How They Fit Together: Bell Curves, Bayes and Black Swans

Probability is defined as the possibility, chance or odds of likelihood that a certain event or occurrence will take place now or in the future.  In a world where business managers like to “know the odds”, how does probabilistic thinking (Frequentism and Bayesian) mesh with extreme events (i.e. Black Swans) that just cannot be predicted?

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Statisticians lament how few business managers think probabilistically. In a world awash with data, statisticians claim there are few reasons to not have a decent amount of objective data for decision making. However, there are some events for which there are no data (they haven’t occurred yet), and there are other events that could happen outside the scope of what we think is possible.

The best quote to sum up this framework for decision making comes from the former US Defense secretary Donald Rumsfeld in February 2002:

“There are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – there are things we do not know we don’t know.”

Breaking this statement down, it appears Mr. Rumsfeld is speaking about Frequentism, subjective probability (Bayes) and those rare but extreme events coined by Nassim Taleb as “Black Swans”.

Author Sharon Bertsch McGrayne elucidates the first two types of probabilistic reasoning in her book “The Theory That Would Not Die”.  Frequentism (conventional statistics), she says, relies on measuring the relative frequency of an event that can be repeated time and again under the same conditions. This is the world of p-values, bell curves, coin flips, casinos and actuaries where data driven decision making is objective based on sampling or computations of large data sets.

The greater part of McGrayne’s tome concentrates on defining Bayesian Inference, or subjective probability also known as a “measure of belief”. Bayes, she says, allows making of predications with no prior information at all (no frequency of events).With Bayes, one makes an educated guess, and then keeps refining that guess based on new information, thus updating and revising the probabilities, and getting “closer to certitude.”

Getting back to Rumsfeld’s quote, Rumsfeld seems to be saying we can guess the probability of  “known knowns” because they’ve happened before and we have frequent data to support objective reasoning. These “known knowns” are Nassim Taleb’s White Swans. There are also “known unknowns” or things that have never happened before, but have entered our imaginations as possible events (Taleb’s Grey Swans). We still need probability to discern “the odds” of that event (e.g. dirty nuclear bomb in Los Angeles), so Bayes is helpful because we can infer subjective probabilities or “the possible value of unknowns” from similar situations tangential to our own predicament.

Lastly, there are “unknown unknowns”, or things we haven’t even dreamed about (Taleb’s Black Swan).  Dr. Nassim Nicholas Taleb labels this “the fourth quadrant” where probability theory has no answers.  What’s an illustration of an “unknown unknown”? Dr. Taleb gives us an example of the invention of the wheel, because no one had even though or dreamed of a wheel until it was actually invented. The “unknown unknown” is unpredictable, because—like the wheel—had it been conceived by someone, it would have been already invented.

Rumsfeld’s quote gives business managers a framework for thinking probabilistically. There are “known knowns” for which Frequentism works best, “unknown knowns” for which Bayesian Inference is the best fit, and there is a realm of “unknown unknowns” where statistics falls short, where there can be no predictions. This area outside the boundary of statistics is the most dangerous area, says Dr. Taleb, because extreme events in this sector usually carry large impacts.

This column has been an attempt to provide a decision making framework for how Frequentism, Bayes and Black Swans fit together—by using Donald Rumsfeld’s quote.

What say you, can you improve upon this framework?

The Human Factor Continually Confounds Probability Models

With four weeks to go in the 2011 Major League Baseball season, the probability of the Boston Red Sox of making the playoffs was 99.6%. And most of us know the story; in one of the biggest collapses in baseball history, the Red Sox tanked a nine game lead and served the wild card slot to the Tampa Bay Rays. In creating “one for the record books”, the 2011 Red Sox show us that the human factor continually confounds probability models.

