Martin’s Blog

Some slides and a movie

Talks I gave last semester:

- A talk about LDPC codes. This explains why I was making those perplexing pictures a few posts down.
- A talk about using a neural network to solve the 8 queens problem. These might not make a whole bunch of sense as the talk was for a course so requires background and when I gave the talk my network didn’t actually work. (Since then I managed to make it pretty much work so I’ve changed the ending of the talk).
- A movie of my neural network converging to a solution. It’s mp4 format which I can play in quicktime or VLC so hopefully you can play it in something*. The exciting swirly part is slightly before halfway.

*It’s unlikely you guys will be able to help me with this but do you know a good way to do video on a mac? I get a 430mb uncompressed avi file out of matlab and I used QTamateur then iSquint to compress it down to 600k. I’m sure there’s a better way to do this.

Animated gifs

You might say that I could have spent my time better but after a few hours of work I was able to make this rather inscrutable picture. Any guesses as to what it’s doing?

Quiz answer & discussion

1/3 is ‘correct’ by exactly Dave’s solution. I was going to provide a more detailed and motivated exposition but I don’t think I can be bothered. My knowledge of Bayesian inference comes exclusively from reading the first half of section two of McKay’s textbook Information Theory, Inference and Learning Algorithms so you can read that and know as much as I do.

So, points for Dave for doing what the book did, Yuliya for getting approximately the right answer first and Leesa for having both the worst and possibly the best answer.

The answer of 0.3 is clearly terrible: if you are a frequentist that doesn’t believe in probability representing belief then your real answer should be that the 10 flips are not at all significant.

But the 0.3485 might be pretty good. What was your prior Leesa? I can certainly believe that a prior that is normal with mean 0.5 and some smallish standard deviation is perhaps more reasonable than a uniform prior and that should give us an answer above 1/3.

A quiz

You toss a (not necessarily fair) coin 10 times and get three heads. What is the probability of getting a head on the next toss?

Guesses and answers with some calculations are both acceptable.

The argument of a complex number

The argument of a complex number is defined to be a number which represents the angle of the vector corresponding to the number measured from the positive real axis. If you draw a picture of this setup your first thought is that . Unfortunately this can't work because the range of is . It is true that but ever since I learnt about complex numbers 8 years ago or so I just knew that you had to draw the picture and think about it to solve this equation for .

But just a few days ago I discovered* that there does exist a formula! (well, it can’t give you an answer of but apart from that it works). It is the rather mysterious statement . Wikipedia says that it’s related to half angle trigonometry but looking at the formula you should also be able to get it from a picture of a circle.

*reading something on Wikipedia is a form of discovery, right?

So here’s one way to come up with it: Draw the point in the complex plane and then draw the circle centred at the origin that goes through the point. Now consider the following diagram.

The angle at the boundary of the circle is half of that at the centre (that’s a theorem from circle geometry but it’s pretty easy to see from the picture in this case). The radius of the circle is so you have and this time you can invert with the standard function.

This seems quite weird to me: that this function works but the other doesn’t. It’s a stereographic projection I guess and is a better coordinate than …

This formula isn’t actually that useful but it was just what I wanted to find to do a question from my complex analysis homework 0, so it was helpful to someone somewhere.