Category: unfinished

Eulogy to the Living

Once upon a time, I read a friend’s anonymous letter thanking everyone who played a major role in her life. She mentioned no names and addressed no one in particular. I have a hunch that few ever read it. That I did was something between an accident and fateful randomness. This is my version of a letter to the people who mean so much to me because, why wait until we are dead?

You once told me that the closest friendships form after fighting, so I suppose it is not surprising that you are the only person I have ever actually fought. I do not know if we are closer because of it, but we are something.

You taught me that words matter. Thank you for giving me strength, courage, wisdom, and most importantly, for showing me the light, even if you yourself could not reach it.

Many people still think I am angelic, or something close to it. You know better. It is you who has seen the most of my cowardice, my anger, my weakness, my foolishness. I regret lying when I should have told the truth and telling the truth when I should have lied. I am sorry for not being better.

As my oldest friend, we have seen quite a bit of each other’s lives over the last sixteen years. I have no doubt that the decades to come will be just as exciting.

My luck turned for the better when you waltzed back into my life, and let me assure you, I take no credit for that.

I think, for a time, I worshipped you. And then when that passed, adored you as a big brother. We now walk different paths of our own choosing. I don’t know what went wrong. Maybe nothing?

Of all the people in my life, I am most delighted in you. May you become the best of us all.

You were a godfather to me, in more senses than one. Piece by piece, I have begun to see that you too have your flaws. But you are right, heroes tend to be that way.

You made my life beautiful. Despite all the sorrow and pain you have caused me, and even if I never see you again, I will never regret dancing with you all these years.

If [redacted] is my big brother, I have always thought of you as my little brother. Despite having a front-row seat to my most spectacular failures, you somehow continue to ask for my advice. I hope I do not lead you astray.

You were there for me during the two darkest periods of my life, and I will always be grateful for that. I do not think this was an accident. Without you to save me, perhaps I would have drowned.

Without you, I wouldn’t be half the crazy motherf***** I am today. We have had our disagreements over the years. But man, we bled together. And that will always mean something to me.

I have never met anyone who was as patient with me as you, and think I may not ever again. My only hope is that I repaid you at least partly with entertainment. Thanks for putting up with all my shit.

Thanks for the laughs, the late night discussions, the support when I faltered. You gave me my favorite name: a leftist in denial. Time will tell whether you are as good of a prophet as you are at ping pong.

Living with you and the others at the [redacted] was one of the most delightful experiences of my life. You were an oasis amidst my storm. Thank you for taking me into your fold.

You appeared in my life like a bolt from the blue and disappeared in almost the same way. I did not expect either, though if I was wiser, I should have expected the latter. Whatever happens, I will always be grateful for the time we had together.

unfinished

Heroes

I suppose it is telling that almost everyone on this list either lived or died violently (often both).

Martin Luther King Jr. and Malcolm X
- for fighting the darkness
章北海 (Zhang Beihai)
- for doing what was necessary
Kvothe
- for giving me the strength to escape my despair
Shusaku
- for being my first hero
Anne Frank
- for holding up a light in the darkness
Jesus
- for sacrificing himself for lesser souls
Madeline
- for conquering demons
Alexander Hamilton
- for taking his shot
Howard “Bunny” Colvin
- for striving towards righteousness, even at great personal cost
Jeremy Lin
- … for being Jeremy Lin

Predicting Chaos

Here is a proposal to predict deterministic chaotic systems, assuming they are continuous and reversible (you can “undo” the chain).

Setup

Assume your chaotic system evolves according to some function $f$ and that your initial state $a_0$ is drawn uniformly from the set of all possible states. Denote each future state $a_t = f(a_0, t)$ where $t$ can be any positive real. Since our chain is reversible, define $g$ such that $a_0 = g(f(a_0, t), -t)$ for all positive real $t$ . Assume that every unit of time (denoted by the positive integers) we measure the state of our chaotic system. Because our tools are imperfect, our measurement $x_t$ is sampled from $\mathcal{N}(a_t, \sigma^2)$ , the normal distribution centered at the true state with variance $\sigma^2$ . Assume $\sigma$ is known.

Prediction

After collecting observations from $T$ time steps, we can use our knowledge of how the system evolves and our imperfect measurements to narrow down the possibilities of what the true state is. For each possible true state $a_T$ , we compute a likelihood score based on how consistent that state would be with our measurements $x_1 \dots x_T$ . Because we $p(x_1 \cdots x_T)$ is unknown, our likelihood score results in an unnormalized energy distribution. Even so, using standard MCMC methods, we can draw (semi-correlated) samples from the distribution and use that to estimate future trajectories.

The assumption that this chain is continuous means that for any fixed time horizon $\delta$ , getting a better estimate of the true state $a_T$ means we get better at predicting $f(a_T, T+\delta)$ . The assumption that the chain is reversible allows us to calculate the probability of each value of $a_T$ as opposed to only being able to deduce the value of $a_0$ , which is much less valuable for predicting the future. Because of these two properties, we should be able to come up with a relationship along the lines of: accurately predicting $N$ timesteps into the future with $Y\%$ confidence is guaranteed if we observe at least $H(N, Y)$ previous time steps, though as [redacted] pointed out, there is a possibility that this will asymptote at some point.

Follow up: can this method predict a double pendulum?