How Do I Create a Complex Gaussian Joint PDF Plot?

How do I create a complex Gaussian joint PDF plot (example figure)?

Thanks to User-12464257877228681648 for catching an important typo corrected below. In addition to Fred Feinberg and Justin Rising’s excellent theoretical answers, I would add a practical point. The assumption of a joint Gaussian distribution is among the most dangerous and unlikely ones you can make, more dangerous and unlikely than assuming a univariate Gaussian distribution. For example, the monthly real total return on the S&P500 since 1871 has a mean of 0.64% and a standard deviation of 4.09%. If returns were Gaussian, you’d expect only one month with a loss of more than 12.7%. In fact there have been 10. For a 10-year treasury the mean was 0.23% and the standard deviation was 1.65%, so you’d expect one monthly loss of 5.1% or worse, but there have been 6. But “fat tails” in real data are well known, and so its no surprise to find extreme events happened 10 or 6 times as often as Gaussian prediction. You get much more extreme deviations when you look at combinations of events. Months like February 1980 and December 2008 saw joint moves in stocks and bonds that should happen less than once per billion years. Even allowing for the excess extreme individual events, the combinations were nearly impossible under the dependence structure allowed by joint Gaussian distributions. This is why copula fixes fail so spectacularly—t correct for the marginal distributions, but not for the dependence structure. The problem multiplies rapidly with more variables. It’s easier to illustrate with Bernoulli random variables than Gaussian, but the principle is the same. Suppose you have 10 uncorrelated events, each with 10% probability. What is the chance that all ten will occur? People are tempted to take 0.1 to the 10th power and answer one chance in 10,000,000,000. This is basically the principle used in the joint Gaussian distribution, in which “uncorrelated” implies “independent.” But imagine if you put ten pieces of paper in a hat. On one you put all ten events. Then for each of the 10 events, you add nine pieces of paper listing it only. Toss in 9 blank sheets. Draw one piece of paper, and the events listed on it happen. The chance of any one event is 10%, since it is written on 10 of the 100 sheets of paper. The chance of any pair of events is 10% squared, since the pair occurs only on one sheet of paper. So the events are uncorrelated (their joint probability is the product of their individual probabilities). But the chance of all ten events is 1 in 100, not 1 in 10,000,000,000. A practical example of this is when people do risk assessments by listing all the things that have to go wrong for a disaster, and multiplying their probabilities—such as a fire has to start, the person monitoring fires has to miss it, when it’s noticed the fire extinguisher has to be missing or defective, the call to the fire department must fail to go through, and so on (a spectacular real example is the estimate of the probability of serious fires in the Chunnel connecting England and France). Major disasters with multiple apparently unrelated causes are far more common than this kind of analysis suggests because even though individual components of the disaster may be uncorrelated in normal times, some abnormal situation can arise that affects all of them at once. That’s impossible under joint Gaussian assumptions in which only pairwise correlations matter. The real world has lots of higher-order and complex dependence among variables, and these cause more great disasters, and great opportunities, than fat tails.