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Go to my personal blog if you want more detailed illustrations and tutorials. I do not update the theoretical reviews in this repository any more. I still update various references such as solutions of interesting problems, proof of useful relations or equations, IPython notebooks, Python codes, and Mathematica notebooks.
During I study machine learning and python coding for it, the biggest trouble for me is that it's hard to find the python code for the core of the algorithm. It is easy to look for completed package, but it's never been so helpful for understanding how the real algorithm works.
This repository contains the materials made by myself in order to help myself. Most essential theories and algorithms are given by a graduate level-book Thus. the contents here will be much more helpful for you with David Mackay's book.
Especially, I followed Mackay's roadmap he introduced at the first of the book. Inference, Bayesian theory, maximum likelihood and clustering, various Monter carlo method, neural network will be topics.
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Metropolis-Hastings sampling Monte Carlo method ipython notebook and figure 29.12
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Slice sampling Monte Carlo method ipython notebook and exercise 29.10
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Hamiltonian Monte Carlo method ipython notebook and figure 30.2
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Gibbs sampling Monte Carlo method and over-relaxation and figure 30.3
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Rectangular Ising model and Monte Carlo method, and figure 31.2 and figure 31.3, ipython notebook
Will be updated soon.
Here I display the figures I gained from the above python codes and mathematica notebooks. I also solved some of challenging exercises in the book, which can be found in the LateX file.
ipython notebook for clustering
I felt there's no good python tutorial for spectral clustering. The theory for clustering and soft k-means can be found at the book of David Mackay. Especially I have read chapter 20 ~ 22 and used the algorithm in the book to obtain the following figures.
Here are the data points. I chose random 50 data from the two boxes with 0.5 lengthy edges. We know the proper K-number for K-means clustering is 2. However, when we set K>2, the K-means clustering gives a ugly result. Besides, the clustering strongly depends on the initial choice of the 4 assignment points.
If we set the stiffness of the soft clustering well, we can obtain the following figure.The python code for above 2 plots is here
It still looks like 4 colored clusters, but the circles show the 4 clusters reduced to 2 clusters. the size of the circle is decided by the stiffness. The python code is here and here
The enhanced soft K-means allow each cluster to have independent size of the circle. In the figure, two circles vanish and 2 clear clusters remain.
The radii of the two main circles are about 0.142(red) and 0.149(yellow). The standard deviations of the two clusters are about 0.02 and 0.02. My soft K-means clustering alogrithm gaussian distribution assume Gaussian distribution. And I chose uniformly random data points. The python code is here and here
If the data are aligned, the circular deviation is not so good. We can split the deviations about each axis. Should be careful of overfitting. It is a weak point of this clustering. The code for above two results is here
The python code is here
This exercise is an extension of exercise 22.15. I think exercises 22.15 and 24.3 are very good examples to better understanding Bayesian theory. The disscusion for both exercises are completed in the LateX file.
Posterior probability distribution function from given data. The distribution functions are not normalized yet. However, this plot already shows the maximum likelihood should be around mu=10. The small bumps around Xn=(-27, 3.6, 8) are also seen in the graph.
The mathematica analysis is found here
The mathematica analyis is here
The mathematica analysis is here
ipython notebook for reject sampling Monte Carlo method
The mathematica analysis is here
The python code is here
What I understand about importance sampling is, when we numerically integrate some function, importance sampling can reduce the cost. The detail can be found in the next link.
A practical difficulty with importance sampling is that it is hard to estimate how reliable the estimator is.
ipython notebook for important sampling
Metropolis method for a toy problem, reproduced figure 29.12, ipython notebook
Histogram of occupancy of the states after 100, 400, and 1200 iterations from Metropolis-Hastings sampling
Histogram of occupancy of the states after 100, 400, and 1200 iterations from independent sampling
Markov chain Monte Carlo sampling, reproduced the figure 29.14, Mathematica notebook
Transition matrix
The probability densities
Slice sampling, exercise 29.10, ipython notebook notebook
The exercise 29.10 is to find the turns from the peak to tail and from tail to peak.
The probability density is given as the follwoing.
Using slice sampling (the code is found in the ipython notebook link), the number of turns it takes from the peak to tail is as follows.
The average of the number of turns is about 5.2362.
On the other hand, the average of the number of the turns from tail to peak is about 5.2967 from similar analysis.
They are quite close, and it makes sense because the area of the peak and the tail in the density function are matched.
Hamiltonian Monte Carlo method, figure 30.2, ipython notebook
One of the weak points of Monte Carlo sampling comes up with random walks. Hamiltonial Monte Carlo method (HMC) is an approach to reduce the randomzing in algorithm of the sampling.
When we regard the energy of the system in probability distribution as potential energy term, the dynamical energy, the sum of the kinetic energy and the potential energy restrains the system. In other words, the dynamical energy conservation reduce the degree of the freedom of the system. And the phase space (x(t), p(t)) followed by Hamiltonian mechanism. As a result, the samples are to be chosen less randomly than Metropolis-Hastings sampling.
The algorithm of Hamiltonian Monte Carlo sampling is as follows.
Octave source code for the Hamiltonian Monte Carlo method. Note that it is made of two main algorithms, leapfrog algorithm and Metropoils-Hastings Monte Carlo sampling. Leapfrog algorithm is nothing but finitesimal movement of the position and the momentum by Hamiltonian equation.
One of the main result is as a following picture.
I would not display all the figures I obtained. You can find more pictures and python code in the ipython notebook link. They dramatically strengthen the efficiency of the sampling. The long narrow ellipse is a contour plot of the 2 dimensional probability density function. Look how fast the points at (-1,1) converges inside the ellipse. Much less random than random walk.
Gibbs sampling and over-relaxation, figure 30.3, ipython notebook
Think about narrow probability density like the above figures. All the 4 samples start at (-1.3, -1.3). Left two figures are sampled by Gibbs sampling, and the right two subplots are by over-relaxation. Thus, it reduces random walk behaviour in Gibbs sampling.
Rectangular Ising model and Monte Carlo method, figure 31.2 and figure 31.3, ipython notebook
We consider spin 1/2 ferromagnetic systems. There are spin-ups(blue) and downs(red).
The first subplot is by random spins. The 2nd one is by Ising model at low temperature. The spins are aligned and the symmetry(randomness) is broken(lost).
Mean Energy and mean square magnetization can be obtained from the Monte Carlo samples. Similarly, we can plot the energy fluctuation and Heat capacity, too, but the samples should be much better to obtain these plots because it include dervatives. I can handle it, but it is quite time consuming, so skip it.