihadist death toll dwarfs that from white extremists.
But that’s not the most egregious error in her logic – and that of most politicians armed with numbers and a cause. The flagrant abuse of data is what Kahneman and Tversky termed base-rate neglect. Omar, in discussing profiling (sampling a population subset) is arguing about frequencies while citing raw incident counts. The base rate (an informative prior, to Bayesians) is crucial. Even if white extremists caused most – as she claimed – terrorism deaths, there would have to be about one hundred times more deaths from white men (terrorists of all flavors are overwhelmingly male) than from Muslims for her profiling argument to hold. That is, the base rate of being Muslim in the US is about one percent.
The press overwhelmingly worked Rubio over for his vicious smear. 38 of the first 40 Google search results on “Omar Rubio” favored Omar. One favored Rubio and one was an IMDb link to an actor named Omar Rubio. None of the news pieces, including the one friendly to Rubio, mentioned Omar’s bad facts (bad data) or her bad analysis thereof (bad math). Even if she were right about the data – and she is terribly wrong – she’d still be wrong about the statistics.
I disagree with Trump about Omar. She should not go back to Somalia. She should go back to school.
For business reasons I’ve started a separate blog – “on risk of” – for topics involving risk analysis, probability , aerospace and process engineering, and the like.
Tonight I wrote a post there on logical fallacies that come up in medicine and in court. For example, confusing the probability of a match between characteristics of a perpetrator as reported by witnesses and those of a specific suspect with the probability of a match with anyone in a large population – particularly when the probability of a match is claimed by prosecution to be the probability that a defendant is not guilty. I also look at cases involving confusion between the conditional probability of A given B vs. the probability of B given A, e.g., the chance of the disease given a positive test result vs. the chance of a positive test result given the disease – classic Bayes Theorem stuff.
Please join me at onriskof.com. Thanks for your interest.
In past consulting work I’ve wrestled with subjective probability values derived from expert opinion. Subjective probability is an interpretation of probability based on a degree of belief (i.e., hypothetical willingness to bet on a position) as opposed a value derived from measured frequencies of occurrences (related posts: Belief in Probability, More Philosophy for Engineers). Subjective probability is of interest when failure data is sparse or nonexistent, as was the data on catastrophic loss of a space shuttle due to seal failure. Bayesianism is one form of inductive logic aimed at refining subjective beliefs based on Bayes Theorem and the idea of rational coherence of beliefs. A NASA handbook explains Bayesian inference as the process of obtaining a conclusion based on evidence, “Information about a hypothesis beyond the observable empirical data about that hypothesis is included in the inference.” Easier said than done, for reasons listed below.
Bayes Theorem itself is uncontroversial. It is a mathematical expression relating the probability of A given that B is true to the probability of B given that A is true and the individual probabilities of A and B:
P(A|B) = P(B|A) x P(A) / P(B)
If we’re trying to confirm a hypothesis (H) based on evidence (E), we can substitute H and E for A and B:
P(H|E) = P(E|H) x P(H) / P(E)
To be rationally coherent, you’re not allowed to believe the probability of heads to be .6 while believing the probability of tails to be .5; the sum of chances of all possible outcomes must sum to exactly one. Further, for Bayesians, the logical coherence just mentioned (i.e., avoidance of Dutch book arguments) must hold across time (synchronic coherence) such that once new evidence E on a hypothesis H is found, your believed probability for H given E should equal your prior conditional probability for H given E.
Plenty of good sources explain Bayesian epistemology and practice far better than I could do here. Bayesianism is controversial in science and engineering circles, for some good reasons. Bayesianism’s critics refer to it as a religion. This is unfair. Bayesianism is, however, like most religions, a belief system. My concern for this post is the problems with Bayesianism that I personally encounter in risk analyses. Adherents might rightly claim that problems I encounter with Bayes stem from poor implementation rather than from flaws in the underlying program. Good horse, bad jockey? Perhaps.
