by Ian Tregillis
Well, writing this post isn’t quite as carefree as I might have hoped.
In my previous blog post (which you can read here if you’d like a refresher), written in May of 2019, I presented some simple thought experiments related to the statistics of our fictional in-universe pandemic of Xenovirus Takis-A (aka the Wild Card virus). Since then, reality has overtaken fiction: as I write this in August of 2020, we’re in the midst of the very real and very deadly global COVID-19 pandemic. It feels a bit twee, perhaps even a bit disrespectful, to while away an idle Saturday afternoon with ultrasimplistic mathematical models of a fictional virus when people in the real world are risking their lives, even as I type this, to understand the dishearteningly real novel coronavirus. It’s grim out there.
(I doubt that I’m unique among Wild Cards writers when I say the current global situation has drastically changed the way I think about the history of our fictional shared universe. Given the real-world social and economic disruptions arising from the COVID-19 pandemic, I struggle to envision the aftermath of the Wild Cards release in 1946 as anything short of post-apocalyptic.)
On the other hand, sometimes fiction can provide a bit of escapism and relief from dark times. And because I am a gigantic nerd, I happen to enjoy playing around with simple math problems. It isn’t as fun as going out and chatting with a pal over a cup of coffee (a simple pleasure I sorely miss in these days of social distancing), but there are worse ways to spend the afternoon. So in this post I’m going to pick up where I left off, in the probably vain hope that somebody else will enjoy the game, too.
Three Not-Entirely Crazy Axioms
Previously, I argued that
- Human populations within the Wild Cards universe must be rife with “crypto-jokers” and “crypto-aces”: people mistakenly thought to be nats, even by themselves. These cryptos, in other words, are walking cases of unidentified of Takis-A expression. (For instance, if the virus painted ultraviolet racing stripes on your aorta, you’d technically become a joker. But how would anybody, including you, ever know that?)
- The 90 : 9 : 1 rule of thumb (black queens : jokers : aces) can only be observed to apply to identified cases of Takis-A expression. After all, if you don’t know how many cards have actually turned—and that number is unknown, thanks to the cryptos—you can’t calculate the absolute outcome percentages, only percentages of the known turns.
- The effects of the Wild Card virus arise from a multivariate probability distribution, rather than from a univariate distribution. That is to say, the fate of a turned card is not determined by a single die roll: RPG roots notwithstanding, it makes little sense to place jokerness and aceness on the same continuum.
I continue to stand by these assertions. Check out my previous post if you’d like to see the underlying arguments; they’re really pretty logical. We can think of these as the three foundational axioms for this exercise.
The Uncorrelated Cartesian Models
From there I constructed a very simple toy model demonstrating how these things might come together into a single coherent picture consistent with the population statistics established by fiat in the Wild Cards universe. That model, which here I’ll call the “Uncorrelated Cartesian” model, is summarized as follows. Let’s say X represents the fraction of Takis-A expressions (i.e., the fraction of turned card outcomes) that are correctly identified as such. For instance, if X=0.75, then Wild Card effects are recognized about 75% of the time. The remainder, 1-X, represents people who are thought to be (and who believe themselves to be) nats: the crypto-jokers and crypto-aces who are unnoticeably altered by the virus. This scenario can still produce the empirically observed 90 : 9 : 1 rule within that X% of the turned-card population by combining outcomes determined by separate “Jokerism” and “Ability” probability distributions, like so:
Here the Jokerism distribution has three regimes: black queen, joker, and crypto-joker, while the Ability distribution has two regimes: ace and crypto-ace. It isn’t hard to prove these regimes must have the following mathematical relationships:
\begin{aligned} f_{black queen} &= 0.90\,X \\ %% f_{joker} &= 0.09\,X\,\left(\frac{1 – 0.90\,X}{1 – 0.91\,X}\right) \\ %% f_{crypto-joker}&=\left(1 – X\right)\,\left(\frac{1 – 0.90\,X}{1 – 0.91\,X}\right) \\ %% f_{ace} &= \frac{0.01 \, X}{1 – 0.90 \, X} \\ %% f_{crypto-ace} &= \frac{1 – 0.91 \, X}{1 – 0.90 \, X}\end{aligned}where f refers to the fraction of the area under the probability distribution curves (which is why the two Ability fractions add up to unity, as do the three Jokerism regime fractions). I didn’t bother to do this last time, but it’s a piece of cake to plot those fractions as a function of X. They give rise to the following trends:
where here the plot on the left shows the individual fractions on the separate Jokerism and Ability axes, while the plot on the right shows the population percentages that result from combining those distributions.
