People always want an explanation of Friston’s Free Energy that doesn’t have any maths. This is quite a challenge, but I hope I have managed to produce something comprehensible.
This is basically a summary of Friston’s Entropy paper (available here). A friend of jellymatter was instrumental in its production, and for this reason I am fairly confident that my summary is going in the right direction, even if I have not emphasised exactly the same things as Friston.
I’ve made a point of writing this without any maths, and I have highlighted what I consider to be the main assumptions of the paper and maked them with a P.
A while ago I wrote a little rant on the (mis)interpretation of P-values. I’d like to return to this subject having investigated a little more. First, this post, I’m going to point to an interesting little subtlety pointed out by Fisher that I hadn’t thought about before, in the second post, I will argue why P-values aren’t as bad as they are sometimes made out to be.
So, last time, I stressed the point that you can’t interpret a P-value as a probability or frequency of anything, unless you say “given that the null hypothesis is true”. Most misinterpretations, e.g. “the probability that you would accept the null hypothesis if you tried the experiment again”, make this error. But there is one common interpretation that is less obviously false: “A P-value is the probability that the data would deviate as or more strongly from the null hypothesis in another experiment, than they did in the current experiment, given that the null hypothesis is true”. This is something that you might think is a more careful statement, but the problem is that in fact when we calculate P values we take into account aspects of the data not necessarily related to how strongly they deviate from the prediction of the null hypothesis. This could be misleading, so we’ll build it up more precisely in this post.
I’ve been reading this article in the independent: “Proofs of God in a photon”. The article is ultimately about some anthropic principle stuff. But the comments are full of silly things that make reluctant to call myself a scientist in case I am associated with the authors. So, as therapy, I shall call a number of the commenters on their bullshit. First, a well meaning guy called Dan,
The latest Jellymatter poll has been up for a while now, time to discuss what the correct solution is. As well as sounding like a question from a Voight-Kampff test, it is a “double trick question”, based on the Monty Hall problem. It was a little mean of me to post it with my own agenda in mind.
For me, the interesting thing about the Monty Hall problem is vehemency of those who argue for “switch”, option. The argument is nearly always unjustified. Whilst arguing this I will talk about how the problem has been stated in the past: It’s history shows how quickly someones brief, informal argument can change into an unintuitive answer to a ill-posed question and then into a dogmatic belief.
You’re walking down a back alley and find a man with the archetypal three cups and pea. You decide to gamble with him in a game of ‘guess where the pea is’; after all the odds are reasonable and he has assured you that he will demonstrate that at the pea is under one of the cups. He places the pea under one of the cups and shuffles them rapidly and you choose one of the cups. At this point the man overturns one of the cups you did not choose – there is no pea underneath it. He then asks you whether you would like to choose the other upright cup instead…
I came across a nice paper called “The Null Ritual” by Gerd Gigerenzer et al recently, it’s an excellent read in my opinion, and sums up a lot of the things that are wrong with null-hypothesis testing. This process is pervasive in many areas of science, particularly psychology (which is what these authors are mainly talking about), and it’s flawed in too many ways to count. Gigerenzer’s paper is worth reading, I’m going to attempt to summarise it, focussing on the things that really bother me. A good point that this paper makes is that its not actually the test itself that is intrinsically “wrong”, it’s more to do with the way that it has permeated in scientific culture. This is what they are calling the “Null Ritual” – the process of more or less automatically doing a null hypothesis test, without there necessarily being a good reason to, except perhaps that some journals or reviewers seem to require it. Before reading this, you might want to try filling in the Jellymatter poll on the subject (which is taken from “The Null Ritual”), before I discuss the correct answer below.
Pretty much everyone (including me) thinks that the world is round, but very rarely is it obvious that this is the case: In our every day lives the world is for all intents and purposes flat. It strikes me as a little absurd that thinking that world is flat is taken to be so incredibly ignorant, when there is so little direct sensory evidence for it. If you haven’t gone to the sea and observed ships crossing the horizon, or performed some experiment to demonstrate it, thinking the world is round is really only taking other peoples word for it. The roundness of the earth is almost completely detached from what we experience and the fact that most of us have this knowledge is a real tribute to our ability for abstract thought.
The intangibility of the roundness of the earth is an excellent example for my ongoing campaign to make people aware of just how far our understanding of the world is from our direct experience of it. Considering the world at the level of our physical interactions with it seems to be ignored by our post-enlightenment mindset – I think it needs setting straight. And the best way of doing this is to represent familiar but abstract concepts in ways that relate to our basic interactions with the world (ecologically, to borrow J. J. Gibson‘s term)
This is my first demonstration of how unnatural thinking the world is round is: I will calculate how big a circle I need to walk in to decide with confidence that the world is round. We all have an intuitive grasp of how far it is to walk somewhere, so the question is: how affected should this understanding be by the curvature of the earth. This is based on the principle that circles are “smaller” on a sphere than they are on a flat surface, reflecting the formal definition of curvature (almost) exactly.
Out in the pacific there are two islands named Foo and Bar. Two ferries, the good ship Fizz and the good ship Buzz pass between them once per day each (that is, if Fizz starts the day on Foo it will end the day on Bar). For each ship then, we can write out the list of islands it ends the day on using the initials: F for Foo and B for Bar:
Fizz: F B F B F B F B F B F B F B F B
Buzz: B F B F B F B F B F B F B F B F
However, on Foo island, Fry is currently searching for his friend Bender. But Bender is on Bar. Fry learns of this and resolves to hop aboard the good ship Buzz and be reunited with Bender in the evening. But alas! Bender has similarly reasoned that Fry is on Foo and therefore set sail with the Fizz. Having passed each other during the day, Fry is stranded on Bar and Bender on Foo, they will have to wait until tomorrow before they can do anything about it.
In fact, what Gould has mistaken for “reification” is neither more nor less than the common practice in every science of hypothesizing explanatory models or theories to account for the observed relationships within a given domain. Well-known examples include the heliocentric theory of planetary motion, the Bohr atom, the electromagnetic field, the kinetic theory of gases, gravitation, quarks, Mendelian genes, mass, velocity, and so forth. None of these constructs exists as a palpable entity occupying physical space.
I’ve been thinking more about this idea of reification, that I brought up in my last post. I was originally going to respond to the slightly confusing discussion that got going about it, but I didn’t want to hijack a thread by going on about Spearman’s g again.
So as I understand it one argument against this reification idea, is that everything in science is “reified”. In a way, if you are willing to go to slightly absurd sounding extremes, your concept of there being a coffee cup in front of you may be a reification, because you can’t prove a coffee cup is physically there just from the photons hitting your eyes. Arthur Jensen’s reply to Gould, which I’ve quoted above, sort of makes this point. Spearman said that g may be the result of a “mental energy”, but according to Jensen this is just a scientific hypothesis, and therefore valid. The heliocentric model of the solar system that most of us accept as pretty basic science could also be said to be a reification. Even if the planets are physically there, the model of the planets is no more a physical thing than Spearman’s g.
When I try to think about this problem I have to admit that the whole reification notion is a bit confusing if you try and get philosophical about it. I think essentially it comes down to the fact that actual scientific hypotheses make testable predictions, which after some time (in human history) get investigated and a consistent theory gets worked out. The hypothesis that “Spearman’s g is a result of some kind of energy in your brain” doesn’t, which is what makes it so silly.