The fake rigor of the mighty dollar

Are we more uncertain about our predictions or our cost-benefit analyses?

Our model for outcome optimization required a specification of a probabilistic model of the world, the process in which we measure, and the value of each of our actions. Moving to a probabilistic model lets us characterize our uncertainty in measurements and predictions. But none of this accounts for the uncertainty in our modeling itself. What happens when our model of the environment, measurement, and actuation is inaccurate? This is a question of uncertainty quantification.

Let’s go back to our main decision rule from yesterday.

if odds(condition given data) > (C(1, 0)-C(0,0))/(C(0, 1)-C(1, 1)):
     take action
else:
     don't take action

To execute this rule, we need to compute the odds of the condition being true. We also need to compute the ratio of costs of action to inaction. Let’s call that term the cost-benefit threshold.

Which of these is uncertain? By and large, in machine learning and statistics, we spend our time quantifying the errors of odds. In casino games like roulette, we can precisely compute the odds of winning. But even for betting on sporting events, it’s hard to nail down the chances that one team will beat another. You might even say that a probabilistic model is ill-suited to predict one-off events (I would say this!). But gamblers love to think they are smarter than the oddsmakers in Vegas. And, as we see in the obsession with prediction markets, many haughty statisticians think gambling is a valid path to computing odds.

In non-gambling contexts, we might quantify the uncertainty in our odds by calibration adjustment or by using techniques for “prediction intervals.” If we can build error models for probability models, we might solve our decision problem using distributionally robust stochastic optimization or some other highly sophisticated search method.

I’ll come back to prediction intervals and markets in future blogs, as there has been a lot of noise about this in machine learning and statistics. That crowd is promising much more than they can deliver. But today I want to note how statistical overmodeling is losing the forest for the trees. In our algorithm, we know that we should definitely act when our odds are much greater than the cost-benefit threshold. And we should definitely not act when the odds are much lower than the threshold. As we get near the cost-benefit threshold, that’s when uncertainty quantification of odds might be useful. But maybe we should step back and ask ourselves how well specified that threshold is in the first place.

Do we really think we can precisely come up with costs to build an absolute line in the sand for decisions? This is, of course, absurd. The uncertainty in the costs usually dwarfs the uncertainty in odds.

Let’s again turn to medical decision-making, as it provides relatable examples of how daunting action in the face of uncertainty can be. There is always a cost associated with not treating a sick person. But every treatment comes with some associated harm. Can we ever compute a precise cost-benefit threshold?

Start with antibiotics. There is some associated harm of antibiotics. They kill your gut biome. They might create resistant bacteria. But it’s not hard to argue that the cost of untreated a nasty bacterial infection is much greater than the associated negative effects of an antibiotic regimen. Perhaps we can’t quantify these costs to the dollar amount, but if a bacterial infection carries a high risk of death, then it’s pretty clear that the treatment’s costs are far subsumed by its benefit.

Let’s take a step up in uncertainty and think about the case of Paxlovid from Monday. Given recent announcements, the harm associated with Paxlovid might just be its cost! Pfizer has decided to charge 2500 dollars per regimen. But Paxlovid also has complex interactions with other medications and known harmful side effects. Taking the drug if you are not covid positive is definitely very costly. But what is the expected reward if you have covid and take Paxlovid? Again, this is unfathomably unclear. We actually have no idea what happens if a healthy, vaccinated, young person takes this drug. Obviously, preventing hospitalization would be great, but it is completely unclear if Paxlovid prevents this. The value of the action of this treatment has been impossible to quantify. So then what is the cost-benefit threshold?

And it unfortunately gets worse as we move into more life-threatening conditions that are harder to detect. Think now about cancer screening. I could spend (and have spent!) a longer time diving into the complexity of screening for cancer. It makes sense that we want to detect bad cancers early. But what is the cost-benefit analysis? What is the cost of an unnecessary mastectomy for a patient who is misdiagnosed with malignant breast cancer? Associating this with a dollar amount is grotesque and heartless. The existence of medical malpractice settlements does not let us equate medical harm with financial payouts. Similarly, what is the cost of not treating cancer before it is symptomatic? For many cancers, the prognosis is the same whether it’s detected early or late. Should we screen all 20 year olds for cancer or not? What about all 50 year olds? We can’t compute a cost-benefit threshold for these questions.

When we have error bars on costs and error bars on odds, it becomes clear that cost-benefit analyses are made up to align with what people want to believe. As Jordan Ellenberg loves to remind us, multiplying badly quantified values with other badly quantified values and summing them up is just vibes. It feels like rigor, but we’re using math to formalize hand waving. Hypothesizing odds and costs of the unknowable doesn’t make you rigorous and rational. It just makes you a pedantic bullshit artist.

And yes, this is part of the reason why utilitarianism and its mutant children effective altruism and existential-risk millenarianism are not serious philosophies. Some things are not mathematically frameable as optimization problems. Morality is one of them.

Perhaps that’s enough fanning of the flames for today. The point of the posts this week is setting the stage for the rest of the semester. We’re going to talk about optimizing a lot. Optimization can be a powerful and valuable framework, but a system based on optimization alone is deeply flawed. The pure optimization mindset leads to solutions both grotesque and fragile. Still, it’s important to teach it. Optimization helps us explore the vast complexity of acting under uncertainty. And it is by far the most common mathematical framework for decision-making. We can work with deeply flawed systems only if we know their inherent limitations. Let’s keep this in mind as we dive into the challenging technical content to come over the next few weeks.