Notes on Bayesian Analysis

This afternoon I encountered one of those rare moments when I had little to do since I was waiting on input from a number of collaborators, so I read through half of this site on intuitively understanding Bayes' Theorem. Here are some things that I learned or that were made clearer than my previous understanding:

1. The outcome of Bayesian analysis is a modification of the prior (the probability of an event that is known before the analysis) which produces the posterior (the probability of an event given certain conditions and dependent upon the prior).
2. Bayesian analysis requires three pieces of information: the prior and two conditional probabilities (a true positive and a false positive outcome).
3. If the two conditional probabilities are equal, the prior is unmodified and equals the posterior. This is because the result of the test is uncorrelated with the outcome.
4. The degree to which the prior is modified can be described by the concept of differential pressure, i.e. the relative difference between the two conditional probabilities. The process of changing the prior due to differential pressure is known as selective attrition.
5. People use spatial intuition to grasp numbers. Teachers can use this and the idea of natural frequencies to their advantage to teach difficult concepts.
6. Related to number 5, the way in which given information is presented in a word problem (e.g. percentages vs. ratios) will affect the percentage of correct scores.

Is Bayesian analysis related to Kant's a priori and a posteriori knowledge?

A day in the life

It's almost 5:00 PM on a Friday, which means I'm tired and don't want to do anymore work. While staring blankly at my computer screen for ten minutes waiting for happy hour and the beer that awaits, I reflected upon my day and what I accomplished. During this time of meditation, it struck me that my day, a fairly typical day in my grad student life, might be a bit unusual. Here's my summary:

8:00 AM: Get into office after working out. Check e-mail/Facebook. So far so good.
8:30 AM: Get first cup of coffee.
8:35 AM: Begin 12 page derivation of expression for decorrelation times of a time series of images of microparticles undergoing anisotropic Brownian motion.
9:30 AM: Get second cup of coffee. The caffeine's really producing some crazy math now.
10:10 AM: Finish derivation. Try and place all the scratch paper strewn about the office in order. Fail at this.
10:20 AM: Tweak Matlab code for simulating anisotropic Brownian motion. Become upset when a function I want to use is in the Spline Toolbox. We don't have a license for the Spline Toolbox.
10:30 AM: Get bored of this. Get more coffee.
10:35 AM: Read journal article on causality and the Kramers-Kronig relations. Spend too much time wondering why differential equations with higher order time derivatives of the force on a system than the system response are not causal. (P. Kinsler, "How to be causal," arXiv:1106.1792v1 (2011)
11:45 AM: Eat lunch.
12:15 PM: Check Facebook. Join the collective graduate student world in celebrating the announcement of the PhD movie.
12:30 PM: Build a setup using Ni:Chrome wire, a current regulated supply, and spare lab parts for cutting slots into plastic Petri dishes.
1:00 PM: Cut slots into the sides of the Petri dishes. Try not to breath the fumes from the molten plastic.
2:00 PM: Glue fibers that were previously tapered with a CO2 laser into the slots with a silicone-based glue that is normally used for gluing colostomy bags to people.
2:30 PM: Become dizzy from the plastic/glue fumes. Take a walk outside.
3:00 PM: Go to biology and observe HeLa cells growing in a petri dish on a quartz surface I prepared last week. Take minor awe in noting that this exact cell line can be traced to a woman who died in 1951.
3:15 PM: Debate with a biology professor why structured networks can induce stress in the actin filaments of the cytoskeleton and why ex vivo observations can facilitate in vivo understandings of cell motility.
3:30 PM: Return to office, which now smells like colostomy bag glue.
3:50 PM: Read notes that came with glue. Make note that isopropyl alcohol-the very chemical I use to sterilize dishes-will dissolve the glue and has rendered this work useless. State an expletive rather loudly at this finding.
3:55 PM: Go to lab and play with the alignment of my imaging fiber setup.
4:05 PM: Try and find the other imaging fiber (needed to be prepared by Monday for the biology professor) but realize my groupmate took it and put it in his own setup.
4:10 PM: Attempt to contact groupmate. He left for the day already and is not answering his cell phone. State another expletive at the realization that I will have to come in on the weekend to prepare the fiber.
4:20 PM: Return to lab. Note that the Ti:Sapphire laser head cooling system reports an error due to lack of cooling water. Check water tank. It's full.
4:30 PM: Return to office. Note giant glob of colostomy bag glue on desk. Luckily it's removed with isopropyl alcohol.
4:35 PM: Summarize life.
5:15 PM: Go drink beer.

Random thought on random walks

I've been working a lot lately with both models and experiments concerning Brownian motion of microparticles in suspension and had a plot of particle trajectories that underwent a 2D random walk up on my computer screen. My officemate commented that it looked like the scribbles that her niece does with crayons and it got me thinking: can the scribble trajectories of children be modeled as some sort of random walk? I imagine it'll likely be anisotropic since they seem to scribble either primarily up-and-down or left-and-right. But it would be an interesting sociological study to try and correlate the statistics of the scribbles (fractal dimension, diffusion coefficient, etc.) with some societal factor.

Yes, this would likely be frivolous science but I'd find it fun anyway =)

The coffee-spilling uncertainty principle

If, when carrying your cup of coffee back from the coffee machine to your office, you hold the full cup close to you, you're less likely to spill any over the sides of the cup, but if you do spill, it is more likely to spill on your clothes. However, if you hold it further out from you, the coffee is much more likely to spill over the side of the cup, but much less likely to spill on your clothes.

The uncertainty in spilling your coffee multiplied by the uncertainty in spilling your coffee on you is always greater than Planck's constant times the amount of time spent at work.

When is too much data necessary?

Ben Goldacre wrote an interesting article last week about research findings that never receive the attention or scrutiny of the academic arena or the public. His point was a bit blurred by the article's strange introduction, but I believe that he was trying to say that our prejudices selectively filter results that we find interesting. This in turn leads to "walled gardens" of knowledge that can negatively impact not just ourselves but society.

However, without any type of filtering I suspect that the amount of information that needs to be processed and weighed is simply too much to handle. Goldacre alludes to this in the concluding paragraph.

The most interesting questions aren’t around individual nuggets of data, but rather how we can corral it to create an information architecture which serves up the whole picture.

What is a good information architecture for society to adopt? Clearly a single individual can not process everything, so the task must be performed by a complex body of professionals and academics.