1 math trick for better predictions

Warning, this is “I watched one YouTube video” level of expertise. Also, some graphs have truncated y-axis.

Predictions are fun. Will a dice roll four or greater? Will it rain tomorrow? Will this company be worth more money tomorrow, next month, next year? An event does or doesn’t happen. We get to predict an outcome.

If an NFL team wins six of their first seven games how many games will they win in total? Well 6/7 is ~85%, and there are seventeen games therefore they’ll win ~14.5 games. But in 2021 there was a team that won six of their first seven games and one math trick could predict it.

Pierre-Simon Laplace gives us the “rule of succession”. That sounds complicated but it’s simple: For any number of outcomes add one to the observed cases and two to the total cases.

Here are four coin flips: heads, heads, tails, heads. The observed rate for heads is 0.75 (3/4). The ‘Laplace’ rate for heads is 0.66 (4/6). Laplace’s addition shifts predictions away from ‘never’ and ‘always’. This is the secret. ‘Never’ and ‘always’ are rare for sequential events.

Here is what the Laplace rate looks like compared to the observed rate for eighteen coin flips.

Here is what the Laplace rate looks like compared to the observed rate for the “six of the first seven” football team, the 2021 Tampa Bay Buccaneers.

Laplace starts at .500. Tampa wins six of their first seven games (.857) but Laplace only increases to .777. Their final winning percentage was .764.

Then there’s the 2021 Detroit Lions, a team that lost their first eight games.

The Laplace rate doesn’t know anything. It doesn’t know coins are 50/50. It doesn’t know about Tom Brady. It doesn’t know the Lions are bad. It’s just a formula that slowly adjusts to extreme events.

Laplace (b. 1749- d. 1827) didn’t have the NFL, so he made predictions about something else, the sunrise. The observed rate is 1.00. The Laplace rate, after 10,000 observed sunrises, is 0.99990002. So you’re saying there’s a chance?

No. That’s a simple wrinkle. Laplace called the sunrise a special “phenomena” which “nothing at present moment can arrest the course of.”

Coin flips, dice rolls, and drawn playing cards are random and have an expected rate.

Sunrises are special phenomena and Laplace’s rate is less helpful.

Football outcomes are a mix. They’re like the sunrise, in that teams have inherent principles. They’re like coin flips in that predictions are difficult, a sign of randomness.

Math helps: relative vs absolute saving rates, people live longer the longer they live, what the mean age means, the vaccine friendship paradox, how many ants long is Central Park?, or how many rolls of toilet paper do the residents of Columbus Ohio use in a week?

Math can be simple. Technique (add one to the numerator, add two to the denominator) and a bit of explanation (extreme events are rare without explanatory phenomena) is all we need.

Marshmallow moods

Recap: Mood affiliation is when an attitude unduly influence our perception, for example cruise ships. Bayesian thinking is updating our beliefs relative to the information.

“The marshmallow study,” Andrew Huberman told Shane Parrish, “was when they gave kids the option to have one now or two if they wait. It’s fun to watch the videos where the kids sit there and use all sorts of distractions and strategies (to keep from eating the marshmallows).”

It’s enjoyable to like the marshmallow study.

We must discount it.

Being Bayesian means updating on new information and liking is information.

Selling is information too. I believe in meditation, vegetarianism, and exercise because they are hard to sell. If someone said: Sit a room and focus on your breathing and you’ll feel better, I would believe them because one of their incentives is NOT financial. Kinda. Health as a product: vitamins, beds, bells, rings, bands and so on, fails this test. Regulating my sleeping temperature (which Huberman helpfully explains) may be helpful, but the bar of persuasion is higher. That’s being Bayesian.

Deferred gratification works. It makes sense, it shows up in the lab though “the studies aren’t as robust as we once thought,” and “it’s obvious deferred gratifiers do better over the long pull than these impulsive children.” But we must raise the bar when when we want to like it – a form of deferred gratification itself.