These are interesting, though not particularly exciting trends. (If you’d like, you can go back to that previous post and verify that these plots are consistent with the calculations presented there for X=100% and X=75%.)
You’ve probably noticed that because the 90 : 9 : 1 rule has only three categories—black queen, joker, and ace—I somewhat arbitrarily lumped joker-aces in with the aces. (I think of this as the “Ace-Biased Model,” or “Model-A” for short.). However, there’s no reason we couldn’t instead lump joker-aces with the jokers (the “Joker-Biased Model,” or “Model-J.”) The chart for the Joker-Biased Uncorrelated Cartesian Model would look like this:
which gives rise to the following mathematical relationships among the regimes:
\begin{aligned} f_{black queen} &= 0.90\,X \\ %% f_{joker} &= 0.09\,X \\ %% f_{crypto-joker} &= 1 – 0.99\,X \\ %% f_{ace} &= \frac{0.01 \, X}{1 – 0.99 \, X} \\ %% f_{crypto-ace} &= \frac{1 – X}{1 – 0.99 \, X}\end{aligned} and trends that look like this:
Naturally, the population percentage plot is the same for the Ace- and Joker-Biased models, because both models were explicitly derived to adhere to the 90 : 9 : 1 rule, subject to the unknown fraction of unidentified card turns. The joker distribution fractions are also very similar for both models across all X values. On the other hand, while the ace distribution fractions are quite similar in the two models when X is below 80%, they different significantly above this threshold.
When slapping together the Uncorrelated Cartesian toy model, I leaned on two dubious but undeniably handy simplifying assumptions:
- The black queen is always the fatal limit of extreme Jokerism.
- The Jokerism and Ability distributions are completely uncorrelated.
What can I say? I’m a theoretician by trade. Solving a “spherical cow” problem in order to gain insights about non-spherical cows is a venerable tradition in my field. So sue me. But as I discussed previously, it’s pretty easy to devise counterarguments for these assumptions. (However, I emphasize that our three foundational axioms are independent of these assumptions. These assumptions were merely tools for simplifying the journey from axioms to equations.)
Let’s build a superior model by relaxing these assumptions.
The Correlated Polar Model
For starters, let’s change the coordinate system of our parameter space, like so:
Now the key random variables aren’t “Jokerism” and “Ability” but rather the quantities I call “S,” which represents the “severity” of a given Takis-A expression, and “\theta,” which can be thought of as a “mixing angle” between pure joker-dom and pure ace-dom. I’ve cribbed this mixing-angle concept from my friends in the particle physics racket. Strange as it might seem, it’s tremendously useful.
For instance, you’ve probably noticed that unlike the Cartesian models, our revised map no longer accommodates the dreaded black queen. Or does it? Above, I’ve shown a single quadrant of the parameter space, corresponding to mixing angles between 0^\circ and 90^\circ. But what about mixing angles between 90^\circ and 360^\circ? Those would correspond to “negative” values of acedom and/or jokerdom, and what the hell would that even mean? Well, I speculate that’s where the black queen lurks. So the FULL parameter space of our new model looks like this:
Okay, maybe that’s a tad lurid. But this approach has many advantages over the Cartesian models:
The random variables in this model represent intuitively simple concepts in contrast to the Cartesian models’ more abstract and arbitrarily distinct variables.
The black queen now has a very logical interpretation within the model, no longer requiring an ad hoc treatment. As shown below, this has very interesting implications.
Joker-aces no longer present a categorization problem. The mixing angle means that except for extraordinarily rare individuals who land precisely atop the jokerdom or acedom axes, every non-fatal Takis-A expression involves a mixture of joker and ace traits.