Problem 1. Subjectively objective
Bayesianism is an interesting mix of subjectivity and objectivity. It imposes no constraints on the subject of belief and very few constraints on the prior probability values. Hypothesis confirmation, for a Bayesian, is inherently quantitative, but initial hypotheses probabilities and the evaluation of evidence is purely subjective. For Bayesians, evidence E confirms or disconfirms hypothesis H only after we establish how probable H was in the first place. That is, we start with a prior probability for H. After the evidence, confirmation has occurred if the probability of H given E is higher than the prior probability of H, i.e., P(H|E) > P(H). Conversely, E disconfirms H when P(H|E) < P(H). These equations and their math leave business executives impressed with the rigor of objective calculation while directing their attention away from the subjectivity of both the hypothesis and its initial prior.
2. Rational formulation of the prior
Problem 2 follows from the above. Paranoid, crackpot hypotheses can still maintain perfect probabilistic coherence. Excluding crackpots, rational thinkers – more accurately, those with whom we agree – still may have an extremely difficult time distilling their beliefs, observations and observed facts of the world into a prior.
3. Conditionalization and old evidence
This is on everyone’s short list of problems with Bayes. In the simplest interpretation of Bayes, old evidence has zero confirming power. If evidence E was on the books long ago and it suddenly comes to light that H entails E, no change in the value of H follows. This seems odd – to most outsiders anyway. This problem gives rise to the game where we are expected to pretend we never knew about E and then judge how surprising (confirming) E would have been to H had we not know about it. As with the general matter of maintaining logical coherence required for the Bayesian program, it is extremely difficult to detach your knowledge of E from the rest of your knowing about the world. In engineering problem solving, discovering that H implies E is very common.
4. Equating increased probability with hypothesis confirmation.
My having once met Hillary Clinton arguably increases the probability that I may someday be her running mate; but few would agree that it is confirming evidence that I will do so. See Hempel’s raven paradox.
5. Stubborn stains in the priors
Bayesians, often citing success in the business of establishing and adjusting insurance premiums, report that the initial subjectivity (discussed in 1, above) fades away as evidence accumulates. They call this washing-out of priors. The frequentist might respond that with sufficient evidence your belief becomes irrelevant. With historical data (i.e., abundant evidence) they can calculate P of an unwanted event in a frequentist way: P = 1-e to the power -RT, roughly, P=RT for small products of exposure time T and failure rate R (exponential distribution). When our ability to find new evidence is limited, i.e., for modeling unprecedented failures, the prior does not get washed out.
6. The catch-all hypothesis
The denominator of Bayes Theorem, P(E), in practice, must be calculated as the sum of the probability of the evidence given the hypothesis plus the probability of the evidence given not the hypothesis:
P(E) = [P(E|H) x p(H)] + [P(E|~H) x P(~H)]
But ~H (“not H”) is not itself a valid hypothesis. It is a family of hypotheses likely containing what Donald Rumsfeld famously called unknown unknowns. Thus calculating the denominator P(E) forces you to pretend you’ve considered all contributors to ~H. So Bayesians can be lured into a state of false choice. The famous example of such a false choice in the history of science is Newton’s particle theory of light vs. Huygens’ wave theory of light. Hint: they are both wrong.
7. Deference to the loudmouth
This problem is related to no. 1 above, but has a much more corporate, organizational component. It can’t be blamed on Bayesianism but nevertheless plagues Bayesian implementations within teams. In the group formulation of any subjective probability, normal corporate dynamics govern the outcome. The most senior or deepest-voiced actor in the room drives all assignments of subjective probability. Social influence rules and the wisdom of the crowd succumbs to a consensus building exercise, precisely where consensus is unwanted. Seidenfeld, Kadane and Schervish begin “On the Shared Preferences of Two Bayesian Decision Makers” with the scholarly observation that an outstanding challenge for Bayesian decision theory is to extend its norms of rationality from individuals to groups. Their paper might have been illustrated with the famous photo of the exploding Challenger space shuttle. Bayesianism’s tolerance of subjective probabilities combined with organizational dynamics and the shyness of engineers can be a recipe for disaster of the Challenger sort.
