Alfred Harwood and I were working through this as part of a Dovetail project and unfortunately I think we’ve found a mistake. The Taylor expansion in Step 2 has the 3rd order term o(δ3)=16[2(√P[X])3](−δ[X])3. This term should disappear as δ[X] goes to zero, but this is only true if √P[X] stays constant. The Γ transformation in Part 1 reduces (most terms of) P[X] and Q[X] at the same rate, so √P[X] decreases at the same rate as δ[X]. So the 2nd order approximation isn’t valid. For example, we could consider two binary random variables with probability distributions P(x=0)=zp and P(X=1)=1−zp and Q(X=0)=zq and Q(X=1)=1−zq. If δ[X]=√P(X)−√Q(X), then δ[X]→0 as z→0. But consider the third order term for X=0 which is 13(√Q(0)−√P(0)√P(0))3=13(√zq−√zp√zp)3=13(√q−√p√p)3 This is a constant term which does not vanish as z→0. We found a counterexample to the whole theorem (which is what led to us finding this mistake), which has KL(X2→X1→Λ′)max[KL(X1→X2→Λ),KL(X2→X1→Λ)]>10, and it can be found in this colab. There are some stronger counterexamples at the bottom as well. We used sympy because we were getting occasional floating point errors with numpy. Sorry to bring bad news! We’re going to keep working on this over the next 7 weeks, so hopefully we’ll find a way to prove a looser bound. Please let us know if you find one before us!

Things I'd like to know: 1. What is the baseline things-going-tits-up-mentally rate for people similarly situated to those who take on meditation, and how does that compare to the rate for people who begin to meditate? There's a smell of "the plural of anecdote is data" about this post. People at high risk for mental illness can go around the bend while meditating? Well, they can go around the bend watching TV or chatting with Claude too. How much more dangerous is meditation than the typical range of alternative activities? 2. There's bad, and then there's Bad. Ask a reformed alcoholic whether there were any negative side effects of giving up alcohol, and they'll tell you it was a bit like an anal probe of the soul with the devil's pitchfork for a month or two at least. Is that a cautionary tale that should steer you away from sobriety, or just par for that otherwise worthy course? Some of the practitioners of meditation (esp. in the Buddhist tradition) think we're most of us addicts to the chimerical delights of the senses, and that of course it'll be a struggle to overcome that. FWIW. 3. Is this like psychedelics, where if you take them in the context of a long-standing ritual practice with lots of baked-in wisdom, things will probably go okay or at least they'll know how to give you a soft pillow to land on if you get too far out there; but if you take them in some arbitrary context there's no telling how it'll turn out? How do outcomes look for people who meditate in an institutional context with feedback from a seasoned veteran vs. those who meditate based on e.g. enthusiastic blog posts? Not saying you're wrong, but answers to things like this would help me know what to do with your observations.

I think that properly understanding the psychology of BDSM might provide the key to understanding psychology in general (in ways that are pretty continuous with the insights of early pioneers of psychology, e.g. Freud and particularly Jung). My current model is: * The process of learning to be "good" typically involves renouncing and suppressing your "antisocial" desires, some of which are biologically ingrained (e.g. many aspects of male aggression) and some of which are learned idiosyncratically (e.g. having a traumatic childhood which teaches you that the world is zero-sum and you can only gain by hurting others). It also involves renouncing and suppressing parts of yourself which are "pathetic" or "weak" (e.g. the desire to not have to make any choices, the belief that you are bad and unworthy of existing). * These desires/beliefs aren't removed from your psyche (since internal subagents have strong survival instincts, making it difficult to fully destroy them) but rather coagulate into a "shadow": a coalition of drives and desires which mostly remains hidden from your conscious thinking, but still influences your behavior in various ways. The influence of your shadow on your behavior is typically hard for you to detect yourself, but often easy for (emotionally intelligent) others to detect in you. * People who have a very strong "will-to-Goodness" don't necessarily have very strong/extreme shadows, but often do, because they created the very strong will-to-Goodness by strongly suppressing their antisocial desires, which then strongly polarized those desires. * Many types of BDSM are a fairly straightforward manifestation of the desires in your shadow. Participating in BDSM can be good for one's psyche in the sense that it represents a partial reconciliation with one's shadow, reducing internal conflict. I.e. rather than having a shadow that's fully repressed, you can have a "bargain" between your ego and your shadow that's something like "the ego is (mostly) in charge almost all the time, while the shadow is (mostly) in charge during kinky sex". It feels really somatically nice for parts of your psyche which are almost always repressed and shamed to be allowed to act for once. * However, BDSM can also be bad for one's psyche in the sense that positive reinforcement during BDSM causes your shadow to grow, thereby increasing internal conflict longer-term. Also, doing BDSM with others can cause their shadow to grow too. "Healthy" BDSM probably looks more like an outlet which gradually helps you to accept and integrate your shadow then move on, rather than a lifestyle or a part of your long-term identity. My guess is that BDSM communities end up instantiating similar "crab in a bucket" dynamics as incel communities—i.e. holding people back from developing healthier psychologies. * Young children are rightly horrified by BDSM when they stumble upon it, because it's an indication that there's something twisted/perverse going on in the world. However, I suspect that almost all adults who feel horrified by BDSM are in part reacting to their own shadow. My guess is that the few people who have actually integrated their shadows in a healthy way are neither very interested in nor very horrified by BDSM, but rather mostly sad about it (like they're sad about suffering more generally). When I say that they've "integrated" their shadows, I mean that their BDSM-like desires are cooperating strongly enough with their other desires that they're a little bit present most of the time, rather than driving them to create simulacra of highly transgressive behavior. This might sound scary, but I expect that fully experiencing the ways in which we all have power over each other in normal life provides enough fodder to satisfy the BDSM-like desires in almost all of us. (For example, if you really allowed yourself to internalize how much power being a westerner gives you over people in developing countries, or the power dynamics in friendships where one person is more successful than the other, I expect that thought process to feel kinda like BDSM.) * Trying to evoke and deal with your shadow is a difficult and fraught process, since (by definition) it involves grappling with the parts of yourself that you're most ashamed about and most scared of giving control to. I recommend doing so gradually and carefully. My most direct engagement with shadow work was regrettably intense (analogous to a bad psychedelic trip) and came very close to having very bad effects on my life (though I've now wrestled those effects into a positive direction, and find shadow work very valuable on an ongoing basis). * As you can probably infer, most of the points above are informed by my own past and ongoing experiences.