The logically inevitable crypto-jokers and crypto-aces now have a simple and obvious interpretation. Cryptos are individuals who underwent a “subtle” low-severity transformation: S < S_c.
Of course, the subtlety threshold S_C arises from subjective considerations that are difficult to quantify. Similarly, the distinction between “jokers” and “aces” as unique categories is essentially artificial. The somewhat arbitrary boundary between those regimes, \theta_0, is determined by subjective perceptions of whether somebody is predominantly deformed or superpowered. In fact any multivariate model will blur those distinctions, yet because the notion of distinct aces and jokers is so fundamental to the Wild Cards universe, every such model must provide a method for delineating those categories.
The crucial point underlying all of these games is this: the Wild Card virus has its own statistical distribution of expression outcomes regardless of whatever conceptual model we use to study those outcomes. There is an underlying reality (i.e., the “truth” of the virus), as distinct from the empirical reality that we infer from observations of the non-crypto population.
Let’s denote the true mortality fraction of the virus, which is unknown, by \mu. We know 0 < \mu < 1 because some people, but not all people, die when their card turns. The conventional wisdom, inferred from the empirically observed 90 : 9 : 1 rule of thumb is that \mu = 0.90, or equivalently that the mortality rate of Xenovirus Takis-A is 90%. Let’s see if that’s true.
Our new model is based on separate probability distributions for the severity and mixing angle random variables: P(S) and P(\theta). Because these are probability distributions we naturally require
\int\limits_{0}^{\infty} P(S) \, \mathrm{d}S = 1 \qquad\qquad \int\limits_{0^\circ}^{360^\circ} P(\theta) \, \mathrm{d}\theta = 1
where now for the sake of rigor I’ve had to slip in a little bit of calculus, but these expressions are merely saying that there’s a 100% chance of landing somewhere within the domain of the problem. The mixing angle can spin around as much as it wants (because angles exceeding 360^\circ are equivalent to angles in 0^\circ-360^\circ, but let’s put a pin in that for later), while the transformation severity can be anything from zero to infinite (more on that in a moment, too). Our “map” of the parameter space tells us the categorical probabilities in this Polar Correlated Model are
\begin{aligned} P_{BQ} &= \int\limits_{90^\circ}^{360^\circ} P(\theta) \, \mathrm{d}\theta \cdot \int\limits_0^\infty P(S) \, \mathrm{d}S = \int\limits_{90^\circ}^{360^\circ} P(\theta) \, \mathrm{d}\theta \equiv \mu \\ P_{\sim BQ} &= \int\limits_{0^\circ}^{90^\circ} P(\theta) \, \mathrm{d}\theta \cdot \int\limits_0^\infty P(S) \, \mathrm{d}S = \int\limits_{0^\circ}^{90^\circ} P(\theta) \, \mathrm{d}\theta = 1 – \mu \\ P_{C} &= \int\limits_{0^\circ}^{90^\circ} P(\theta) \, \mathrm{d}\theta \cdot \int\limits_{0}^{S_c} P(S) \, \mathrm{d}S = \left(1 – \mu\right) \cdot \int\limits_{0}^{S_c} P(S) \, \mathrm{d}S \\ P_{J} &= \int\limits_{\theta_0}^{90^\circ} P(\theta) \, \mathrm{d}\theta \cdot \int\limits_{S_c}^{\infty} P(S) \, \mathrm{d}S \\ P_{A} &= \int\limits_0^{\theta_0} P(\theta) \, \mathrm{d}\theta \cdot \int\limits_{S_c}^{\infty} P(S) \, \mathrm{d}S\end{aligned}
The empirically observed 90 : 9 : 1 rule places strong constraints on various “sectors” of the probability distributions. It isn’t hard to show
\begin{aligned} \frac{N_J}{N_A} = 9 &\implies \int\limits_{0^\circ}^{\theta_0} P(\theta) \, \mathrm{d}\theta = \frac{1 – \mu}{10} \,\,\, \mathrm{\&} \,\,\, \int\limits_{\theta_0}^{90^\circ} P(\theta) \, \mathrm{d}\theta = \frac{9\left(1 – \mu\right)}{10} \\ \frac{N_J}{N_{BQ}} = \frac{1}{10} &\implies \int\limits_{S_c}^{\infty} P(S) \, \mathrm{d}S = \frac{1}{9}\left(\frac{\mu}{1 – \mu}\right) \,\,\, \mathrm{\&} \,\,\, \int\limits_{0}^{S_c} P(S) \, \mathrm{d}S = \frac{9 – 10\,\mu}{9\left(1-\mu\right)}\end{aligned}
(People who know their stats might be grinding their teeth, because I’ve glossed over the details that justify equating population ratios with probability ratios. After all, the category populations have their own statistical distributions. Here I’m using the average number of people in each category because, for the distributions at play here (more on those below), that ratio really is the same as the probability ratio. Furthermore, when the number of trials (turned cards) grows large – and at this point in the Wild Cards universe it is surely in the millions – these probability distributions squish themselves very tightly to the mean.)