All opinions welcome.
In a post on Richard Feynman and philosophy of science, I suggested that engineers would benefit from a class in philosophy of science. A student recently asked if I meant to say that a course in philosophy would make engineers better at engineering – or better philosophers. Better engineers, I said.
Here’s an example from my recent work as an engineer that drives the point home.
I was reviewing an FMEA (Failure Mode Effects Analysis) prepared by a high-priced consultancy and encountered many cases where a critical failure mode had been deemed highly improbable on the basis that the FMEA was for a mature system with no known failures.
How many hours of operation has this system actually seen, I asked. The response indicated about 10 thousand hours total.
I said on that basis we could assume a failure rate of about one per 10,001 hours. The direct cost of the failure was about $1.5 million. Thus the “expected value” (or “mathematical expectation” – the probabilistic cost of the loss) of this failure mode in a 160 hour mission is $24,000 or about $300,000 per year (excluding any secondary effects such as damaged reputation). With that number in mind, I asked the client if they wanted to consider further mitigation by adding monitoring circuitry.
I was challenged on the failure rate I used. It was, after all, a mature, ten year old system with no recorded failures of this type.
Here’s where the analytic philosophy course those consultants never took would have been useful.
You simply cannot justify calling a failure mode extremely rare based on evidence that it is at least somewhat rare. All unique events – like the massive rotor failure that took out all three hydraulic systems of a DC-10 in Sioux City – were very rare before they happened.
The authors of the FMEA I was reviewing were using unjustifiable inductive reasoning. Philosopher David Hume debugged this thoroughly in his 1738 A Treatise of Human Nature.
Hume concluded that there simply is no rational or deductive basis for induction, the belief that the future will be like the past.
Hume understood that, despite the lack of justification for induction, betting against the sun rising tomorrow was not a good strategy either. But this is a matter of pragmatism, not of rationality. A bet against the sunrise would mean getting behind counter-induction; and there’s no rational justification for that either.
In the case of the failure mode not yet observed, however, there is ample justification for counter-induction. All mechanical parts and all human operations necessarily have nonzero failure or error rates. In the world of failure modeling, the knowledge “known pretty good” does not support the proposition “probably extremely good”, no matter how natural the step between them feels.
Hume’s problem of induction, despite the efforts of Immanuel Kant and the McKinsey consulting firm, has not been solved.
A fabulously entertaining – in my view – expression of the problem of induction was given by philosopher Carl Hempel in 1965.
Hempel observed that we tend to take each new observation of a black crow as incrementally supporting the inductive conclusion that all crows are black. Deductive logic tells us that if a conditional statement is true, its contrapositive is also true, since the statement and its contrapositive are logically equivalent. Thus if all crows are black then all non-black things are non-crow.
It then follows that if each observation of black crows is evidence that all crows are black (compare: each observation of no failure is evidence that no failure will occur), then each observation of a non-black non-crow is also evidence that all crows are black.
Following this line, my red shirt is confirming evidence for the proposition that all crows are black. It’s a hard argument to oppose, but it simply does not “feel” right to most people.
Many try to salvage the situation by suggesting that observing that my shirt is red is in fact evidence that all crows are black, but provides only unimaginably small support to that proposition.
But pushing the thing just a bit further destroys even this attempt at rescuing induction from the clutches of analysis.
If my red shirt gives a tiny bit of evidence that all crows are black, it then also gives equal support to the proposition that all crows are white. After all, my red shirt is a non-white non-crow.
The quality of risk management has mostly fallen for the past few decades. There are signs of change for the better.
Risk management is a broad field; many kinds of risk must be managed. Risk is usually defined in terms of probability and cost of a potential loss. Risk management, then, is the identification, assessment and prioritization of risks and the application of resources to reduce the probability and/or cost of the loss.
