Linear Algebra II: Eigenvectors and Diagonalisability

This post continues the discussion of the Oxford first-year course Linear Algebra II. We’ve moved on from determinants, and are now considering eigenvalues and eigenvectors of matrices and linear maps.

A good question to ask is: what’s the point of knowing about eigenvectors? I can think of a quick answer and a longer answer. The quick answer is that whenever we have a mapping of any kind, it is natural to ask about its fixed points. And since we are thinking about vector spaces and linear maps, if we can’t find any fixed points, we might nonetheless be able to find the best thing, some vectors whose direction is fixed by the map. In general, knowing about fixed points of a mapping might tell us other more qualitative properties, including the behaviour seen when you apply the map iteratively a large number of times. (Indeed a recent post discusses this exact problem for positive matrices in a context relevant to a chapter of my thesis…)

A more specific answer concerns bases. Recall that a linear map is defined independently of any basis: it’s just a map from the vector space to itself. We can express the linear map via a matrix with respect to some basis, but how to choose the basis? We could always choose the canonical basis in \mathbb{R}^n, since it’s easy to do vector and matrix calculations when most of the entries of all the vectors are zero. We also have a good visual idea (at least in up to three dimensions) of what a matrix might mean with respect to that basis. If we needed to divide the three-dimensional world around us into small volumes, we’d tend to describe it with small cubes rather than small arbitrary parallelopipeds.

But once we know something about the linear map, we might want to choose a basis of vectors on which the behaviour of the map is particularly easy to describe. And eigenvectors fulfil precisely this role. If we are able to choose a basis of eigenvectors, describing the map’s action, either abstractly, or via a (diagonal) matrix, is very straightforward. If we are given a matrix to begin with, we know how to do a change of basis, and changing to the basis of eigenvectors is precisely what’s going when we write A=P^{-1}DP, where D is a diagonal matrix. We construct P by taking its columns to be these eigenvectors. In particular, for a given vector x, y=Px is the vector giving the coefficients of x in the basis of eigenvectors.

So the case where we have a basis of eigenvectors is particularly useful, and in this case, we say the matrix or the map is diagonalisable. Remember how we find eigenvalues. If there exists a non-zero vector x satisfying Ax=\lambda x, then x is in the kernel of A-\lambda I. As we discussed last time, introducing the determinant gives a much more manageable way to verify which values of \lambda result in A-\lambda I having a non-trivial kernel. In particular, if non-zero x is in the kernel, we have \mathrm{det}(A-\lambda I)=0, and this leads to a polynomial of degree n (the dimensional of the vector space / size of the matrix) for \lambda, called the characteristic polynomial \chi_A(z), which has the eigenvalues as its roots.

If we agree to work over the complex field, then this is good, because it means we always have eigenvalues, and so it becomes sensible to talk about exactly how many eigenvalues and eigenvectors we have. Observe that if we restrict to real vector spaces, this might not be the case. In the plane, the rotation by \pi/2 for example has no fixed vectors.

Multiplicities of eigenvalues

We call the algebraic multiplicity \alpha(\lambda) of an eigenvalue \lambda to be the exponent of the factor (z-\lambda) in the factorisation of the characteristic polynomial. To define the geometric multiplicity, observe that all the eigenvectors with eigenvalue \lambda form a subspace, and so it is meaningful to talk about the dimension of this subspace (‘eigenspace’), which is the geometric multiplicity \gamma(\lambda). There are two facts that one needs to remember. The slightly less obvious one is that \gamma(\lambda)\le \alpha(\lambda) for all \lambda. One can see this by, for example, working in a basis that extends a basis of the \lambda-eigenspace. Observe at this stage that the sum of the algebraic multiplicities has to be n by definition, while the sum of geometric multiplicities is at most n. And this makes sense, because the space spanned by all the eigenvectors is a subspace, and so has dimension at most n.

The more obvious, but more frequently forgotten result is that

\alpha(\lambda)\ge 1 \quad \iff \quad \gamma(\lambda)\ge 1,

which is simply a consequence of the property discussed a few paragraphs previously concerning the kernel of A-\lambda I.

In particular, we might make the heuristic observation that ‘most’ polynomials of degree n have n distinct roots. This is certainly true for quadratics: there is only one value that the discriminant can take such that we see a repeated root. Alternatively, imagine shifting the quadratic up and down (in a complex way if necessary); again there is only one moment at which it might have a repeated root. This observation can be generalised easily to higher degree polynomials in a number of ways.

So if we lift this observation across to matrices, we see that most matrices have n distinct eigenvalues, and thus have n linearly independent eigenvectors which form a basis, hence the matrix is diagonalisable. I think it’s really worth reflecting on this, since much of a first exploration into linear algebra ends up treating exactly the case where the matrix is not diagonalisable.

The principal example of a non-diagonalisable matrix is \begin{pmatrix}2&1\\0&2\end{pmatrix}, where the 2s can be replaced by any value, and the 1 can be replaced by an non-zero value. There’s plenty to learn about to what extent versions of this matrix of higher size represent all non-diagonalisable matrices, but such an exposition of Jordan normal form comes next year for the students taking this course.

It probably is worth saying now though, that this example gives a good sanity check for whether a method is actually using diagonalisability correctly. For example, it is easily seen that elementary row operations to not preserve diagonalisability by starting from \begin{pmatrix}2&0\\0&2\end{pmatrix} and ending up at our counter-example. One could also argue from this that the set of non-diagonalisable matrices are dense within the set of matrices with a repeated eigenvalue. That is, having a repeated eigenvalue but full eigenspace is doubly-infinitely-unlikely.

Cayley-Hamilton theorem

Anyway, among other results, we also saw the Cayley-Hamilton theorem, which states that a matrix A satisfies its own characteristic equation. That is \chi_A(A)=0, where the zero on the right-hand side is the zero matrix. It’s tempting to substitute A into the expression \mathrm{det}(A-\lambda I), but of course this is not valid. Indeed imagine a typical eigenvalue determinant matrix with terms like (7-\lambda) on the diagonal; it doesn’t make sense to substitute a matrix for \lambda as one of the entries of the overall matrix!

Fortunately, we can argue convincingly in the case where A is a diagonalisable matrix. Remember that \chi_A(A) is a matrix. Now looki at the action of \chi_A(A) on any eigenvector v, corresponding to eigenvalue \lambda. Applying some power of A to v gives v multiplied by the same power of \lambda, and so we end up with

\chi_A(A)v = \chi_A(\lambda)v = 0.

This only worked when v was an eigenvector, but fortunately there is a basis of eigenvectors if A is diagonalisable, and so \chi_A(A)v=0 for all v, hence \chi_A(A)=0.

But \chi_A(A) is just a matrix-valued function of A. If you think about it, \chi_A is a monic polynomial, all of whose non-leading coefficients are multinomials of degree at most n-1 in the entries of A. Furthermore, these multinomials have (non-negative) integer coefficients. Therefore the entries of \chi_A(A) are multinomials of degree at most 2n-1 in the entries of A, and again have (non-negative) integer coefficients.

Even without the integrality of the coefficients, this says that, under any reasonable definition of continuity of matrices (which could be induced from any topology on \mathbb{R}^{n\times n}) the function \chi_A(A) should be continuous as a function of A. But we’ve shown \chi_A(A)=0 for all diagonalisable A, and also argued that most complex-valued matrices are diagonalisable. Turning this into a formal statement about denseness means that we’ve shown the Cayley-Hamilton theorem for non-diagonalisable matrices also. It feels that because the coefficients are non-negative integers, we might also have shown the result for other fields too, but I have minimal knowledge or recollection at the moment of the things one has to check for this sort of result.