Any probability distributions for the severity and mixing angle that conform to the above constraints will generate, in the Correlated Polar Model, empirical observations consistent with the 90 : 9 : 1 rule. Putting it all together, the resulting categorical probabilities are quite simple:
\begin{aligned} P_{BQ}&= \int\limits_{90^\circ}^{360^\circ} P(\theta) \, \mathrm{d}\theta \equiv \mu \\ P_{C} &= 1 – \frac{10\,\mu}{9} \\ P_{J} &= \frac{\mu}{10} \\ P_{A} &= \frac{\mu}{90} \\ P_{\sim BQ} = P_{C} \,\, + \,\, &P_{J} + P_{A} = 1 – \mu = 1 – P_{BQ}\end{aligned}
A couple of things worth pointing out here:
First, the probability of drawing the black queen is determined by the mixing angle distribution. Given P(\theta) we can compute \mu alongside everything else—it’s integral to the model. That’s fantastic.
Second, in order for all of the probabilities to be physically meaningful, they must of course be non-negative. Therefore, if cryptos exist, as dictated by pretty simple logic (remember, this is one of our axioms), then obviously P_C > 0, which means \mu < 0.90. In other words, the existence of crypto-jokers and crypto-aces immediately tells us the true mortality rate of Xenovirus Takis-A is less than 90%! Hence the title of this blog post. One immediate corollary is that the probability of becoming an ace is less than 1%: contrary to popular wisdom, aces must constitute less than 1% of the total turned-card population. This is consistent with the findings in my previous post (titled “The Fallacy of the One Percent” for this very reason).
Candidate Distributions: A Rogues’ Gallery
As always, the overall problem is under-constrained: we can’t reverse-engineer the virus’s true underlying distributions P(S) and P(\theta) from these simplistic considerations. There exist an infinite number of possibilities for each, even after imposing the sector constraints. But we can play around with some interesting candidates.
For instance, the relative rarity of the most powerful aces—your Fortunatos, your Tesseracts, your Golden Boys, your Curveballs and Amazing Bubbleses—suggests perhaps the transformation severity follows an exponentially decaying probability distribution:
\begin{aligned} P\left(S\right) = \alpha \, \mathrm{e}^{-\alpha \, S} \qquad S_c = -\frac{1}{\alpha} \, \mathrm{ln} \left[ \frac{1}{9} \, \left(\frac{\mu}{1 – \mu}\right) \right]\end{aligned}
where that funky expression for the subtlety threshold comes from imposing the sector constraints. Here \alpha > 0 is an unknown parameter; the larger it is, the harder it is to randomly generate a really powerful ace. A strong argument in favor of exponential distributions and their ilk for P(S) is these functions have the very nice property that they can satisfy the normalization requirement (i.e., the aggregate probability of all possible S values must be unity) without placing an arbitrary upper limit on S—the severity of any individual transformation can theoretically go to infinity. This allows for the possibility that some extraordinarily lucky individual receives a super-high S value and transcends into, I don’t know, say, a godlike energy being. That’s a great thing to build into our model, because within the Wild Cards universe it’s completely unknown if there’s an upper limit to the transformation severity, and in fact such a limit might not exist at all.