The earliest and most accessible example of risk management is insurance, first documented in about 1770 BC in the Code of Hammurabi (e.g., rules 23, 24, and 48). The Code addresses both risk mitigation, through threats and penalties, and minimizing loss to victims, through risk pooling and insurance payouts.
Insurance was the first example of risk management getting serious about risk assessment. Both the frequentist and quantified subjective risk measurement approaches (see recent posts on belief in probability) emerged from actuarial science developed by the insurance industry.
Risk assessment, through its close relatives, decision analysis and operations research, got another boost from World War II. Big names like Alan Turing, John Von Neumann, Ian Fleming (later James Bond author) and teams at MIT, Columbia University and Bletchley Park put quantitative risk analyses of several flavors on the map.
Today, “risk management” applies to security guard services, portfolio management, terrorism and more. Oddly, much of what is called risk management involves no risk assessment at all, and is therefore inconsistent with the above definition of risk management, paraphrased from Wikipedia.
Most risk assessment involves quantification of some sort. Actuarial science and the probabilistic risk analyses used in aircraft design are probably the “hardest” of the hard risk measurement approaches, Here, “hard” means the numbers used in the analyses come from measurements of real world values like auto accidents, lightning strikes, cancer rates, and the historical failure rates of computer chips, valves and motors. “Softer” analyses, still mathematically rigorous, involve quantified subjective judgments in tools like Monte Carlo analyses and Bayesian belief networks. As the code breakers and submarine hunters of WWII found, trained experts using calibrated expert opinions can surprise everyone, even themselves.
A much softer, yet still quantified (barely), approach to risk management using expert opinion is the risk matrix familiar to most people: on a scale of 1 to 4, rate the following risks…, etc. It’s been shown to be truly worse than useless in many cases, for a variety of reasons by many researchers. Yet it remains the core of risk analysis in many areas of business and government, across many types of risk (reputation, credit, project, financial and safety). Finally, some of what is called risk management involves no quantification, ordering, or classifying. Call it expert intuition or qualitative audit.
These soft categories of risk management most arouse the ire of independent and small-firm risk analysts. Common criticisms by these analysts include:
1. “Risk management” has become jargonized and often involves no real risk analysis.
2. Quantification of risk in some spheres is plagued by garbage-in-garbage-out. Frequency-based models are taken as gospel, and believed merely because they look scientific (e.g., Fukushima).
3. Quantified/frequentist risk analyses are not used in cases where historical data and a sound basis for them actually exists (e.g., pharmaceutical manufacture).
4. Big consultancies used their existing relationships to sell unsound (fluff) risk methods, squeezing out analysts with sound methods (accused of Arthur Anderson, McKinsey, Bain, KPMG).
5. Quantitative risk analyses of subjective type commonly don’t involve training or calibration of those giving expert opinions, thereby resulting in incoherent (in the Bayesian sense) belief systems.
6. Groupthink and bad management override rational input into risk assessment (subprime mortgage, space shuttle Challenger).
7. Risk management is equated with regulatory compliance (banking operations, hospital medicine, pharmaceuticals, side-effect of Sarbanes-Oxley).
8. Some professionals refuse to accept any formal approach to risk management (medical practitioners and hospitals).
While these criticisms may involve some degree of sour grapes, they have considerable merit in my view, and partially explain the decline in quality of risk management. I’ve worked in risk analysis involving uranium processing, nuclear weapons handling, commercial and military aviation, pharmaceutical manufacture, closed-circuit scuba design, and mountaineering. If the above complaints are valid in these circles – and they are – it’s easy to believe they plague areas where softer risk methods reign.
