> Throughout the last five years, we have failed repeatedly. We have mislabeled tubes and used ethanol instead of water to dilute DNA. We have dropped flasks, shattering glass and spilling bacteria on the floor. We have been frustrated, again and again. But we continued on, inching closer to mastery over the molecular wiring of genomes. In the future, we’d like to extend this work to other organisms; perhaps even humans.
This. I love the honesty about the scientific process. This is completely lost to most science students while they cram facts before an exam. The road to most beautiful discoveries is often paved with lots of fumbles.
I agree. Unfortunately, the typical undergraduate experience in biology involves a lot of rote memorization and cookbook experiments, not actual discovery. Probably filters out some people who would actually make good scientists.
It filtered me out. I spent several years in a non-science field and always had that itch I couldn't quite scratch with my day-to-day work. I left my career earlier this year to pursue a new one in science. Went back to school for the first time in a while, not cramming but actually taking the time to learn. I love it.
Definitely not an expert but there's a fundamental hurdle to this.
One way to think of this is to realize how easy it is to multiply numbers but how much more work it takes to divide numbers.
For something like automatic differentiation, you're essentially applying the chain rule for partial derivatives repeatedly. This is analytically pretty straightforward to do for most applications. All you need is an analytical derivative for the simple functions your more-complex function is comprised of (e.g. a neural network).
For integration, the analogue of the chain rule is integration by substitution [1]. The toolbox for solving integration problems is more limited than for differentiation. You run into issues where the answer cannot even be expressed using standard mathematical notation [2]. Sometimes you get lucky and the answer can be expressed via an alternating Taylor series so you can estimate the answer within some margin of error [3].
Stan is a piece of software that runs state-of-the-art MCMC methods to basically just compute fancy integrals. A Stan model will take an order of magnitude more time to run than a simple neural network via something like PyTorch on the same dataset. But they answer different questions.
Slightly tangential fact I heard recently on a podcast [0]: historically the problem of integration preceded that of differentiation. But because differentiation is so much easier, we teach that to kids first. It's one of the many ways that schools obscure the intuition of calculus and turn it into so many formulae to be memorised.
You're assuming that you want to express the integral as an elementary function, or as a member of another narrow class of functions. It's only in that case that the lack of a chain rule is a problem. What you actually want is the ability to evaluate the integral to within epsilon numerically, which is a much more flexible problem.
But then again, it's not clear what the OP meant by "automatic integration" anyway.
Can you please elaborate on "you never get the expression"?
When you input the variable values into a symbolic derivative you just get a value at the end. d/dx x^2 = 2x. If x = 0.5 then d/dx = 1. The same is true for symbolic integrals. For most practical applications, we don't really care about the full symbolic expression. We just want the answer, or at least a good approximation. This post uses a specific example of the difference between two Beta distributions. We want to get that 0.71. It is very hard to "automatically" make that happen.
Just to clarify: This result follows directly from the definition of the derivative:
f'(x) = lim_{e->0} (f(x+e) - f(x))/e
If you're able to express f(x+e) on the form (f(x) + y e) then it follows that y is the derivative.
It also should be noted that auto-diff doesn't let you skip the rules for derivation and you're using the same calculation as you would to show that e.g. f'(x^n)=n*x^{n-1}.
Closed form integration is fundamentally different.
What's the integral of exp( sqrt ( 1 + (tan^(3/2) X)2 ) ) ) ?
We only know a handful of forms that can be integrated in closed form and its down to our creativity to discover new forms that can be integrated (same deal with solving differential equations and the reasons are the same).
The forms that we know how to integrate can be done by a computer. CAS tools will do that for you. For example Mathematica.
The reason why auto-diff is exact and amazing is because (1) the derivative operation is "decomposable" through the chain rule and (2) we've found the symbolic derivative of the basic functions. There's not a mathematical trick in auto-diff which we recently discovered. It's mainly a reformulation of the properties of the derivative which turns out to be useful for computers.
Finding an "auto-integrate method" would probably involve finding a way of calculating the integral in a decomposable way, and that indeed would be amazing, but I don't really see that happening any time soon.
Are there any active studies on intermittent fasting in healthy young adults? (NY area)
I’m interested in intermittent fasting but I’d like to objectively track the effects beyond fat/weight loss. For example: ketone body blood concentration, respiratory exchange ratio, heart rate variability, etc.
I remember the frequentist approach taught in introductory stats classes never making sense to me. I didn't want to shove "stats" into the back of my brain and just focus on graduating. I genuinely wanted to understand the world a little better.
I began to research alternative approaches to modeling and conducting inference a few years ago. Discovering Bayesian Inference has had a large impact on the way I think and conduct research. There's a lot of hype and uncertainty about what "Bayesian" actually means. Here's a compact definition that I hope will attract some interest:
Bayesian Inference allows you to explicitly quantify your prior beliefs and get a more complete picture of uncertainty when modeling something.
If you'd like to learn more, the links below should be helpful.
If you're interested in spending some time learning about applied Bayesian Inference, I highly recommend Statistical Rethinking. The book doesn't assume a strong mathematical background and its filled with practical examples.
https://xcelab.net/rm/statistical-rethinking/
https://www.youtube.com/watch?v=tUX-frlNBJY&t=122s