Ok, fine. But what about that mixing angle?
First off, consider this: what if the mixing angle probability distribution were completely uniform? That is, what if all angles had an equal chance of being pulled from the distribution? In that case, we’d know immediately \mu = 0.75 simply because fatal transformations cover 3/4 of the parameter plane. That’s a seductively simple result… but unfortunately it’s disallowed: without going through the details, it turns out the uniform distribution violates our sector constraints.
The simplest functional form for a mixing angle probability distribution compatible with all requirements is something linear:
\begin{aligned} P\left(\theta\right) &= \beta\,\theta + \gamma \\ \beta = \frac{8\mu-6}{3\,\pi^2} \qquad \gamma = \frac{15-16\mu}{6\pi} &\qquad \theta_0 = -\frac{\gamma}{\beta} + \sqrt{ \left(\frac{\gamma}{\beta}\right)^2 + \frac{1 – \mu}{5 \, \beta} }\end{aligned}
This will represent a probability that increases as the mixing angle swings away from acedom and toward jokerdom while simultaneously allowing for the existence of cryptos as long as
\frac{3}{4} < \mu < \frac{9}{10}
Qualitatively, the exponential P(S) and linear P(\theta) would look something like this:
where the statistical likelihood decreases from yellow to purple in these contour plots. Together, they’d give rise to a combined distribution of non-fatal transformations akin to this:
where here I’ve marked the S_C and \theta_0 boundaries for illustrative purposes. Notice that an exponentially decaying severity distribution implies a large crypto population.
This is fine and all, but honestly a little boring. Plus, it fails to showcase the versatility of this model. After all, I’ve been calling it the Correlated Polar Model. How might we get a nontrivial correlation between acedom and jokerdom?
One approach is to use a “bell curve” for the probability distribution of mixing angles: \begin{aligned} P\left(\theta\right) = \delta \, \mathrm{e}^{-\frac{\left(\theta-\theta^*\right)^2}{2\,\sigma^2}} \qquad \delta^{-1} = \sqrt{\frac{\pi}{2}}\,\sigma\, \left[ \mathrm{erf}\left(\frac{2\pi-\theta^*}{\sqrt{2}\sigma}\right) + \mathrm{erf}\left(\frac{\theta^*}{\sqrt{2}\sigma}\right) \right]\end{aligned} where here \theta^* is the peak of the distribution, \sigma is related to the width of the distribution, and the normalization constant \delta contains a special mathematical function called an “error function” (hi, fellow nerds), the details of which are immaterial to this discussion. The sector constraints establish a similarly complicated relationship between the ace/joker boundary angle, the peak, and the width. But, qualitatively, such a mixing angle distribution might look like this:
where here for the sake of illustration I set the joker/ace boundary angle at 30^\circ, the peak of the mixing angle distribution at 60^\circ (so it tends to favor jokers over aces), and the distribution width to a fairly narrow 10^\circ. Combining this with the exponential severity distribution from above gives a combined distribution of non-fatal Takis-A expressions like this:
Now there’s a very clear nontrivial correlation between acedom and jokerdom, even though the underlying probability distributions are completely independent of one another. Beautiful! (Alas, this particular bell curve is an unlikely candidate for the mixing angle distribution. When integrated all the way around the circle, through the Black Queen Badlands, it yields an absurdly low mortality fraction. After all, the action is concentrated in the non-fatal quadrant, with the fatal portions of the plane relegated to the statistical boondocks. While I do think the Wild Card virus could be more benign than we realize, I don’t believe it’s harmless.)
Meta-Powered Aces: Lies, Damn Lies, and Statistics
The Correlated Polar Model also provides a simple framework for describing what I’ll call “metapowered” aces: abilities that somehow subvert the usual Takis-A statistics.
For instance, consider Genetrix. Her very unusual power enables her to produce clutches of eggs that hatch into short-lived homunculi with abilities that fly in the face of the typically observed outcomes. That is, she produces far more aces (or at least deuces) than jokers. How is this possible? It’s a worthy question: in-universe, she was a research subject at the BICC for this very reason. But the Correlated Polar Model makes it extremely easy to understand (or, at least, describe mathematically) what her ace power is doing.