Several books and scores of papers specifically address the problems of simple risk-score matrices, often dressed up in fancy clothes to look rigorous. The approach has been shown to have dangerous flaws by many analysts and scholars, e.g., Tony Cox, Sam Savage, Douglas Hubbard, and Laura-Diana Radu. Cox shows examples where risk matrices assign higher qualitative ratings to quantitatively smaller risks. He shows that risks with negatively correlated frequencies and severities can result in risk-matrix decisions that are worse than random decisions. Also, such methods are obviously very prone to range compression errors. Most interestingly, in my experience, the stratification (highly likely, somewhat likely, moderately likely, etc.) inherent in risk matrices assume common interpretation of terms across a group. Many tests (e.g., Kahneman & Tversky and Budescu, Broomell, Por) show that large differences in the way people understand such phrases dramatically affect their judgments of risk. Thus risk matrices create the illusion of communication and agreement where neither are present.
Nevertheless, the risk matrix has been institutionalized. It is embraced by government (MIL-STD-882), standards bodies (ISO 31000), and professional societies (Project Management Institute (PMI), ISACA/COBIT). Hubbard’s opponents argue that if risk matrices are so bad, why do so many people use them – an odd argument, to say the least. ISO 31000, in my view, isn’t a complete write-off. In places, it rationally addresses risk as something that can be managed through reduction of likelihood, reduction of consequences, risk sharing, and risk transfer. But elsewhere it redefines risk as mere uncertainty, thereby reintroducing the positive/negative risk mess created by economist Frank Knight a century ago. Worse, from my perspective, like the guidelines of PMI and ISACA, it gives credence to structure in the guise of knowledge and to process posing as strategy. In short, it sets up a lot of wickets which, once navigated, give a sense that risk has been managed when in fact it may have been merely discussed.
A small benefit of the subprime mortgage meltdown of 2008 was that it became obvious that the financial risk management revolution of the 1990s was a farce, exposing a need for deep structural changes. I don’t follow financial risk analysis closely enough to know whether that’s happened. But the negative example made public by the housing collapse has created enough anxiety in other disciplines to cause some welcome reappraisals.
There is surprising and welcome activity in nuclear energy. Several organizations involved in nuclear power generation have acknowledged that we’ve lost competency in this area, and have recently identified paths to address the challenges. The Nuclear Energy Institute recently noted that while Fukushima is seen as evidence that probabilistic risk analysis (PRA) doesn’t work, if Japan had actually embraced PRA, the high risk of tsunami-induced disaster would have been immediately apparent. Late last year the Nuclear Energy Institute submitted two drafts to the U.S. Nuclear Regulatory Commission addressing lost ground in PRA and identifying a substantive path forward: Reclaiming the Promise of Risk-Informed Decision-Making and Restoring Risk-Informed Regulation. These documents acknowledge that the promise of PRA has been stunted by distrust of the method, focus on compliance instead of science, external audits by unqualified teams, and the above-mentioned Fukushima fallacy.
Likewise, the FDA, often criticized for over-regulating and over-reach – confusing efficacy with safety – has shown improvement in recent years. It has revised its decades-old process validation guidance to focus more on verification, scientific evidence and risk analysis tools rather than validation and documentation. The FDA’s ICH Q9 (Quality Risk Management) guidelines discuss risk, risk analysis and risk management in terms familiar to practitioners of “hard” risk analysis, even covering fault tree analysis (the “hardest” form of PRA) in some detail. The ASTM E2500 standard moves these concepts further forward. Similarly, the FDA’s recent guidelines on mobile health devices seem to accept that the FDA’s reach should not exceed its grasp in the domain of smart phones loaded with health apps. Reading between the lines, I take it that after years of fostering the notion that risk management equals regulatory compliance, the FDA realized that it must push drug safety far down into the ranks of the drug makers in the same way the FAA did with aircraft makers (with obvious success) in the late 1960s. Fostering a culture of safety rather than one of compliance distributes the work of providing safety and reduces the need for regulators to anticipate every possible failure of every step of every process in every drug firm.
This is real progress. There may yet be hope for financial risk management.