It’s worth ending with the brief comment that Cayley-Hamilton is useful, among other reasons because it enables us to write the inverse of A as a polynomial of degree at most n-1 in terms of A. In many settings this is a lot easier to work with in terms of calculations than an argument with minors.

Linear Algebra II: Determinants 2

In the previous post, we introduced determinants of matrices (and by extension linear maps) via its multilinearity properties, and as the change-of-volume factor. We also discussed how to calculate them, via row operations, or Laplace expansion, or directly via a sum of products of entries over permutations.

The question of why this is ever a useful quantity to consider remains, and this post tries to answer it. We’ll start by seeing one example where this is a very natural quantity to consider, and then the main abstract setting, where the determinant is zero, and consider a particularly nice example of this.

Jacobeans as a determinant

We consider integration by substitution. Firstly, in one variable: when it comes to Riemann integration of a function g(x) with respect to x, we view dx as the width of a small column which approximates the function near x. Now, if we reparameterise, that is if we write x=f(y) for some well-behaved (in particular differentiable) function f, then the width of the column is dx= dy.(dx/dy)=f'(y) dy. This may be negative, if y is decreasing while x is increasing, but for now let’s not worry about this overly, for example by assuming the function g is non-negative. Thus if we want to integrate with y as the variable, we multiply the integrand by this factor |f'(y)|.

What about in higher dimensions? We have exactly the same situation, only instead of two-dimensional columns, we have (n+1)-dimensional columns. We then multiply the n-dimensional volume of the base by the height, again given by g(\mathbf{x}). If we have a similar transformation of the base variable \mathbf{x}=f(\mathbf{y}), we differentiate to get

\mathrm{d}x_i = \sum_{j=1}^n\frac{\mathrm{d}f_i}{\mathrm{d}y_j} \mathrm{d}y_j.

In other words

\mathrm{d}\mathbf{x}= J \mathrm{d}\mathbf{y},

where J is the Jacobean matrix of partial deriatives. In particular, we know how to relate the volume [0,\mathrm{d}x_1]\times\ldots\times [0,\mathrm{d}x_n] to the volume [0,\mathrm{d}y_1]\times \ldots\times [0,\mathrm{d}y_n]. It’s simply the determinant of the Jacobean J. So if we want to integrate with respect to \mathbf{y}, it only remains to pre-multiply the integrand by |\mathrm{det}J| and proceed otherwise as in the one-dimensional case.

Det A = 0

A first linear algebra course might well motivate the introducing matrices as a notational shortcut for solving families of linear equations, Ax=b. The main idea is that generally we can solve this equation uniquely. Almost all of the theory developed in such a first linear algebra course deals with the case when this fails to hold. In particular, there are many ways to characterise this case, and we list some of them now:

  • Ax=b has no solutions for some b;
  • A is not invertible;
  • A has non-trivial kernel, that is, with dimension at least one;
  • A does not have full rank, that is, the image has dimension less than n;
  • The columns (or indeed the rows) are linearly dependent;
  • The matrix can be row-reduced to a matrix with a row of zeroes.

It is useful that these are equivalent, as in abstract problems one can choose whichever interpretation from this list is most relevant. However, all of these are quite hard to check. Exhibiting a non-trivial kernel element is hard – one either has to do manual row-reduction, or the equivalent in the context of linear equations. But we can add the characterisation

  • det A = 0;

to the list. And this is genuinely much easier to check for specific examples, either abstract or numerical.

Let’s quickly convince ourselves of a couple of these equivalences. Determinant is invariant under row-reductions, and by multilinearity it is certainly the case that det A = 0 if A has a row of zeroes. We also said that A is the change-of-volume factor. Note that A is a map from the domain to its image, so if A has less than full rank, then any set in the image has zero volume.

The Vandermonde matrix

This is a good example of this theory in practice. Consider the Vandermonde matrix where each row is a geometric progression:

V=\begin{pmatrix}1&\alpha_1&\ldots&\alpha_1^{n-1}\\1&\alpha_2&\ldots&\alpha_2^{n-1}\\ \vdots&\vdots&\ddots&\vdots\\ 1&\alpha_n&\ldots&\alpha_n^{n-1}\end{pmatrix}.

Now suppose we attempt to solve

V\begin{pmatrix}a_0\\a_1\\ \vdots\\ a_{n-1}\end{pmatrix}=\begin{pmatrix}b_1\\b_2\\ \vdots \\ b_n\end{pmatrix}.

There’s a natural interpretation to this, that’s especially clear with this suggestive notation. Each row corresponds to a polynomial, where the coefficients are given by the (a_0,a_1,\ldots,a_{n-1}), and the argument is given by \alpha_i.

So if we try to solve for (a_0,a_1,\ldots,a_{n-1}), given (\alpha_1,\ldots,\alpha_n) and (b_1,\ldots,b_n), we are asking whether we can find a polynomial P with degree at most n-1 such that P(\alpha_i)=b_i for $i=1. Lagrange interpolation gives an argument where we just directly write down the relevant polynomial, but we can also deploy our linear algebraic arguments too.

The equivalence of all these statements means that to verify existence and uniqueness of such a polynomial, we only need to check that the Vandermonde matrix has non-zero determinant. And in fact there are a variety of methods to show that

\mathrm{det}V=\prod_{1\le i< j}\le n(\alpha_j-\alpha_i).

For the polynomial question to be meaningful, we would certainly demand that the (\alpha_i) are distinct, and so this determinant is non-zero, and we’ve shown that n points determine a degree (n-1) polynomial uniquely.

If we multiply on the left instead, suppose that we are considering a discrete probability distribution X that takes n known values (\alpha_1,\ldots,\alpha_n) with unknown probabilities (p_1,\ldots,p_n). Then we have

(p_1,\ldots,p_n) V = (1,\mathbb{E}X, \mathbb{E}[X^2],\ldots, \mathbb{E}[X^{n-1}]).

So, again by inverting the Vandermonde matrix (which is know is possible since its determinant is non-zero…) we can recover the distribution from the first (n-1) moments of the distribution.

A similar argument applies to show that the Discrete Fourier Transform is invertible, and in this case (where the \alpha_is are roots of unity), the expression for the Vandermonde determinant is particularly tractable.

Linear Algebra II: Determinants 1

This term, I’m giving tutorials on a course that’s new to me, the apparently notorious ‘Linear Algebra II’ for first year undergraduates. I can appreciate how it might have ended up with this reputation, but as always, every challenge is also an opportunity, So I’m going to (try to) write a short series of blog posts about what we’ve discussed in the tutorials.

The first problem sheet-and-a-half concerned determinants of matrices. There are three things worth addressing here:

  1. What are abstract definitions, and which is most useful in each setting?
  2. How to actually calculate them?
  3. What’s the overall point?

The answers are obviously not completely unrelated, but we’ll probably defer the third question to a second post.

The determinant is a map from the set of matrices \mathcal{M}_n to the base field (hereafter assumed to be \mathbb{R},\mathbb{C}). The Oxford course defines it through its properties:

  • Multilinear in the columns of the matrix.
  • Equal to zero if two columns are equal.
  • Equal to one if the matrix is the identity.

One then checks that there is a unique such map, and so from now on it’s reasonable to call it the determinant of the matrix. It will follow from pretty much any consequence that we can replace ‘columns’ with ‘rows’ throughout and get the same map.

Other definitions

We have a closed form expression for the determinant given via permutations of n

\mathrm{det}(A)=\sum_{\sigma\in \Sigma_n} \mathrm{sign}(\sigma) a_{1\sigma(1)}\ldots a_{n\sigma(n)}.