TTID Restaurant

Restaurants are an interesting case study. In part because of the accessibility, everyone has can cook something. So much like making movies or winning wines there’s a fair bit of “I’ve seen that done so I could do that”.

But restaurants are difficult businesses. The pricing power belongs to the landlord not the chef. Staffing is brutal. Inventory expired expediently. Examples like Chez Panisse, Five Guys, McDonald’s, and In-N-Out provide great history through industry, but are outliers in business.

I’m reminded of Sir David Spiegelhalter’s (OBE FRS) comments about health news. Basically the Brit wants us budding Bayesians to not update. Health news, David said, is only news because it is novel.

But.

Maybe.

This time is different.

One template for TTID is to ask if the technology has changed the system in an important way. For hobbies it was the internet. For air travel it was deregulation. For tickets it might be NFTs. For high jump it was the landing materials.

Another way to think of technology is: rules of the system.

For restaurants it might be robot. A restaurant rule of thumb is that food, labor, and real estate each tend to eat up 30% of the costs. Yet just with location is the issue of “wholesale transfer pricing“. It’s the same idea behind Netflix’s original content: Can my suppliers raise prices faster for me than I can for my customers? If the answer is yes then we don’t need Admiral Ackbar to note it’s a trap.

But the pasta robot changes that:

“The robot means Cala saves 60% on real estate costs, which it says it puts into spending more on the cost of food ingredients, allowing it, Richard says, to deliver higher quality meals at a better price. The company’s labour costs are similar to other restaurants — they still have staff serving the meals to customers.” – Freya Pratty, Sifted, October 2021

Cala, like ghost kitchens, has shifted the 30/30/30 economic equation. If the JTBD of food has changed then maybe the economics have changed too. Maybe future restauranteurs will feel a little more full.

Even TTID is subject to Spiegelhalters’s scorn. It’s attention grabbing to say that things really are different this time. Hopefully our series helps us figure out when it truly is.

Tailing Rodgers (part two)

For each thing that happens there is a field of potential things which could happen. Those potential events fill out a distribution where some events are more likely than others. A daughter’s height for instance, could be between four and eight feet but it’s very very likely that her height will be between her mother’s and father’s heights.

Thinking about these distributions of potential outcomes can be helpful because the areas which are not compact, like daughter’s height, are interesting.

Our annual NFL example (last year was Tom Brady passing yards) is Aaron Rodgers over/under 38.5 touchdowns. Here’s how we visualized it in September 2021:

The thinking then, as now, was that Rodgers would throw between twenty and fifty touchdowns but not with equal odds. The number of touchdowns would be asymmetrical. It was much more likely Rodgers threw half of 38.5 than double it.

Five games into the season offers a chance to be Bayesians and update our forecast. In addition to the preseason line of 38.5, his career average is 33.4, and his current pace is 32. Mix in the chance of injury, and he could also finish the year with the ten touchdowns he’s tossed thus far.

Let’s tack this on to the 2021 predictions:
– +10 TD, 90%
– +20 TD, 85%
– +30 TD, 50%
– +38.5 TD, 10%

I wanted to go lower on the 38.5 percentage, but one lesson from Cade Massey is to be less certain about extreme events. So in the same way that online doesn’t equate to real life and we should adjust for that, I will adjust my percentages as well.

Daughter height is top of mind because I have eleven and thirteen year old daughters. 😬

Machiavellian framing

“The thing that makes The Prince such a timeless and scandalous work,” explained Stacy Vanek Smith, “is the same exact thing, Machiavelli removes morality from the situation.”

Smith is out to talk about her book, Machiavelli for Women and the book’s seed came about when Smith was stuck on her salary. Rather, her salary disparity. In her first job out of college, Smith and two classmates both ended up at the same organization in roughly the same jobs. But, not with the same pay. Rather than plead her case, pound the table, and present data, “I was in an emotional spiral of unjustness and upset and I never asked for a raise.” Smith needed some unemotional advice.