In a recent post I mentioned that probabilistic failure models are highly vulnerable to wrong assumptions of independence of failures, especially in redundant system designs. Common-mode failures in multiple channels defeats the purpose of redundancy in fault-tolerant designs. Likewise, if probability of non-function is modeled (roughly) as historical rate of a specific component failure times the length of time we’re exposed to the failure, we need to establish that exposure time with great care. If only one channel is in control at a time, failure of the other channel can go undetected. Monitoring systems can detect such latent failures. But then failures of the monitoring system tend to be latent.
For example, your car’s dashboard has an engine oil warning light. That light ties to a monitor that detects oil leaks from worn gaskets or loose connections before the oil level drops enough to cause engine damage. Without that dashboard warning light, the exposure time to an undetected slow leak is months – the time between oil changes. The oil warning light alerts you to the condition, giving you time to deal with it before your engine seizes.
But what if the light is burned out? This failure mode is why the warning lights flash on for a short time when you start your car. In theory, you’d notice a burnt-out warning light during the startup monitor test. If you don’t notice it, the exposure time for an oil leak becomes the exposure time for failure of the warning light. Assuming you change your engine oil every 9 months, loss of the monitor potentially increases the exposure time from minutes to months, multiplying the probability of an engine problem by several orders of magnitude. Aircraft and nuclear reactors contain many such monitoring systems. They need periodic maintenance to ensure they’re able to detect failures. The monitoring systems rarely show problems in the check-ups; and this fact often lures operations managers, perceiving that inspections aren’t productive, into increasing maintenance intervals. Oops. Those maintenance intervals were actually part of the system design, derived from some quantified level of acceptable risk.
Common-mode failures get a lot press when they’re dramatic. They’re often used by risk managers as evidence that quantitative risk analysis of all types doesn’t work. Fukushima is the current poster child of bad quantitative risk analysis. Despite everyone’s agreement that any frequencies or probabilities used in Fukushima analyses prior to the tsunami were complete garbage, the result for many was to conclude that probability theory failed us. Opponents of risk analysis also regularly cite the Tacoma Narrows Bridge collapse, the Chicago DC-10 engine-loss disaster, and the Mount Osutaka 747 crash as examples. But none of the affected systems in these disasters had been justified by probabilistic risk modeling. Finally, common-mode failure is often cited in cases where it isn’t the whole story, as with the Sioux City DC-10 crash. More on Sioux City later.
On the lighter side, I’d like to relate two incidents – one personal experience, one from a neighbor – that exemplify common-mode failure and erroneous assumptions of exposure time in everyday life, to drive the point home with no mathematical rigor.
I often ride my bicycle through affluent Marin County. Last year I stopped at the Molly Stone grocery in Sausalito, a popular biker stop, to grab some junk food. I locked my bike to the bike rack, entered the store, grabbed a bag of chips and checked out through the fast lane with no waiting. Ninety seconds at most. I emerged to find no bike, no lock and no thief.
I suspect that, as a risk man, I unconsciously model all risk as the combination of some numerical rate (occurrence per hour) times some exposure time. In this mental model, the exposure time to bike theft was 90 seconds. I likely judged the rate to be more than zero but still pretty low, given broad daylight, the busy location with lots of witnesses, and the affluent community. Not that I built such a mental model explicitly of course, but I must have used some unconscious process of that sort. Thinking like a crook would have served me better.
If you were planning to steal an expensive bike, where would you go to do it? Probably a place with a lot of expensive bikes. You might go there and sit in your pickup truck with a friend waiting for a good opportunity. You’d bring a 3-foot long set of chain link cutters to make quick work of the 10 mm diameter stem of a bike lock. Your friend might follow the victim into the store to ensure you were done cutting the lock and throwing the bike into the bed of your pickup to speed away before the victim bought his snacks.
After the fact, I had much different thought thoughts about this specific failure rate. More important, what is the exposure time when the thief is already there waiting for me, or when I’m being stalked?
My neighbor just experienced a nerve-racking common mode failure. He lives in a San Francisco high-rise and drives a Range Rover. His wife drives a Mercedes. He takes the Range Rover to work, using the same valet parking-lot service every day. He’s known the attendant for years. He takes his house key from the ring of vehicle keys, leaving the rest on the visor for the attendant. He waves to the attendant as he leaves the lot on way to the office.