We’ll come back to a discussion of when this particular definition is useful. It can be derived by carefully transforming the identity matrix into A, using the operations which are mentioned in the original definition of the determinant, in particular, keeping track of the number of transpositions of columns.

It’s clear from any definition that the determinant is a polynomial of degree n in the entries of the matrix, but this definition will be useful if you want to make some more precise comment on the nature of this polynomial. For example, if entries of the matrix are polynomials in x of various degree (think of the eigenvalue equation for example) this allows you to control (or at least bound) the overall degree of the determinant as a polynomial in x.

The determinant is also the volume of the n-dimensional parallelopiped formed by the column vectors of the matrix. This is easy to check in two dimensions, for the matrix \begin{pmatrix}a&b\\c&d\end{pmatrix}:

20160225_172206To calculate the area of the central parallelogram, we have to subtract the area of two small rectangles and four small triangles from the outer rectangle, obtaining

(a+b)(c+d)-2bc - \frac12(ac+ac+bd+bd)=ad-bc,

as we expect. This calculation is harder to execute in higher dimensions, and certainly harder to visualise.

Maybe, though, we don’t have to, so long as we can reassure ourselves that this volume satisfies the implicit definition of the determinant map at the start. Multilinearity in the columns is not that hard to see. If we multiply the jth column by some constant, we are stretching the parallelopiped by the same constant factor in one direction, and so the volume grows appropriately. The additivity property can similarly be thought of as joining together two parallelopipeds at their common face (which is common since the other column vectors have to be constant in this construction). If two column vectors are equal, then clearly this volume actually has dimension at most n-1, and thus volume zero, so the final two conditions are genuinely easy to check.

The challenge here is that there is a direction involved. Determinants can be negative, but in our classical viewpoint, areas generally are not. In 2D, we can think of this as saying that the area is positive if the vector (b,d) lies anti-clockwise from (a,c) in the parallelogram, while is it negative otherwise. Again, this is harder to visualise in higher dimensions, but it is at least plausible that one could develop a similar decomposition. Ultimately, we are happy with the notion of directed lengths (ie vectors on the real line), and these are easy to add up without having to separate into cases, and the case holds for areas and higher-dimensional volumes.

Evaluating determinants

If we actually want to compute the determinant of a given matrix, the sum over permutations is intractable since it doesn’t have any natural splits into stages. The implicit definitions and this area consideration are clearly useless for all but the most special of examples.

The Laplace expansion is the usual algorithm to calculate the determinant of an n x n matrix. You pick a row (or a column), and evaluate the determinants of the (n-1) x (n-1) minor matrices given by deleting this row (or column), and each column (or row) in turn. This leaves us with n determinants of smaller matrices, which we pre-multiply by the entries in the original deleted row (or column), and add up in an alternating way (*). This is highly computationally intensive for large matrices, but for 3×3 and 4×4 can be done by hand with probability of an error bounded away from 1.

There is the flexibility to choose the reference row or column. Since the entries of these affect the sum through small products, it is highly convenient to choose a row or column with a lot of zeros. In particular, if there’s a row or column with exactly one non-zero entry, this is an ideal candidate.

The sum over permutations also works well when a lot of the entries are zero, because then a lot of the permutations give a summand which is zero. Upper-triangular matrices are a good example: only one permutation (the identity permutation) avoids all the zero elements underneath the diagonal.

One can also observe from the multilinearity property of the determinant map that there are lots of operations we can apply to the matrix which leave the determinant fixed. These are often called elementary row operations, though obviously we can apply them to the columns as well. To summarise, if we interchange two rows, the sign of the determinant is reversed. And if we add some multiple of one row to any other row, the determinant stays the same.

When matrices are not square, it’s quite important to be specific about exactly what form you can reduce a general matrix to via such row operations, but in this context, it’s not hugely important. Reduced echelon form (without the condition that leading coefficients will be one) is achievable, but this is a special case of an upper triangular matrix, for which the determinant is given by the product of the diagonal entries, ie is easy.

Whether this is substantially easier than Laplace expansion depends on the matrix itself and taste, both to do manually, and to code.

(*) I’m not a fan personally of this alternating definition. It seems to me much more natural to define the minor as

M^{i,j}=(a_{i+k,j_\ell})_{1\le k,\ell\le n-1},

with indices taken modulo n. Then you don’t have any \pm 1s in the Laplace expansion.

Using determinants in abstract problems

So the determinant gives directly the area of the image (under A) of the unit hypercube. By linearity (of A), it is easy to see that it also gives the scale factor of the area change (under A) of any hyper-cuboid, parallel to the conventional axes, anywhere in the space. Then, eg by approximating any sensible n-dimensional shape (*) as a union of such hyper-cuboids, we can show that in fact the area of any sensible shape increases by a factor (det A) under application of A.

This is a good thing to remember, because it is an excellent heuristic for seeing why the determinant of a linear map is basis-independent. It also gives a much easier proof of the key result

\mathrm{det}(AB)=\mathrm{det}(A) \mathrm{det}(B),

than that given by fixing B and viewing det(AB) as a map from matrices to the field, just like the original definition of determinant.

Some of the theory in the course is proved using elementary row operations. But these invite complicated notation, so are best used only in simple arguments, or when things are fairly explicit to begin with. Given an abstract problem about determinants of matrices, it is often tempting to induct on the size of the matrix in some way. I think it’s worth saying that even though the Laplace expansion is explicitly set up in this way, the notation involved is also likely to be annoying here, while permutations are easy to describe inductively: eg let \sigma(1)=k, then view the remainder of the permutation as a bijection \{2,3,\ldots,n\}\rightarrow [n]\backslash \{k\}.

Shortly, we’ll have a second post answering the final question: what’s the point of working with determinants? We’ve already seen half an answer, in that they describe the change-of-volume factor of a matrix (or linear map), but this can be substantially developed.

Hitting Probabilities for Markov Chains

This continues my previous post on popular questions in second year exams. In the interest of keeping it under 2,500 words I’m starting a new article.

In a previous post I’ve spoken about the two types of Markov chain convergence, in particular, considering when they apply. Normally the ergodic theorem can be used to treat the case where the chain is periodic, so the transition probabilities do not converge to a stationary distribution, but do have limit points – one at zero corresponding to the off-period transitions, and one non-zero. With equal care, the case where the chain is not irreducible can also be treated.

A favourite question for examiners concerns hitting probabilities and expected hitting times of a set A. Note these are unlikely to come up simultaneously. Unless the hitting probability is 1, the expected hitting time is infinite! In both cases, we use the law of total probability to derive a family of equations satisfied by the probabilities/times. The only difference is that for hitting times, we add +1 on the right hand side, as we advance one time-step to use the law of total probability.

The case of hitting probabilities is perhaps more interesting. We have:

h_i^A = 1,\; i\in A, \quad h_i^A=\sum_{j\in S}p_{ij}h_j^A,\; i\not\in A.

There are two main cases of interest: where the chain is finite but has multiple closed communicating classes, and where the chain is infinite, so even though it is irreducible, a trajectory might diverge before hitting 0.

For the case of a finite non-irreducible Markov chain, this is fairly manageable, by solving backwards from states where we know the values. Although of course you could ask about the hitting probability of an open state, the most natural question is to consider the probability of ending up in a particular closed class. Then we know that the hitting probability starting from site in the closed class A is 1, and the probability starting from any site in a different closed class is 0. To find the remaining values, we can work backwards one step at a time if the set of possible transitions is sparse enough, or just solve the simultaneous equations for \{h_i^A: i\text{ open}\}.