Some things aren’t supposed to happen. The 2011 Boston Red Sox certainly should not have missed the playoffs with a nine game lead, and the 1995 Anaheim Angels should not have finished their year 12-26 (losing a nine game lead and missing the playoffs). Moreover, probability models said the stock market (DJIA) should not have lost 54% of its value in the 2008 “Great Recession”.

There’s definitely a danger in too much reliance on normal distribution probability models, especially when humans are concerned says Financial Times writer John Authers.

Studying the 2011 Boston Red Sox, Authers suggests the team may have been overconfident in statistics since few teams in baseball history had collapsed with such a lead.

Authers also believes bell curve probabilistic models would not have been a reliable indicator of possible failure because such models assume event independence where one event should not affect another. But those who follow sports understand the concept of “momentum in a game”, or even from game-to-game where a team can feed off past success to gain confidence.

In reference to the 2008 market crash, Steven Solmonson, head of Park Place Capital Ltd said; “Not in a million years would we have expected this gyration to be as vicious and enduring as it has been.”  And I’m sure that Boston Red Sox fans didn’t believe their team could lose a significant lead over the Tampa Bay Rays with just a few games left in the season.

Whenever humans are involved, the lesson is clear: don’t get over confident in normal distribution probability models. Next thing you know, you might get slapped (or worse) by the fat tail.

Blasphemy? Quantitative Approaches Don’t Always Work Best

Ray Dalio’s Bridgewater Associates hedge fund, Pure Alpha II, is up 25% in year that hasn’t been kind to competitors. How did he do it?  Hint: it wasn’t through a purely quantitative approach.

Hedge fund manager Ray Dalio is a rare breed in financial investing. Dalio is known as a “macro” investor, or someone who takes a “big picture” approach to investing as opposed to math whiz “quants” who rely on quantitative/numerical techniques.

The July 25, 2011 issue of New Yorker, highlights Dalio’s investment methods as he looks for hidden profit opportunities; “(Dalio) spends most of his time trying to figure out how economic and financial events fit together in a coherent framework. His constant goal (is to) understand how the economic machine works.”

Dalio isn’t concerned with the nuts and bolts of companies. He doesn’t want to scrub the bowels of the machine to see how it works. And he shuns frequency based probability techniques used by financial quants to estimate whether stocks will move up or down in penny increments.

While other hedge funds and investment banks control risks with sophisticated Value at Risk (VAR) models and use of derivatives, Dalio suggests that studying the big picture is a better approach. “Risky things are not in themselves risky if you understand them and control them,” he says. Instead of statistical distributions, it appears Dalio is more focused on what he calls the “probability of knowing”.  He never places all his eggs in one basket, especially because he understands that a complex and global world can shift course in a moment’s notice.

This is not to say, however, that Dalio doesn’t use analytical techniques. Of course, Dalio crunches the numbers and uses computers for much of his work. But he’s not driven by making money with techniques such as high frequency trading, where super computers trade liquid instruments at near light speed. Instead, his algorithmic trading models are written with his investment philosophy of components and relationships in mind, and help supplement decision making for broad and big bets.

Dalio is doing much more than guesswork here, but it’s a different kind of analysis based on a rules based framework codified in thirty years of investment experience. “It’s the commitment to systematic analysis and investment (within the boundaries of his mental framework) that makes the difference,” he says.

The contrast between Dalio’s approach and those of data driven quants couldn’t be clearer. Quants model investment decisions based on math and use computers to move volumes of liquid securities thus making money on tight spreads. Dalio seeks to understand “larger underlying forces”, interrelationships and historical context. His main advantages appear to be a “top down” rather than “bottom up” approach to investing and the pursuit of a longer time line for decision making.

In 2008 during the worst of the Great Recession, Dalio was up 9.5%, in 2010 the fund was up 45%, and Dalio’s $122B fund is up 25% this year (2011) based on macro bets for Treasuries, Japanese Yen and Gold.

It may be blasphemy, but for one investor, a macro “big picture” approach is proving much more profitable than one that’s (normal distribution) probability driven.