“What Machiavelli does is remove all that. He would probably look at (the salary disparity) as great information to use. Now what’s the best way to go about getting a raise? What’s the best way to ask? What do I do now? That’s why it is timeless, because it’s so smart.” – Stacy Vanek Smith, September 2021

Good framing is a design choice that affects behavior. We can frame self talk by having multiple ‘jobs’. We can frame vaccines as better than being bulletproof. We can frame decisions by asking, would I want this even if it were free? Each prompt changes the reference point and possibly the behavior.

As needed then, maybe some people should be Machiavelli Bayesians. Be slightly more princely, if that works do it again until it doesn’t.

Being (even more) Bayesian

Bayesianism has become my favorite math-idea-that-doesn’t-involve-math. It’s three simple steps. Step 1: have an idea about a thing. Step 2: observe the thing. Step 3: have a new idea based on the observation. (repeat)

There are two tricks to make this work for you. The first is how much to update. Being Bayesian means changing your mind in proportion to the change. Try the expression, “I’m slightly more sympathetic to X,” for example. Saying this acknowledges the new information and massages the ego.

The second trick is where to start (Step one), and we have to start somewhere.

“By not taking advantage of the accumulated knowledge that we have as a scientific community, we are artificially leveling the playing field. We are giving theories with no basis in scientific fact too great of a chance to prove themselves through the data.” – Aubry Clayton, The Conversation, August 2021

Clayton’s context is Covid19, but he touches on a larger point too. How much coordination and decentralized command a system allows.

A decentralized command iterates quickly. From the front lines of fast food to fashion to fights. If an organization wants to move fast, the decentralized command structure works better than coordination.

But while individual agents may be fast, the whole may be slow. Why? No coordination. The scientists in a medical research lab will do more experiments with no oversight or collaboration but they may not make more progress.

Coordination and decentralized command apply to both knowledge and people. Having accurate base rates and priors means coordinating our existing knowledge with the accumulated.

Bayesians even frame things beautifully. It’s not “changing your mind” bur rather it is “updating your beliefs”.

3 prompts to being Bayesian

How much should new information matter? Or, is this time different? Because sometimes it’s not.

“Merlin (Heidemanns) said that essentially the polls gain more weight. It’s not that we construct a model and weight the polls. We don’t take a weighted average of the polls, we estimate latent parameters and the polls are data. That said, you can roughly approximate the estimate as a weighted average.”

Andrew Gelman

According to Gelman, priors like “the economy, stupid” never exit the model. According to Nate Silver, the final poll removes all prior data.

In college sports priors matter more than Gelman allows for politics. Nearly two-thirds of college basketball teams who start a season ranked in the top-25, finish in the top-25. College football is the same.

How much new information should matter is a tricky question, but it’s helpful and why the Wharton Moneyball co-hosts encourage each other to become more Bayesian.

At the start of a football season we can guess (or hope) on a team’s chances. With more information, each play, game, and season, we update our idea. Eventually our guess at the start of the year gives way to the information of the year.

It’s hard to do though because we don’t know is this time different. Most of the time it’s not different enough, and base rates work best. But there are three general frameworks which might help us become more Bayesian.

First is to ask a Marc Andreessen like question: is software eating the world? How has the system changed and what does that mean? From FAANG to Testa joining the S&P, it seems like a systemic shift toward technology. Ditto for passing in football.

Second is to ask the Michael Mauboussin question: how much of this was luck? There’s a lot more luck in a single football play than an entire football game. It’s always a mix of skill and luck, but in what ratio?

Third is to consider our identity: am I attached to a position for unacknowledged reasons? This category includes biases like sunk cost and personal influences like ego or status.

The 2021 vaccine rollout is a good instance of practicing Bayesianism. Start with the base rate for vaccines. Watch for evidence. Adjust accordingly.