We therefore care mainly about an infinite state-space that might be transient. Typically this might be some sort of birth-and-death chain on the positive integers. In many cases, the hitting probability equations can be reduced to a quadratic recurrence relation which can be solved, normally ending up with the form

h_i=A+B\lambda^i,

where \lambda might well be q/p or similar if the chain is symmetric. If the chain is bounded, typically you might know h_0=1, h_N=0 or similar, and so you can solve two simultaneous equations to find A and B. For the unbounded case you might often only have one condition, so you have to rely instead on the result that the hitting probabilities are the minimal solution to the family of equations. Note that you will always have h^i_i=1, but with no conditions, h^i_j\equiv 1 is always a family of solutions.

It is not clear a priori what it means to be a minimal solution. Certainly it is not clear why one solution might be pointwise smaller than another, but in the case given above, it makes sense. Supposing that \lambda<1, and A+B=1 say, then as we vary the parameters, the resulting set of ‘probabilities’ does indeed vary monotically pointwise.

Why is this true? Why should the minimum solution give the true hitting probability values? To see this, take the equations, and every time an h_i^A appears on the right-hand side, substitute in using the equations. So we obtain, for i\not\in A,

h_i^A=\sum_{j\in A}p_{ij}+\sum_{j\not\in A} p_{ij}h_j^A,

and after a further iteration

h_i^A=\sum_{j_1\in A}p_{ij_1}+\sum_{j_1\not\in A, j_2\in A}p_{ij_1}p_{j_1j_2}+\sum_{j_1,j_2\not\in A}p_{ij_1}p_{j_1j_2}h_{j_2}^A.

So we see on the RHS the probability of getting from i to A in one step, and in two steps, and if keep iterating, we will get a large sum corresponding to the probability of getting from i to A in 1 or 2 or … or N steps, plus an extra term. Note that the extra term does not have to correspond to the probability of not hitting A by time N. After all, we do not yet know that (h_{i}^A) as defined by the equations gives the hitting probabilities. However, we know that the probability of hitting A within N steps converges to the probability of hitting A at all, since the sequence is increasing and bounded, so if we take a limit of both sides, we get h_i^A on the left, and something at least as large as the hitting probability starting from i on the right, because of the extra positive term. The result therefore follows.

It is worth looking out for related problems that look like a hitting probability calculation. There was a nice example on one of the past papers. Consider a simple symmetric random walk on the integers modulo n, arranged clockwise in a circle. Given that you start at state 0, what is the probability that your first return to state 0 involves a clockwise journey round the circle?

Because the system is finite and irreducible, it is not particularly interesting to consider the actual hitting probabilities. Also, note that if it is convenient to do so, we can immediately reduce the problem when n is even. In two steps, the chain moves from j to j+2 and j-2 with probability ¼ each, and stays at j with probability ½. So the two step chain is exactly equivalent to the lazy version of the same dynamics on n/2.

Anyway, even though the structure is different, our approach should be the same as for the hitting probability question, which is to look one step into the future. For example, to stand a chance of working, our first two moves must both be clockwise. Thereafter, we are allowed to move anticlockwise. There is nothing special about starting at 0 in defining the original probability. We could equally well ask for the probability that starting from j, the first time we hit 0 we have moved clockwise round the circle.

The only thing that is now not obvious is how to define moving clockwise round the circle, since it is not the case that all the moves have to be clockwise to have experienced a generally clockwise journey round the circle, but we definitely don’t want to get into anything complicated like winding numbers! In fact, the easiest way to make the definition is that given the hitting time of 0 is T, we demand that the chain was at state n at time T-1.

For convenience (ie to make the equations consistent) we take h_0=0, h_n=1 in an obvious abuse of notation, and then

h_j=\frac12h_{j-1}+\frac12 h_{j+1},

from which we get

h_j=a+bj \Rightarrow h_j=\frac{j}{n}.

Of course, once we have this in mind, we realise that we could have cut the circle at 0 (also known as n) and unfolded it to reduce the problem precisely to symmetric gambler’s ruin. In particular, the answer to the original problem is 1/2n, which is perhaps just a little surprising – maybe by thinking about the BM approximation to simple random walk, and that BM started from zero almost certainly crosses zero infinitely many times near we might have expected this probability to decay faster. But once it is unfolded into gambler’s ruin, we have the optimal stopping martingale motivation to reassure us that this indeed looks correct.

Avoiding Mistakes in Probability Exams

Over the past week, I’ve given several tutorials to second year undergraduates preparing for upcoming papers on probability and statistics. In particular, I’ve now seen a lot of solutions to a lot of past papers and specimen questions, and it’s worthwhile to consider some of the typical mistakes students can make on these questions. Of course, as with any maths exam, there’s always the possibility of a particularly subtle or involved question coming up, but if the following three common areas of difficulty can be avoided, you’re on track for doing well.

Jacobians

In a previous course, a student will learn how to calculate the pdf of a function of a random variable. Here, we move onto the more interesting and useful case of finding the (joint) density of function(s) of two or more random variables. The key thing to remember here is that manipulating pdfs is not a strange arbitrary exercise – it is just integration. It is rarely of interest to consider the value of a pdf at a single point. We can draw meaningful conclusions from a pdf or from comparison of two pdfs by integrating them.

Then the question of substituting for new random variables is precisely integration by substitution, which we are totally happy with in the one-dimensional case, and should be fairly happy with in the two-dimensional case. To get from one joint density to another, we multiply by the absolute value of the Jacobian. To ensure you get it right, it makes sense to write out the informal infinitesimal relation

f_{U,V}(u,v) du dv = f_{X,Y}(x,y)dx dy.

This is certainly relevant if we put integral signs in front of both sides, and explains why you obtain f_{U,V} = \frac{d(x,y)}{d(u,v)} f_{X,Y} rather than the other way round. Note though that if \frac{d(u,v)}{d(x,y)} is easier to calculate for some reason, then you can evaluate this and take the inverse, as your functions will almost certainly be locally bijective almost everywhere.

It is important to take the modulus of the Jacobian, since densities cannot be negative! If this looks like a fudge, then consider the situation in one dimension. If we substitute for x\mapsto f(x)=1-x, then f’ is obviously negative, BUT we also end up reversing the order of the bounds of the integral, eg [1/3, ¾] will become [2/3,1/4]. So we have a negative integrand (after multiplying by f'(x)) but bounds in the ‘wrong’ order. These two factors of -1 will obviously cancel, so it suffices just to multiply by |f'(x)| at that stage. It is harder to express in words, but a similar relation works for the Jacobian substitution.

You also need to check where the new joint density is non-zero. Suppose X, Y are supported on [0,1], then when we write f_{X,Y}(x,y) we should indicate that it is 0 off this region, either by splitting into cases, or adding the indicator function 1_{\{x,y\in[0,1]\}} as a factor. This is even more important after substitutions, as the range of the resulting random variables might be less obvious than the originals. Eg with X,Y as above, and U=X^2, V=X/Y, the resulting pdf will be non-zero only when u\in[0,1], v\ge \sqrt{u}. Failing to account for this will often lead to ludicrous answers. A general rule is that you can always check that any distribution you’ve produced does actually integrate to one.

Convergence using MGFs

There are two main reasons to use MGFs and PGFs. The first is that they behave nicely when applied to (possibly random) sums of independent random variables. The independence property is crucial to allow splitting of the MGF of the sum into the product of MGFs of the summands. Of course, implicit in this argument is that MGFs determine distributions.

A key theorem of the course is that this works even in the limit, so you can use MGFs to show convergence in distribution of a family of distributions. For this, you need to show that the MGFs converge pointwise on some interval [-a,a] around 0. (Note that the moments of the distribution are given by the family of derivatives at 0, as motivation for why this condition might be necessary.) Normally for such questions, you will have been asked to define the MGF earlier in the question, and probably will have found the MGF of a particular distribution or family of distributions, which might well end up appearing as the final answer.

