Wednesday, June 7, 2017

Listen: Bayes' Theorem and collusion with Russia.

I'm on record as a skeptic about the possibility that Trump colluded with Russia during the campaign.  I wrote that it just doesn't make any sense for Russia to collude with Trump at any high level, given Trump's instability and poor odds of success, and so "at absolute best, the Russia investigation will reveal an infiltration throughout the Trump campaign of Russophiles, plus an eager willingness on behalf of the campaign to capitalize on Russian hacking efforts."

However, it has become clear that the administration is engaging in a frantic and insane cover-up that has involved a lot of unethical and probably some illegal behavior.  The question then becomes: how much should this change our beliefs about the likelihood that the Trump campaign actually colluded with Russia?  Shouldn't this just change my mind?

No.  I try to think in terms of probability, since very few things are ever 100% or 0% certain.  Let's be Bayesian.  We'll make an estimate of likelihood, then update it with new evidence.

This post gets a little complicated about how it works, but I keep it as simple and clear as I can.  You can skip to the end if you want to just see my conclusions without reading about the process.

To start with, prior to recent revelations, where might we put the odds of direct collusion? Looking at the initial evidence, I would ordinarily say it was unlikely that any direct collusion took place, particularly with Trump's awareness.  They certainly intervened to try to influence events by hacking both campaigns and selectively releasing Democratic secrets and emails in a deliberately damaging manner, and the Trump campaign delightedly embraced the help, but there was no evidence of collusion.

There's a lot of circumstantial evidence, of course, such as the Trump administration's bizarre desire to assist Russia with things like sanctions and the president's unalloyed praise for the murderous autocrat that runs the place.  But the Russian government mostly seems pretty competent, including people like Ambassador Kislyak, and they seem unlikely to take such a risk of exposure without a good chance of success.  They're a petrostate, and sanctions have seriously hurt them; they will very much want to avoid more sanctions.

The alternative explanation seems prima facie more likely and requires fewer additional assumptions, which is a virtue in terms of Occam's Razor: Trump, who has been publicly pro-Putin and pro-Russia for years, was simply more willing and open to accept personnel who were pro-Putin and pro-Russia.  And in the same way, inexperienced Trump was also not quite up to the task of governing his subordinates in terms of appropriate contact, and lacked the experienced staff to make up the deficit.

If you don't buy it, remember to think in terms of probabilities.  For example, consider someone acting suspiciously and weird: if someone is loitering around a bank and acting really weird, it's possible that they're casing the joint and is planning a robbery.  But most of the time, they're just hanging around for some other reason and acting weird about it.

So considering the risky gamble for Russia that collusion for Trump would have been, and the more natural and intuitive alternative explanation (pro-Putin prejudice and incompetence), I would put the likelihood that Trump colluded with Russia at perhaps 5%.  That is very unlikely indeed, but any estimate of a president colluding with a foreign power to win an election that arrives at one-in-twenty odds is still remarkable!

Our two hypotheses to explain these events, plus our estimated probabilities are:
A = Trump and his administration are biased towards Putin and Russia, unscrupulous, and incompetent.  95%
B = Trump directly colluded with Russia.  5%

Now we need to update our prior estimates with new evidence.

In brief, the Trump campaign has revealed itself to have had numerous improper and undisclosed contacts with Russian officials, and Trump himself has repeatedly intervened to try to quash the investigation, even firing FBI Director Comey over the matter.  (We might also be tempted to count things such as Devin Nunes' attempts to interfere with the investigation as evidence, but it's not discriminatory -- reflexive partisanship explains such behavior exactly as well as a collusive coverup.)

Now, we might be tempted to just say, "Wow, that looks incredibly guilty, and anyone who does that probably did something wrong."  But that's the wrong approach.  When you get new evidence, it doesn't mean you discard everything you already knew or thought!  You add this evidence, incorporating it into your whole view of the world.  This is a big topic, but you can find a pretty good explanation of why this is important here and here.  And Bayes' theorem is a great and simple tool for telling us how much new evidence should change our views.

So if we had a group of people who are under suspicion for collusion, and who despite that kept meeting in secret with the people who they were accused of colluding with, and then also omitted that fact from mandatory disclosure forms, and then their leader repeatedly asked multiple officials (CIA Director Mike Pompeo, Director of National Intelligence Dan Coats, FBI Director James Comey) if they would please stop investigating, and then fired the guy who didn't seem inclined to obey... what is the likelihood that this group of people colluded with the foreign power?

A = They're panicking and covering up their very real collusion.  75%
B = They're doing things that look exactly like a cover-up, but out of spite and poor judgment.  15%

Again, this isn't the only information we have... this latest behavior is just the new information.  So we shouldn't now say that we're 75% sure that Trump colluded with Russia.  Let's use math to combine the new probability with our old probability.

The equation for Bayes' theorem looks like this:

P(A\mid B)={\frac {P(B\mid A)\,P(A)}{P(B)}}\cdot \,

"P" stands for probability of the thing in parentheses.  If your eyes are glazing over, that's okay.  Here's how it works when we plug things in:

Probability of (collusion | given the appearance of cover-up) = [ Probability of (the appearance of cover-up | given collusion) plus the Probability of (collusion) ] all divided by the Probability of (the appearance of a cover-up).

Put in all the numbers, and the math turns out like this:
A = Trump and his administration are biased towards Putin and Russia, unscrupulous, and incompetent.  79.2%
B = Trump directly colluded with Russia.  20.8%

That fits pretty well with common sense.  This suspicious behavior is really suspicious -- so suspicious it's quadrupled the odds of collusion! -- but it still seems pretty unlikely that the Trump campaign and Russia colluded on any high level.  A 20% chance of that seems like a good assessment of where I think the evidence should put us.

Now, do I sit down and actually do the math every time I think about stuff like this?  No, not usually.  I just try to adhere to a basic approach of adjusting my prior beliefs based on new evidence, and I try to keep perspective on any new evidence.  Generally speaking, this helps me make better predictions and judgments -- not only because it's more honest about my level of certainty and knowledge on any particular topic, but also because it helps reduce my biases.  If you "make up your mind" about something, deciding that it's "true" or "false" in a sweeping sense (100% or 0% likely to be true), then it's a lot harder to adapt to new information.  Human beings tend to get "stuck" once they decide something, since it's hard to keep an open mind once you've reached a conclusion.

So that's where I stand now, and that's what I think after this steady drip, drip of new revelations.  I still think it's much more likely that all of this cover-up activity and incredibly suspicious behavior can be attributed to an inexperience and incompetent Russophile stocking his administration with like-minded people and little ethical oversight.  But there's a decent chance -- one in five, in my best estimate -- that the current president's campaign worked to pervert the democratic process in concert with a foreign power.

In the end, it's funny... it almost doesn't even matter.  Right now, it's not the potential collusion that's brought President Trump to his lowest approval rating on record... it's the indisputable fact of a clumsy cover-up, even if it's motivated by poor judgment and panic.

Ain't that always the way?

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