One day last year he erred in thinking the attendant had seen him. Someone else, now quite familiar with his arrival time and habits, got to his Range Rover while the attendant was moving another car. The thief drove out of the lot without the attendant noticing. Neither my neighbor nor the attendant had reason for concern. This gave the enterprising thief plenty of time. He explored the glove box, finding the registration, which includes my neighbor’s address. He also noticed the electronic keys for the Mercedes.
The thief enlisted a trusted colleague, and drove the stolen car to my neighbor’s home, where they used the electronic garage entry key tucked neatly into its slot in the visor to open the gate. They methodically spiraled through the garage, periodically clicking the button on the Mercedes key. Eventually they saw the car lights flash and they split up, each driving one vehicle out of the garage using the provided electronic key fobs. My neighbor lost two cars though common-mode failures. Fortunately, the whole thing was on tape and the law men were effective; no vehicle damage.
Should I hide my vehicle registration, or move to Michigan?
In theory, there’s no difference between theory and practice. In practice, there is.
Last time I started with my friend Willie’s bold claim that he doesn’t believe in probability; then I gave a short history of probability. I observed that defining probability is a controversial matter, split between objective and subjective interpretations. About the only thing these interpretations agree on is that probability values range from zero to one, where P = 1 means certainty. When you learn probability and statistics in school, you are getting the frequentist interpretation, which is considered objective. Frequentism relies on directly equating observed frequencies with probabilities. In this model, the probability of an event exactly equals the limit of the relative frequency of that outcome in an infinitely large number of trials.
The problem with this interpretation in practice – in medicine, engineering, and gambling machines – isn’t merely the impossibility of an infinite number of trials. A few million trials might be enough. Running trials works for dice but not for earthquakes and space shuttles. It also has problems with things like cancer, where plenty of frequency data exists. Frequentism requires placing an individual specimen into a relevant population or reference class. Doing this is easy for dice, harder for humans. A study says that as a white males of my age I face a 7% probability of having a stroke in the next 10 years. That’s based on my membership in the reference class of white males. If I restrict that set to white men who don’t smoke, it drops to 4%. If I account for good systolic blood pressure, no family history of atrial fibrillation or ventricular hypertrophy, it drops another percent or so.
Ultimately, if I limit my population to a set of one (just me) and apply the belief that every effect has a cause (i.e., some real-world chunk of blockage causes an artery to rupture), you can conclude that my probability of having a stroke can only be one of two values – zero or one.
Frequentism, as seen by its opponents, too closely ties probabilities to observed frequencies. They note that the limit-of-relative-frequency concept relies on induction, which might mean it’s not so objective after all. Further, those frequencies are unknowable in many real-world cases. Still further, finding an individual’s correct reference class is messy, possibly downright subjective. Finally, no frequency data exists for earthquakes that haven’t happened yet. All that seems to do some real damage to frequentism’s utility score.
The subjective interpretations of probability propose fixes to some of frequentism’s problems. The most common subjective interpretation is Bayesianism, which itself comes in several flavors. All subjective interpretations see probability as a degree of belief in a specific outcome, as held by a rational person. Think of it as a fair bet with odds. The odds you’re willing to accept for a bet on your race horse exactly equals your degree of belief in that horse’s ability to win. If your filly were in the same race an infinite number of times, you’d expect to break even, based on those odds, whether you bet on her or against her.
Subjective interpretations rely on logical coherence and belief. The core of Bayesianism, for example, is that beliefs must 1) originate with a numerical probability estimate, 2) adhere to the rules of probability calculation, and 3) follow an exact rule for updating belief estimates based on new evidence. The second rule deals with the common core of probability math used in all interpretations. These include things like how to add and multiply probabilities and Bayes theorem, not to be confused with Bayesianism, the belief system. Bayes theorem is an uncontroversial equation relating the probability of A given B to the probability of A and the probability of B. The third rule of Bayesianism is similarly computational, addressing how belief is updated after new evidence. The details aren’t needed here. Note that while Bayesianism is generally considered subjective, it is still computationally exacting.