Sometimes such an argument might involve substituting in something unusual, like t/N, rather than t, into a known MGF. Normally a Taylor series can be used to show the final convergence result. If you have a fraction, try to cancel terms so that you only have to evaluate one Taylor series, rather than lots.

Using the Markov Property

The Markov property is initially confusing, but once we become comfortable with the statement, it is increasingly irritating to have to answer the question: “show that this process has the Markov property.” This question is irritating because in most cases we want to answer: “because it obviously does!” Which is compelling, but unlikely to be considered satisfactory in a mathematics exam. Normally we observe that the random dynamics of the next step are a function only of the present location. Looking for the word ‘independent’ in the statement of the process under discussion is a good place to start for any argument along these lines.

The most developed example of a Markov process in this course is the Poisson process. I’ve written far too much about this before, so I won’t do so again, except to say this. When we think of the Poisson process, we generally have two thoughts going through our minds, namely the equivalent definitions of IID exponential inter-arrival times, and stationary, Poisson increments (or the infinitesimal version). If we draw a sketch of a sample trajectory of this process, we can label everything up and it is clear how it all fits together. But if you are asked to give a definition of the Poisson process (N_t), it is inappropriate to talk about inter-arrival times unless you define them in terms of N_t, since that is the process you are actually trying to define! It is fine to write out

T_k:=\min\{t: N_t=k\},\quad N_t=\max\{k: Y_1+Y_2+\ldots+Y_k\le t\}

but the relation between the two characterisations of the process is not obvious. That is why it is a theorem of the course.

We have to be particularly careful of the difference in definition when we are calculating probabilities of various events. A classic example is this. Find the distribution of N_2, conditional on T_3=1. It’s very tempting to come up with some massive waffle to argue that the answer is 3+Po(1). The most streamlined observation is that the problem is easy if we are conditioning instead on N_1=3. We just use the independent Poisson increments definition of (N_t), with no reference to inter-arrival times required. But then the Markov property applied at time 1 says that the distribution of (N_2) depends only on the value of N_1, not on the process on the interval [0,1). In a sense, the condition that T_3=1 is giving us extra information on the behaviour of the process up to time 1, and the Markov property, which we know holds for the Poisson process, asserts precisely that the extra information doesn’t matter.

Generating Functions for Dice

So last week I was writing an article for Betting Expert about laws of large numbers, and I was trying to produce some representations of distributions to illustrate the Weak LLN and the Central Limit Theorem. Because tossing a coin feels too simplistic, and also because the natural state space for this random variable, at least verbally, is not a subset of the reals, I decided to go for dice instead. So it’s clear what the distribution of the outcome of a single dice roll is, and with a bit of thought or a 6×6 grid, we can work out the distribution of the average of two dice rolls. But what about 100 rolls? Obviously, we need large samples to illustrate the laws of large numbers! In this post, we discuss how to calculate the distribution of the sample mean of n dice rolls.

First we observe that the total set of outcomes of n dice rolls is 6^n. The sum of the outcomes must lie between n and 6n inclusive. The distribution of the sum and the distribution of the sample mean are equivalent up to dividing by n. The final observation is that because the total number of outcomes has a nice form, we shouldn’t expect it to make any difference to the method if we calculate the probability of a given sum, or the number of configurations giving rise to that sum.

Indeed, tying in nicely with the first year probability course, we are going to use generating functions, and there is no difference in practice between the probability generating function, and the combinatorial generating function, if the underlying mechanism is a uniform choice. Well, in practice, there is a small difference, namely a factor of 6 here. The motivation for using generating functions is clear: we are considering the distribution of a sum of independent random variables. This is pretty much exactly why we bother to set up the machinery for PGFs.

Anyway, since each of {1,2,…,6} is equally likely, the GF of a single dice roll is

x+x^2+\ldots+x^6=x\cdot \frac{1-x^6}{1-x}.

So, if we want the generating function of the sum of n independent dice rolls, we can obtain this by raising the above function to the power n. We obtain

x^n(1-x^6)^n(1-x)^{-n}.

Note the factor of x^n at the beginning arises because the minimum value of the sum is n. So to work out the number of configurations giving rise to sum k, we need to evaluate the coefficient of x^k. We can deal with (1-x^6)^n fairly straightforwardly, but some thought it required regarding whether it’s possible to do similar job on (1-x)^{-n}.

We have to engage briefly with what is meant by a binomial coefficient. Note that

\binom{x}{k}=\frac{x(x-1)\ldots(x-k+1)}{1\cdot\ldots\cdot k}

is a valid definition even when x is not a positive integer, as it is simply a degree k polynomial in x. This works if x is a general positive real, and indeed if x is a general negative real. At this stage, we do need to keep k a positive integer, but that’s not a problem for our applications.

So we need to engage with how the binomial theorem works for exponents that are not positive integers. The tricky part with the standard expression as

(a+b)^n=\binom{n}{0}a^n+\ldots + \binom{n}{n}b^n,

is that the attraction of this symmetry in a and b prompts us to work in more generality than is entirely necessary to state the result. Note if we instead write

(1+x)^n=1+\binom{n}{1}x+\binom{n}{2}x^2+\ldots,

we have unwittingly described this finite sum as an infinite series. It just happens that all the binomial coefficients apart from the first (n+1) are zero. The nice thing about this definition is that it might plausibly generalise to non-integer or negative values of n. And indeed it does. I don’t want to go into the details here, but it’s just a Taylor series really, and the binomial coefficients are set up with factorials in the right places to look like a Taylor series, so it all works out.

It is also worth remarking that it follows straight from the definition of a negative binomial coefficient, that

\binom{-n}{j}=(-1)^j \binom{n+j-1}{j}.

In any case, we can rewrite our expression for the generating function of the IID sum as

x^n\left[\sum_{k=0}^n \binom{n}{k}(-1)^k x^{6k}\right]\left[\sum_{j\ge 0} \binom{-n}{j}(-1)^j x^j\right]

By accounting for where we can gather exponents from each bracket, we can evaluate the coefficient of x^m as

\sum_{6k+j=m+n}\binom{n}{k}\binom{n+j-1}{j}(-1)^k.

Ie, k in the sum takes values in \{0,1,\ldots, \lfloor \frac{m+n}{6}\rfloor\}. At least in theory, this now gives us an explicit way to calculate the distribution of the average of multiple dice rolls. We have to be wary, however, that many compilers will not be happy dealing with large binomial coefficients, as the large factorials grow extremely rapidly. An approximation using logs is likely to be more tractable for larger settings.

Anyway, I leave you with the fruits of my labours.

dice30Related articles

Enhanced by Zemanta

Convergence of Transition Probabilities

As you can see, I haven’t got round to writing a post for a while. Some of my reasons for this have been good, and some have not. One reason has been that I’ve had to give a large number of tutorials for the fourth quarter of the second year probability course here in Oxford. The second half of this course concerns discrete-time Markov chains, and the fourth problem sheet discusses various modes of convergence for such models, as well as a brief tangent onto Poisson Processes. I’ve written more about Poisson Processes than perhaps was justifiable in the past, so I thought I’d say some words about convergence of transition probabilities in discrete-time Markov chains.

Just to be concrete, let’s assume the state space K is finite, and labelled {1,2,…,k}, so that it becomes meaningful to discuss

p_{12}^{(n)}:=\mathbb{P}(X_n=2|X_0=1).