I posit that Genetrix’s power subverts the usual joker/ace statistics by effectively rotating the mixing angle distribution, gracing her offspring with an additional phase angle: \begin{aligned} P\left(\theta\right) &\rightarrow P\left(\theta + \phi_{G}\right) \end{aligned}
That’s it! That’s all it takes. For example, say Genetrix’s secret phase angle is 35^\circ. Applying this phase shift to the mixing angle distribution demonstrated previously leads to the following result:
Voila! A statistical distribution skewed in favor of aces over jokers.
Genetrix’s homunculi subvert expectations in another way, too, because they’re always jokers or aces, but never cryptos. This, too, is easy to describe within our model: each of her offspring receives a “boost” to their transformation severity \begin{aligned} S \rightarrow S + S_G \qquad S_G \geq S_C\end{aligned} such that they never roll up a value below the subtlety threshold. The final multivariate distribution for Genetrix’s homunculi might therefore look something like this:
Ta-da! Now we have a statistical distribution that not only prefers aces over jokers, but also is guaranteed to never produce a crypto. Pretty cool. So Genetrix’s power, which seems so unusual at first blush, boils down to “merely” rotating and translating the usual combined probability distribution for Takis-A expression. This is a mathematically well-defined problem that BICC researchers could really sink their teeth into.
For another example of somebody who subverts the usual Wild Card statistics, look no further than our pal Croyd Crenson, aka the Sleeper. The virus in his body resets every time he dozes, thereby greeting him with a completely new form of Takis-A expression when he wakes. And yet somehow, in over 70 years of this cycle, he has never drawn the black queen. That’s hundreds of turned-card trials! If Croyd were pulling from the same random distribution as regular schmoes, he would have pulled a deadly mixing angle decades ago.
How does he manage this trick? That might sound like a vague and hopeless question. But when framed within the context of our model, it’s actually pretty specific: How is it that Croyd’s randomly selected mixing angles never exceed 90^\circ?
I posit the answer lies in what’s known as “modular arithmetic” (not to be confused with Modular Man). Whether you realize it or not, you’re already intimately familiar with this idea. It’s just clock math. Quick, what’s 8 plus 5? On a 12-hour wall clock, the answer isn’t 13, but 1. (If it’s 8 AM now, in five hours it will be 1 PM.) That’s modulo-12 math. Military time is modulo-24. The very idea of modular arithmetic might seem like a cute but ultimately pointless exercise, but in fact it has deep connections to many fields of pure and applied mathematics, including public-key cryptography.
Besides, we’ve been doing modular arithmetic all along. Remember that pin for later? When setting up the model I said angles greater than 360^\circ can always be mapped to values in 0^\circ-360^\circ. That’s just a wordier way of saying we’re working in modulo-360^\circ. (And if you really want to get into the weeds, there’s nothing magical about 360^\circ. For instance, there’s no reason the virus’s true underlying modulus couldn’t be, I don’t know, say 783^\circ, with fatal outcomes spread over mixing angles 90^\circ-783^\circ… but this way lies madness, so let’s all agree to never speak of it again.)
So maybe, just maybe, Croyd’s metapower changes that modulus to impose modulo-90^\circ-or-less arithmetic on the mixing angle distribution: \begin{aligned} P\left(\theta\,\,\rm{mod} \,\, 360^\circ \right) \rightarrow P\left(\theta \,\,\rm{mod} \,\, \hat{\theta}\right) \qquad \hat{\theta} \leq 90^\circ\end{aligned}
Presto! According to our model, a person pulling from this random distribution is guaranteed to never draw the black queen. And by the way, this trick applies to Genetrix, too: her offspring rarely draw the black queen. That again suggests an altered modulus.