The obvious problem with all subjective interpretations, particularly as applied to engineering problems, is that they rely, at least initially, on expert opinion. Life and death rides on the choice of experts and the value of their opinions. As Richard Feynman noted in his minority report on the Challenger, official rank plays too large a part in the choice of experts, and the higher (and less technical) the rank, the more optimistic the probability estimates.
The engineering risk analysis technique most consistent with the frequentist (objective) interpretation of probability is fault tree analysis. Other risk analysis techniques, some embodied in mature software products, are based on Bayesian (subjective) philosophy.
When Willie said he didn’t believe in probability, he may have meant several things. I’ll try to track him down and ask him, but I doubt the incident stuck in his mind as it did mine. If he meant that he doesn’t believe that probability was useful in system design, he had a rational belief; but I disagree with it. I doubt he meant that though.
Willie may have been leaning toward the ties between probability and redundancy in system design. Probability is the calculus by which redundancy is allocated to redundant systems. Willie may think that redundancy doesn’t yield the expected increase in safety because having more equipment means more things than can fail. This argument fails to face that, ideally speaking, a redundant path does double the chance having a component failure, but squares the probability of system failure. That’s a good thing, since squaring a number less than one makes it smaller. In other words, the benefit in reducing the chance of system failure vastly exceeds the deficit of having more components to repair. If that was his point, I disagree in principle, but accept that redundancy is no excuse for lack of component design excellence.
He may also think system designers can be overly confident of the exponential increase in modeled probability of system reliability that stems from redundancy. That increase in reliability is only valid if the redundancy creates no common mode failures and no latent (undetected for unknown time intervals) failures of redundant paths that aren’t currently operating. If that’s his point, then we agree completely. This is an area where pairing the experience and design expertise of someone like Willie with rigorous risk analysis using fault trees yields great systems.
Unlike Willie, Challenger-era NASA gave no official statement on its belief in probability. Feynman’s report points to NASA’s use of numeric probabilities for specific component failure modes. The Rogers Commission report says that NASA management talked about degrees of probability. From this we might guess that NASA believed in probability and its use in measuring risk. On the other hand, the Rogers Commission report also gives examples of NASA’s disbelief in probability’s usefulness. For example, the report’s Technical Management section states that, “NASA has rejected the use of probability on the basis that such techniques are insufficient to assure that adequate safety margins can be applied to protect the lives of the crew.”
Regardless of what NASA’s beliefs about porbability, it’s clear that NASA didn’t use fault tree analysis for the space shuttle program prior to the Challenger disaster. Nor did it use Bayesian inference methods, any hybrid probability model, or any consideration of probability beyond opinions about failures of critical items. Feynman was livid about this. A Bayesian (subjective, but computational) approach would have at least forced NASA to make it subjective judgments explicit and would have produced a rational model of its judgments. Post-Challenger Bayesian analyses, including one by NASA, varied widely, but all indicated unacceptable risk. NASA has since adopted risk management approaches more consistent with those used in commercial and military aircraft design.
An obvious question arises when you think about using a frequentist model on nearly one-of-a-kind vehicles. How accurate can any frequency data be for something as infrequent as a shuttle flight? Accurate enough, in my view. If you see the shuttle as monolithic and indivisible, the data is too sparse; but not if you view it as a system of components, most of which, like o-ring seals, have close analogs in common use, with known failure rates.
The FAA mandated probabilistic risk analyses of the frequentist variety (effectively mandating fault trees) in 1968. Since then flying has become safe, by any measure. In no other endeavor has mankind made such an inherently dangerous activity so safe. Aviation safety progressed through many innovations, redundant systems being high on the list. Probability is the means by which you allocate redundancy. You can’t get great aircraft systems without designers like Willie. Nor can you get them without probability. Believe it or not.