That is, the probability that if we start at state 1, then after n ‘moves’ we are at state 2. We are interested in the circumstances under which this converges to the stationary distribution. The heuristic is that we can view a time-step of a Markov chain as an operation on the space of distributions on K. Note that this operation is deterministic. If this sounds complicated, what we mean is that we specify an initial distribution, that is the distribution of X_0. If we consider the distribution of X_1, this is given by \lambda P, where \lambda is the initial distribution, and P the transition matrix.

Anyway, the heuristic is that the stationary distribution is the unique fixed point of this operation on the space of distributions. It is therefore not unreasonable to assume that unless there are some periodic effects, we expect repeated use of this operation to move us closer to this fixed point.

We can further clarify this by considering the matrix form. Note that a transition P always has an eigenvalue equal to 1. This is equivalent to say that there is a solution to \pi P=\pi. Note it is not immediately equivalent to saying that P has a stationary distribution, as the latter must be non-negative and have elements summing to one. Only the first property is difficult, and relies on some theory or cleverness to prove. It can also be shown that all eigenvalues satisfy |\lambda|\le 1, and in general, there will be a single eigenvalue (ie dimension 1 eigenspace) with |\lambda|=1, and the rest satisfies |\lambda|<1. Then, if we diagonalise P, it is clear why \pi P^n converges entry-wise, as \pi UP^n U^{-1} converges. In the latter, only the entries in the row corresponding to \lambda=1 converge to something non-zero.

In summary, there is a strong heuristic for why in general, the transition probabilities should converge, and if they converge, that they should converge to the stationary distribution. In fact, we can prove that for any finite Markov chain, p_{ij}^{(n)}\rightarrow \pi_j, provided we two conditions hold. The conditions are that the chain is irreducible and aperiodic.

In the rest of this post, I want to discuss what might go wrong when these conditions are not satisfied. We begin with irreducibility. A chain is irreducible if it has precisely one communicating class. That means that we can get from any state to any other state, not necessarily in one step, with positive probability. One obvious reason why the statement of the theorem cannot hold in this setting is that \pi is not uniquely defined when the chain is not irreducible. Suppose, for example, that we have two closed communicating classes A and B. Then, supported on each of them is an invariant distribution \pi^A and \pi^B, so any affine combination of the two \lambda \pi^A+(1-\lambda) \pi^B will give a stationary distribution for the whole chain.

In fact, the solution to this problem is not too demanding. If we are considering p_{ij}^{(n)} for i\in A a closed communicating class, then we know that p_{ij}^{(n)}=0 whenever j\not\in A. For the remaining j, we can use the theorem in its original form on the Markov chain, with state space reduced to A. Here, it is now irreducible.

The only case left to address is if i is in an open communicating class. In that case, it suffices to work out the hitting probabilities starting from i of each of the closed communicating classes. Provided these classes themselves satisfy the requirements of the theorem, we can write

p_{ij}^{(n)}\rightarrow h_i^A \pi^A_j,\quad i\not\in A, j\in A.

To prove this, we need to show that as the number of steps grows to infinity, the probability that we are in closed class A converges to h_i^A. Then, we decompose this large number of steps so to say that not only have we entered A with roughly the given probability, but in fact with roughly the given probability we entered A a long time in the past, and so there has been enough time for the original convergence result to hold in A.

Now we turn to periodicity. If a chain has period k, this says that we can split the state space into k classes A_1,\ldots,A_k, such that p_{ij}^{(n)}=0 whenever n\not\equiv j-i \mod k. Equivalently, the directed graph describing the possible transitions of the chain is k-partite. This definition makes it immediately clear that p_{ij}^{(n)} cannot converge in this case. However, it is possible that p_{ij}^{(kn)} will converge. Indeed, to verify this, we would need to consider the Markov chain with transition matrix P^k. Note that this is no longer irreducible, as it there are no transitions allowed between classes A_1,\ldots,A_k. Indeed, a more formal definition of the period, in terms of the lcd of possible return times allows us to conclude that there is no finer reducibility structure. That is, A_1,\ldots,A_k genuinely are the closed classes when we consider the chain with matrix P^k. And so the Markov chain with transition matrix P^k restricted to any of the A_is satisfies the conditions of the theorem.

There remains one case which I’ve casually brushed over. When we were discussing the irreducible case, I said that if we had at least one communicating classes, then we could work out the limiting transition probabilities from a state in an open class to a state in a closed class by calculating the hitting probability of that closed class, then applying the standard version of the theorem to that closed class. This relies on the closed class being aperiodic.

Suppose otherwise that the destination closed class A has period k as before. If it were to be the case that the number of steps required to arrive at A had some fixed value mod k, or modulo a non-trivial divisor of k, then we certainly wouldn’t have convergence, for the same reasons as in the globally periodic case. However, we should ask whether we can ever have convergence?

In fact, the answer is yes. For concreteness, and because it’s easier to write ‘odd’ and ‘even’ than m \mod k, let’s assume A has size 2 and period 2. That is, once we arrive in A, thereafter we alternate deterministically between the two states. Anyway, for some large time n, we can write p_{ca}^{(n)} for a\in A, c\not\in A as:

p_{ca}^{(n)}=h_i^A(n),

where the latter term is the probability that we arrive in A at a time-step which has the same parity as n. It’s not terribly hard to come up with an example where this holds, and this idea holds in greater generality, where A has period k (and not necessarily just k states), we have to demand that the probability of arriving at a time which is a mod k is equal for all a in [0,k-1].

Of course, for applications, we don’t normally care much about irreducible chains, and we can easily remove periodicity by introducing so-called laziness, whereby on each time-step we flip a coin (biased if necessary) and stay put if it comes up heads, and apply the transition matrix if it comes up tails. Then it’s possible to get from any state to itself in one step, and so we are by construction aperiodic.

Enhanced by Zemanta

Nested Closed Intervals

Th UK team for this year’s International Mathematical Olympiad in Santa Marta, Colombia, has just been selected. For details, see the BMOC website.

During the selection camp, which was hosted at Oundle School near Peterborough, we spent a while discussing analytic questions that typically lie outside the scope of the olympiad syllabus. Furthermore, incorrect consideration of, for example, the exact conditions for a stationary point to be a global maximum, are likely to incur very heavy penalties if a candidate has attempted a solution using, for example, Lagrange multipliers. As a result we have a dilemma about how much analysis to teach during the training process: we want the students to be able to use sophisticated methods if necessary; but we don’t want to spoil the experience of learning this theory in a serious step-by-step manner as first year undergraduates.

This post does not present a solution to this dilemma. Rather, I want to discuss one question that arose on the last day of exams. Because the statement of the question is currently classified, I will have to be oblique in discussion of the solution, but this shouldn’t distract from the maths I actually want to talk about.

The setup is as follows. We have a sequence of nested closed intervals in the reals, that is:

[a_1,b_1]\supset [a_2,b_2]\supset [a_3,b_3]\supset\ldots

We want to demonstrate that there is some real number that lies in all of the intervals, that is \cap_{n\geq 1}[a_n,b_n]\neq \varnothing. This feels intuitively obvious, but some form of proof is required.

First, what does closed mean? Well a closed interval is a closed set, but what about, for example, [a,\infty)\subset\mathbb{R}? It turns out that it is most convenient to define an open set, and then take a closed set to be the complement of an open set.

The best way of thinking about an open set is to say that it does not contain its boundary. This is certainly the case for an open interval or an open ball. It is not immediately clear how to extend this to a general definition. But note that if no point in the set X can be on the boundary, then in all the natural examples to consider there must some finite distance between any point x\in X and the boundary. So in particular there is a small open ball centred on x that is entirely contained within X. This is then the definition of an open set in a metric space (in particular for some \mathbb{R}^d).

