[Content Warning: TERFs, high-level discussion of violence and sexual assault, math.]

TERFs are obsessed with violence. Their core talking point revolves around the gender disparity between sexual assault perpetration, which they somehow think applies to transgender women. It doesn’t matter that even if their assertions were correct, that would be insufficient justification for their policies; nor does it matter that this assertion deeply conflicts with Gender Critical claims about gender; nor does it matter that there’s no evidence transgender people are more violent. Logic and evidence are subservient to vibes and propaganda. Keep saying the libel, loudly and forcefully, and eventually the people on the fence start believing it anyway.

You’ll never sway the fervent true believers with logic and evidence, so I’m more interested it that mushy middle. Nevermind the intent behind or consequences of the claim, I can see people being curious about the evidence. To be specific, do transgender women show patterns of violence more characteristic of cis women, or of cis men?

The US Secret Service just released an analysis of mass attacks in the USA, and it contains the first raw numbers on transgender perpetrators I’ve seen. This is relevant to the question at hand, as there’s evidence for a correlation between committing violence and committing sexual assault, so we can use it as a proxy measure.

The TERF viewpoint is that transgender women exhibit patterns of violence characteristic of cisgender men; this implies that trans women would commit mass shootings at a similar rate to cis men. I’ll take the opposite claim, that trans women are women and thus show similar patterns. Evaluating this is quite easy: every time a trans woman commits a mass attack, that’s one unit of evidence towards them being like cis men and one unit away from them being like cisgender women.

This is an awful lot like a Bernoulli process with an unknown probability, isn’t it? The math for this has been worked out ages ago, and I’ve even invoked it myself. If we only have observations, our belief about that underlying probability is expressed by the Beta distribution, which happens to be its own conjugate prior when paired with the Binomial distribution as a likelihood function. If the shape of the prior Beta distribution is represented by $\alpha$ and $\beta$, then given $k$ expected observations from $n$ total our posterior is defined by a Beta distribution with ${\alpha }_{\text{pos}}=\alpha +k,{\beta }_{\text{pos}}=\beta +n–k$.

This is all terribly binary, so how would we incorporate non-binary people? Technically, I already have. The Beta distribution is also a conjugate prior for the Multinomial distribution, so with $k$ expected observations for category $X$ out of $n$ total our posterior is a Beta defined by ${\alpha }_{\text{pos}}={\alpha }_X+k,{\beta }_{\text{pos}}={\beta }_X+n–k$, where ${\alpha }_X$ and ${\beta }_X$ are the Beta distribution parameters of the prior for category $X$. Since the Secret Service didn’t include any info on non-binary attackers, I’ll go back to the binary for now.

What sort of prior should we use? The easiest approach is to plug the number of cisgender men ($cm$) and cis women ($cw$) into the Beta directly, but I prefer the diachronic approach. Suppose we are aliens freshly landed on this planet. Given no evidence of gun violence or mass shootings, how likely would you think men are to commit either, as opposed to women? Since guns can be any size and are simple enough for small children to operate, you’d figure there would be equal numbers; but being an alien, you’d be pretty open to having your mind changed on the topic. I’d say that’s best represented by ${\alpha }_{\text{prior}}={\beta }_{\text{prior}}=2$. I can then plug in the Secret Services’ actual data on the gender of who perpetrated mass violence and create a Beta posterior; I then declare that to be my prior on transgender patterns of violence, and plug those numbers in. If you’re not a fan of my ur-prior, don’t worry, I’ll leave enough room in the math to use your own prior.

Mathematically, my hypothesis can be expressed as

$$H(\text{mine}|tm,tw)\propto \text{Binomial}(tm|tm+tw,p)\text{Beta}(p|{\alpha }_{\text{prior}}+cm,{\beta }_{\text{prior}}+cw)$$

where $tm$ and $tw$ are the totals for transgender men and trans women, respectively. The TERF version is

$$H(\text{TERF}|tm,tw)\propto \text{Binomial}(tw|tm+tw,p)\text{Beta}(p|{\alpha }_{\text{prior}}+cm,{\beta }_{\text{prior}}+cw)$$

We can greatly simplify the math by invoking the Bayes factor form and dividing the former by the latter, resulting in a lot of cancellations. We still have a nuisance parameter to worry about, $p$ or in this case the probability of a man pulling the trigger in a mass casualty event. The Bayesian way to handle that is to integrate it out.

$$\frac{H(\text{mine}|tm,tw)}{H(\text{TERF}|tm,tw)}=\frac{{\int }_{p=0}^1{p}^{{\alpha }_{\text{prior}}+cm+tm–1}(1-p{)}^{{\beta }_{\text{prior}}+cw+tw–1}}{{\int }_{p=0}^1{p}^{{\alpha }_{\text{prior}}+cm+tw–1}(1-p{)}^{{\beta }_{\text{prior}}+cw+tm–1}}$$

As before, we don’t really need to calculate those integrals, we can swap them out for a set of Gamma functions and do more cancellations.

$$\frac{H(\text{mine}|tm,tw)}{H(\text{TERF}|tm,tw)}=\frac{\mathrm{\Gamma }({\alpha }_{\text{prior}}+cm+tm)\mathrm{\Gamma }({\beta }_{\text{prior}}+cw+tw)}{\mathrm{\Gamma }({\alpha }_{\text{prior}}+cm+tw)\mathrm{\Gamma }({\beta }_{\text{prior}}+cw+tm)}$$

If we’re smart and ensured ${\alpha }_{\text{prior}}\in \mathbb{N}$ as well as ${\beta }_{\text{prior}}\in \mathbb{N}$, then we can simplify further by swapping out the Gamma functions for factorials. Now all we need are the raw numbers, but here’s where I bring some bad news.