Putting it all together, perhaps the mixing angle and transformation severity in the Correlated Polar Model should be thought of like this: \begin{aligned} \theta + \hat{\phi}_{\rm{phase}}&\,\, \rm{mod} \,\, \hat{\theta}_{\rm{modulus}} \\ S &+ \hat{S}_{\rm{boost}}\end{aligned} where only metapowers can alter the hatted variables, which are otherwise fixed to base values \begin{aligned} \hat{\phi}_{\rm{phase}} = 0^\circ, \qquad &\hat{\theta}_{\rm{modulus}} = 360^\circ, \qquad \hat{S}_{\rm{boost}} = 0\end{aligned}
The X-Factor
Our model for Takis-A expression outcomes incorporates the black queen in a very natural way, to the extent that given a mixing angle probability distribution we can compute the mortality fraction, \mu. We can even rule out certain candidate distributions this way. One could argue, though, that the underlying details are somewhat irrelevant. Only \mu matters, because the categorical probabilities are simple functions of the mortality fraction. When your card turns, you have a 100\mu% chance of drawing the black queen. Black-queendom isn’t a smoothly varying function. It’s a binary function with only two output values: Dead and Not Dead. (And despite what Miracle Max might say, for the purposes of this discussion, there is no “Mostly Dead.”)
This is practically the definition of what’s known as a “binomial distribution,” which is the kind of thing you get from flipping a coin a bunch of times. There, if the coin is fair, the heads/tails probabilities are both 50%; here, the Coin of Fate is almost certainly unfair, with Dead/Not Dead fractions of \mu and 1 – \mu. Given \mu (or given the underlying mixing angle probability distribution) the binomial distribution tells us the probability of generating N_{BQ} black queens out of N turned cards. In fact, each category has its own binomial distribution: \begin{aligned} P\left(\mu ; N_k\right) &= \frac{N!}{N_k! \left(N – N_k\right)!} \cdot P_k\left(\mu\right)^{N_k} \cdot \left[1 – P_k\left(\mu\right)\right]^{N – N_k}\end{aligned} where the category k could be black queen, joker, ace, or crypto. (Details aside, our toy model just happens to have the exact properties required for this to be true.) I’ve plotted some examples here:
The vertical line in the cryptos plot simply reiterates our earlier conclusion that a mortality fraction of exactly 0.90 translates into a 100% chance of producing no cryptos. Once again, \sigma measures the width or “blurriness” of the distribution, while the location of each peak is the mean or expectation value for the population in that category. If we ignore the blurriness for a moment and restrict our attention to those peaks (which isn’t a terrible approximation in the limit of large N), it’s easy to compute the (most-likely) categorical population percentages as a function of the mortality fraction in the Correlated Polar Model:
By now, that plot shouldn’t surprise us. Ignoring the blurriness is a minor simplification, but isn’t it nifty (and reassuring) that using the expectation values yields the result that what we, um, expect?
Perhaps unsurprisingly, the overall probability distribution controlling all categories at once is described by what’s known as a “multinomial distribution”: \begin{aligned} P\left(\mu; N_{BQ}, N_J, N_A, N_C\right) &= \frac{N!}{N_{BQ}! \, N_J! \, N_A! \, N_C!} \cdot P_{BQ}\left(\mu\right)^{N_{BQ}} \cdot P_{J}\left(\mu\right)^{N_{J}} \cdot P_{A}\left(\mu\right)^{N_{A}} \cdot P_{C}\left(\mu\right)^{N_{C}}\end{aligned} where we now have the additional constraint that the sum over all category populations must equal the total number of turned cards, N (because again, each turned card must land in exactly one category). Here the multinomial distribution is four-dimensional, thanks to our four categories for possible outcomes, making it a little tricky to visualize on a 2D screen.