Note that it is not a sensible definition to say that a closed set has the property that there is a closed ball containing each point. Any open set has this property also! For if there is an open ball of radius R around a point x, then there is a closed ball of radius R/2 around that same point. So we really do have to say that a set is closed if the complement is open. Note that in \mathbb{R}, a closed interval is closed, and a finite union of closed intervals is closed, though not a countable union as:

(0,1]=\cup_{n\geq 1}[\frac{1}{n},1].

Now we know what a closed set is, we can start thinking about the question.

First we remark that it is not true if we allow the interval to be unbounded. Obviously \cap_n [n,\infty)=\varnothing. Note that even though it appears that these sets have an open upper boundary, they are closed because the complement (-\infty,n) is open. This will not be a problem in our question because everything is contained within the first interval [a_1,b_1] and so is bounded.

Secondly we remark that the result is not true if we move to a general host set. For example, it makes sense to consider open and closed sets in the rationals. For example, the open ball radius 1 either side of 1/2 is just all the rationals strictly between -1/2 and 3/2. We could write this as (-\frac12,\frac32)\cap\mathbb{Q}. But then note that

\cap_{n\geq 1}[\pi-\frac{1}{n},\pi+\frac{1}{n}]\cap\mathbb{Q}

cannot contain any rational elements, so must be empty.

There was various talk that this result might in general be a consequence of the Baire Category Theorem. I think this is overkill. Firstly, there is a straightforward proof for the reals which I will give at the end. Secondly, if anything it is a lemma required for the proof of BCT. This result is often called Cantor’s Lemma.

Lemma (Cantor): Let X be a complete metric space, and let F_1\supset F_2\supset\ldots be a nested sequence of non-empty closed subsets of X with \text{diam}(F_n)\rightarrow 0. There there exists x\in X such that

\cap F_n=\{x\}.

Translation: for ‘complete metric space’ think ‘the reals’ or \mathbb{R}^d. The diameter is, unsurprisingly, the largest distance between two points in the set. For reasons that I won’t go into, the argument for the olympiad question gave \text{diam}(F_{n+1})\leq \frac12\text{diam}(F_n) so this certainly holds here.

Proof: With reference to AC if necessary, pick an element x_n\in F_n for all n. Note that by nesting, x_m\in F_n\;\forall n\leq m. As a result, for m>n the distance d(x_n,x_m)\leq \text{diam}(F_n). This tends to 0 as n grows. The definition of complete is that such a sequence then has a limit point x.

Informally, completeness says that if a sequence of points get increasingly close, they must tend towards a point in the set. This is why it doesn’t work for the rationals. You can have a sequence of rationals that get very close together, but approach a point not in the set, eg an irrational. We use the definition of closed sets in terms of sequences: if the sequence is within a closed set, then the limit is too. This could only go wrong if we ‘leaked onto the boundary’ in the limit, but for a closed set, the boundary is in the set. This shows that x\in F_n for each n, and so $x\in\cap_n F_n$. But if there is another point in \cap_n F_n, then the distance between them is strictly positive, contradicting the claim that diameter tends to 0. This ends the proof.

Olympiad-friendly version: I think the following works fine as a fairly topology definition-free proof. Consider the sequence of left-boundaries

a_1\leq a_2\leq a_3\leq \ldots <b_1.

This sequence is non-decreasing and bounded, so it has a well-defined limit. Why? Consider the supremum. We can’t exceed the sup, but we must eventually get arbitrarily close, by definition of supremum and because the sequence is non-decreasing. Call this limit a. Then do the same for the upper boundaries to get limit b.

If a>b, then there must be some a_n>b_n, which is absurd. So we must have some non-empty interval as the intersection. Consideration of the diameter again gives that this must be a single point.

Bayesian Inference and the Jeffreys Prior

Last term I was tutoring for the second year statistics course in Oxford. This post is about the final quarter of the course, on the subject of Bayesian inference, and in particular on the Jeffreys prior.

There are loads and loads of articles sitting around on the web contributing the debate about the relative merits of Bayesian and frequentist methods. I do not want to continue that debate here, partly because I don’t have a strong opinion, but mainly because I don’t really understand that much about the underlying issues.

What I will say is that after a few months of working fairly intensively with various complicated stochastic processes, I am starting to feel fairly happy throwing about conditional probability rather freely. When discussing some of the more combinatorial models for example, quite often we have no desire to compute or approximate complication normalising constants, and so instead talk about ‘weights’. And a similar idea underlies Bayesian inference. As in frequentist methods we have an unknown parameter, and we observe some data. Furthermore, we know the probability that such data might have arisen under any value of the parameter. We want to make inference about the value of the parameter given the data, so it makes sense to multiply the probability that the data emerged as a result of some parameter value by some weighting on the set of parameter values.

In summary, we assign a prior distribution representing our initial beliefs about the parameter before we have seen any data, then we update this by weighting by the likelihood that the observed data might have arisen from a particular parameter. We often write this as:

\pi(\theta| x)\propto f(x|\theta)\pi(\theta),

or say that posterior = likelihood x prior. Note that in many applications it won’t be necessary to work out what the normalising constant on the distribution ought to be.

That’s the setup for Bayesian methods. I think the general feeling about the relative usefulness of such an approach is that it all depends on the prior. Once we have the prior, everything is concrete and unambiguously determined. But how should we choose the prior?

There are two cases worth thinking about. The first is where we have a lot of information about the problem already. This might well be the case in some forms of scientific research, where future analysis aims to build on work already completed. It might also be the case that we have already performed some Bayesian calculations, so our current prior is in fact the posterior from a previous set of experiments. In any case, if we have such an ‘informative prior’, it makes sense to use it in some circumstances.

Alternatively, it might be the case that for some reason we care less about the actual prior than about the mathematical convenience of manipulating it. In particular, certain likelihood functions give rise to conjugate priors, where the form of the posterior is the same as the form of the prior. For example, a normal likelihood function admits a normal conjugate prior, and a binomial likelihood function gives a Beta conjugate prior.

In general though, it is entirely possible that neither of these situations will hold but we still want to try Bayesian analysis. The ideal situation would be if the choice of prior had no effect on the analysis, but if that were true, then we couldn’t really be doing any Bayesian analysis. The Jeffreys prior is one natural candidate because it removes a specific problem with choosing a prior to express ignorance.

It sounds reasonable to say that if we have total ignorance about the parameter, then we should take the prior to be uniform on the set of possible values taken by the parameter. There are two potential objections to this. The first is that if the parameter could take any real value, then the prior will not be a distribution as the uniform distribution on the reals is not normalisable. Such a prior is called improper. This isn’t a huge problem really though. For making inference we are only interested in the posterior distribution, and so if the posterior turns out to be normalisable we are probably fine.

The second problem is more serious. Even though we want to express ignorance of the parameter, is there a canonical choice for THE parameter? An example will make this objection more clear. Suppose we know nothing about the parameter T except that it lies in [0,1]. Then the uniform distribution on [0,1] seems like the natural candidate for the prior. But what if we considered T^100 to be the parameter instead? Again if we have total ignorance we should assign T^100 the uniform distribution on its support, which is again [0,1]. But if T^100 is uniform on [0,1], then T is massively concentrated near 1, and in particular cannot also be uniformly distributed on [0,1]. So as a minimum requirement for expressing ignorance, we want a way of generating a prior that doesn’t depend on the choice of parameterisation.