Consistent with previous Secret Service analyses of mass attacks, nearly all the 180 attackers (n = 172, 96%) in the study were male. Three attackers were transgender, assigned female at birth but known to identify as male at the time of their attacks. The remaining five attackers were female.

We did all that math crunching, just so we could feed in three datapoints? That seems like a waste, especially since Bayesian stats tends to be more sluggish than frequentist stats. Ah well, we’ve come this far…

$$\begin{array}{rl}\frac{H(\text{mine}|tm,tw)}{H(\text{TERF}|tm,tw)}& =\frac{\mathrm{\Gamma }(2+172+3)\mathrm{\Gamma }(2+5+0)}{\mathrm{\Gamma }(2+172+0)\mathrm{\Gamma }(2+5+3)}\\ & =\frac{176!\cdot 6!}{173!\cdot 9!}=\frac{176\cdot 175\cdot 174}{9\cdot 8\cdot 7}\\ & =\frac{31,900}{3}\approx 10,633\end{array}$$

Surprise! I had no idea the evidence would be so overwhelming until I was cancelling out factorials. And I do mean overwhelming, Bayes factors above 150 are considered extremely strong evidence in favour of one hypothesis. So what gives, didn’t I say Bayesian statistics was sluggish?

It is! Let’s tackle this from the frequentist side of things. Ronald Fisher has stated that given $a$ occurrences over $a+b$ trials, the probability of the next occurrence, well, occurring, was $\frac{a+\frac{1}{2}}{a+b+1}$ given no other information. This implies the odds of a man starting the next mass attack is $\frac{345}{356}\approx 0.97$, and thus (excluding non-binary people) the odds for a woman doing the same are about 0.03. My theory argues that transgender men are men, thus the odds of three of them committing mass shootings is the odds of a man doing so raised to the third power; a TERF would argue trans men are women, and raise that value to the third power. Putting that in odds ratio form, we get

$${\left(\frac{345}{356–345}\right)}^3=\frac{41,063,625}{1331}\approx 30,850,$$

which, as promised, is more extreme. But what about that prior I imposed? Surely I gamed the results a bit by using the subjective prior ${\alpha }_{\text{prior}}={\beta }_{\text{prior}}=2$; how about we swap in the “objective” Bayes/Laplace one instead?

$$\begin{array}{rl}\frac{H(\text{mine}|tm,tw)}{H(\text{TERF}|tm,tw)}& =\frac{175\cdot 174\cdot 173}{8\cdot 7\cdot 6}\\ & =\frac{125,425}{8}\approx 15,678\end{array}$$

I did indeed game the results by my choice of prior, in favour of TERFs. Ah, you say, but what about sex ratios? There was a big meta-analysis that suggested transgender women outnumbered trans men by a 2 to 1 ratio, but more recent reports suggest the gender difference is disappearing. Even if we go with that 2 to 1 figure, we’d compensate for the imbalance by weighting each shooting by a transgender man as worth twice that of a transgender women; here, that means multiplying the above ratios by $2^3=8$, tipping things even more in my favour!

There are three ways to wiggle out of the math: outright reject it, concede that trans women are women…. or discard trans men. Think about it: 0.03 is below 0.05, science’s most common threshold for declaring your findings to be “significant” and thus worthy of consideration. One transgender person going on a rampage hits that threshold, and if that person happens to be a transgender woman then it’s game-set-match for the TERF hypothesis. The catch, of course, is that TERFs are forced to fudge the math by finding ways to dismiss or ignore transgender men.

I don’t doubt that some transphobes behave this way towards trans men out of genuine concern and even a strange sense of love – but that doesn’t make it right. Whatever the intentions behind it, the aim is still to make us cis, to convert us “back” into womanhood – to erase our trans staus. Whichever way you look at it, they’re trying to remove another trans person from the world. …

There is an idea, in transphobic circles, that trans men are all confused butch lesbians. How this works varies from circle to circle. Some think we are “self-hating butch lesbians”, who simply need to be reassured that it’s okay to be a butch lesbian (which, of course it is! Butch lesbians are amazing! It’s just not for me – because I’m a man), and that if we could “accept” that we’re butch lesbians we would somehow “revert” back to being a cis person. On the other hand, some think we “would have been butch lesbians” but were forced into being trans, either by our parents, our friends, or some mysterious transgender cabal (I don’t think I need to point out that this is a ludicrous idea – nobody is forcing cis people to become trans, this rhetoric is a ridiculous attempt at fearmongering using the exact blueprints of the historic Gay Scare).

It’s a rare treat when you can use math to help explain bigotry.