However, we can use this multinomial distribution to compute the corresponding probability distribution for X, the fraction of “correctly identified” card turns (i.e., Takis-A expressions that do not produce a crypto). Previously, when playing around with the Uncorrelated Cartesian Model, we treated this as a free parameter, then looked at how everything else changed in response. But as I said above, the underlying “true” behavior of the Wild Card virus is what it is—everything springs from properties of the virus, not the other way around. So a more meaningful approach is to directly compute the statistical distribution of X values. I’ve plotted the result for three different (and very small) turned-card populations, N, and several different mortality fractions here:
For this exercise I kept the population of turned cards (N) small owing mostly to issues of speed and computational requirements. (The number of possible ways to distribute N turned cards across the four categories grows very very quickly—enumerating them all would be a combinatorial nightmare.) But notice two things. First, even when the population is very small, the expectation or mean value of X—that is, the peak of the probability distribution—zeros in on a particular value: \frac{10}{9}\mu. Second, the distributions get narrower, cleaving more tightly to that expectation value, as N grows larger. Here, the largest value is a measly 150 turned cards. When N is measured in the millions, the X distribution becomes a needle stuck at the mean value.
In fact, using angle brackets to indicate those mean values, the result is highly reassuring: \begin{aligned} x \approx \frac{\langle N_{BQ} \rangle + \langle N_J \rangle + \langle N_A \rangle}{N} \equiv P_{BQ} + P_{J} + P_{A} = \frac{10\,\mu}{9}\end{aligned} What a great result! It agrees not only with the more rigorously computed X distribution plots, above, but also with the logical requirement P_C = 1 – X. Cool. The ratio of cryptos to non-cryptos in the total population of turned cards is \begin{aligned} \frac{N_C}{N_{\sim C}} \approx \frac{P_C}{P_{\sim C}} = \frac{1 – x}{x} = \frac{9}{10\,\mu} – 1\end{aligned}
A fun implication is that if the true-yet-unknown mortality rate of the Wild Card virus is \mu < \frac{9}{20} = 0.45, then the number of cryptos in the turned-card population will exceed the combined number of black queens, aces, and jokers in the “correctly identified” population. According to this picture, Xenovirus Takis-A could be far more benign than we think, despite the empirically observed large ratios of black queens to jokers and jokers to aces.
Final(?) Thoughts from Cloud-Cuckoo Land
Okay. I know I’ve already beaten this poor dead horse practically into dust. But since I’m already homesteading in Cloud-Cuckoo Land, there’s one final observation that I simply cannot resist throwing over the transom. This will be screamingly obvious to people of a certain persuasion, and it will sound like a howl of madness to others, but it’s too juicy to ignore.
Remember those hypothetical mathematicians at the BICC? I think they would take one look at our Correlated Polar Model and say, “Gott im Himmel! Every turned card produced by the Wild Card virus is expressible as a single complex number!” (Apparently I also believe the mathematicians at BICC are all German, but whatever.) What would they mean by that?
In a sense, it’s just a more concise way to describe what we’ve already done. Any location in our turned-card parameter space can be associated with a unique complex number Z like so: \begin{aligned} Z \equiv A + \mathrm{i} \, J \equiv S \, \mathrm{e}^{\, \mathrm{i} \, \theta}\end{aligned} where i is the square root of minus one, A and J are the distances along the acedom and jokerdom axes, and S and \theta are exactly the severity and mixing angle that we’ve been using all along. This might seem like a crazy pointless thing to do, but in fact mathematicians have spent centuries building an incredibly sophisticated and potent theoretical machinery from such objects. There is no doubt in my mind that researchers studying Xenovirus Takis-A would jump at the chance to take advantage of that enormous body of pre-existing work.
So there you have it.
Overall, I think this Correlated Polar Model is a fun little toy for thinking about how the Wild Card virus changes people. But it isn’t without its own simplifying assumptions. For instance, it’s possible the crypto subtlety threshold is a function of the mixing angle, or that the joker/ace boundary location is a function of the severity. Both of those boundaries are hilariously subjective anyway.
What implications might arise from relaxing these assumptions? Or from dispensing with this model and replacing it with something entirely different? The stats experts will notice, and perhaps even roll their eyes at the fact, I’ve taken a strictly “frequentist” (not to mention overwhelmingly amateur) approach to statistics here. What would change, and what might we learn, if instead we approached this problem from a Bayesian direction? Or from mapping Takis-A expression outcomes to complex numbers?
I leave it to interested readers to explore these possibilities. Or perhaps a future blog post.
Special thanks to Deena Warner and Raya Golden for the extra effort getting this overly complicated writeup looking nice online. –IT