The Jeffreys prior has this property. Note that there may be separate problems with making such an assumption, but this prior solves this particular objection. We define it to be \pi(\theta)\propto [I(\theta)]^{1/2} where I is the Fisher information, defined as

I(\theta)=-\mathbb{E}_\theta\Big[\frac{\partial^2 l(X_1,\theta)}{\partial \theta^2}\Big],

where the expectation is over the data X_1 for fixed \theta, and l is the log-likelihood. Proving that this has the property that it is invariant under reparameterisation requires demonstrating that the Jeffreys prior corresponding to g(\theta) is the same as applying a change of measure to the Jeffreys prior for \theta. The proof is a nice exercise in the chain rule, and I don’t want to reproduce it here.

For a Binomial likelihood function, we find that the Jeffreys prior is Beta(1/2,1/2), which has density that looks roughly like a bucket suspended above [0,1]. It is certainly worth asking why the ‘natural’ choice for prior might put lots of mass at the edge of the domain for the parameter.

I don’t have a definitive answer, but I do have an intuitive idea which comes from the meaning of the Fisher information. As the second derivative of the log-likelihood, a large Fisher information means that with high probability we will see data for which the likelihood changes substantially if we vary the parameter. In particular, this means that the posterior probability of a parameter close to 0 will be eliminated more quickly by the data if the true parameter is different.

If the variance is small, as it is for parameter near 0, then the data generated by this parameter will have the greatest effect on the posterior, since the likelihood will be small almost everywhere except near the parameter. We see the opposite effect if the variance is large. So it makes sense to compensate for this by placing extra prior mass at parameter values where the data has the strongest effect. Note that in the previous example, the Jeffreys prior is in fact exactly inversely proportional to the standard deviation. For the above argument to make sense, we need it to be monotonic with respect to SD, and it just happens that in this case, being 1/SD is precisely the form required to be invariant under reparameterisation.

Anyway, I thought that was reasonably interesting, as indeed was the whole course. I feel reassured that I can justify having my work address as the Department of Statistics since I now know at least epsilon about statistics!

Generating Functions for the IMO

The background to this post is that these days I find myself using generating functions all the time, especially for describing the stationary states of various coalescence-like processes. I remember meeting them vaguely while preparing for the IMO as a student. However, a full working understanding must have eluded me at the time, as for Q5 on IMO 2008 in Madrid I had written down in big boxes the two statements involving generating functions that immediately implied the answer, but failed to finish it off. The aim of this post is to help this year’s team avoid that particular pitfall.

What are they?

I’m going to define some things in a way which will be most relevant to the type of problems you are meeting now. Start with a sequence (a_0,a_1,a_2,\ldots). Typically these will be the sizes of various combinatorial sets. Eg a_n = number of partitions of [n] with some property. Define the generating function of the sequence to be:

f(x)=\sum_{k\geq 0}a_k x^k=a_0+a_1x+a_2x^2+\ldots.

If the sequence is finite, then this generating function is a polynomial. In general it is a power series. As you may know, some power series can be rather complicated, in terms of where they are defined. Eg

1+x+x^2+x^3+\ldots=\frac{1}{1-x},

only when |x|<1. For other values of x, the LHS diverges. Defining f over C is fine too. This sort of thing is generally NOT important for applications of generating functions to combinatorics. To borrow a phrase from Wilf, a generating function is a convenient `clothesline’ on which to hang a sequence of numbers.

We need a notation to get back from the generating function to the coefficients. Write [x^k]g(x) to denote the coefficient of x^k in the power series g(x). So, if g(x)=3x^3-5x^2+7, then [x^2]g(x)=-5. It hopefully should never be relevant unless you read some other notes on the topic, but the notation [\alpha x^2]g(x):=\frac{[x^2]g(x)}{\alpha}, which does make sense after a while.

How might they be useful?

Example: binomial coefficients a_k=\binom{n}{k} appear, as the name suggests, as coefficients of

f_n(x)=(1+x)^n=\sum_{k=0}^n \binom{n}{k}x^k.

Immediate consequence: it’s trivial to work out \sum_{k=0}^n \binom{n}{k} and \sum_{k=0}^n(-1)^k \binom{n}{k} by substituting x=\pm 1 into f_n.

Less obvious consequence. By considering choosing n from a red balls and b blue balls, one can verify

\binom{a+b}{n}=\sum_{k=0}^n \binom{a}{k}\binom{b}{n-k}.

We can rewrite the RHS as

\sum_{k+l=n}\binom{a}{k}\binom{b}{l}.

Think how we calculate the coefficient of x^n in the product f(x)g(x), and it is now clear that \binom{a+b}{n}=[x^n](1+x)^{a+b}, while

\sum_{k+l=n}\binom{a}{k}\binom{b}{l}=[x^n](1+x)^a(1+x)^b,

so the result again follows. This provides a good slogan for generating functions: they often replicate arguments via bijections, even if you can’t find the bijection.

Useful for? – Multinomial sums

The reason why the previous argument for binomial coefficients worked nicely is because we were interested in the coefficients, but had a neat expression for the generating function as a polynomial. In particular, we had an expression

\sum_{k+l=n}a_k b_l.

This is always a clue that generating functions might be useful. This is sometimes called a convolution.

Exercise: prove that in general, if f(x) is the generating function of (a_k) and g(x) the generating function of (b_l), then f(x)g(x) is the generating function of \sum_{k+l=n}a_kb_l.

Even more usefully, this works in the multinomial case:

\sum_{k_1+\ldots+k_m=n}a^{(1)}_{k_1}\ldots a^{(m)}_{k_m}.

In many applications, these a^{(i)}s will all be the same. We don’t even have to specify how many k_i’s there are to be considered. After all, if we want the sum to be n, then only finitely many can be non-zero. So:

\sum_{m}\sum_{k_1+\ldots+k_m=n}a_{k_1}\ldots a_{k_m}=[x^n]f(x)^n=[x^n]f(x)^\infty,

provided f(0)=1.

Useful when? – You recognise the generating function!

In some cases, you can identify the generating function as a `standard’ function, eg the geometric series. In that case, manipulating the generating functions is likely to be promising. Here is a list of some useful power series you might spot.

1+x+x^2+\ldots=\frac{1}{1-x},\quad |x|<1

1+2x+3x^2+\ldots=\frac{1}{(1-x)^2},\quad |x|<1

e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots

\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}\pm\ldots

Exercise: if you know what differentiation means, show that if f(x) is the gen fn of (a_k), then xf'(x) is the gen fn of ka_k.

Technicalities: some of these identities are defined only for certain values of x. This may be a problem if they are defined at, say, only a single point, but in general this shouldn’t be the case. In addition, you don’t need to worry about differentiability. You can definition differentiation of power series by x^n\mapsto nx^{n-1}, and sort out convergence later if necessary.

Useful for? – Recurrent definitions

The Fibonacci numbers are defined by:

F_0=F_1=1,\quad F_{n+1}=F_n+F_{n-1},\quad n\geq 1.

Let F(x) be the generating function of the sequence F_n. So, for n=>1,

[x^n]F(x)=[x^{n-1}]F(x)+[x^{n-2}]F(x)=[x^n](xF(x)+x^2F(x)),

and F(0)=1, so we can conclude that:

F(x)=1+(x+x^2)F(x)\quad\Rightarrow\quad F(x)=\frac{1}{1-x-x^2}.

Exercise: Find a closed form for the generating function of the Catalan numbers, defined recursively by:

C_n=C_0C_{n-1}+C_1C_{n-2}+\ldots+C_{n-1}C_0.

Can you now find the coefficients explicitly for this generating function?

Useful for? – Partitions

Partitions can be an absolute nightmare to work with because of the lack of explicit formulae. Often any attempt at a calculation turns into a massive IEP bash. This prompts a search for bijective or bare-hands arguments, but generating functions can be useful too.

For now (*), let’s assume a partition of [n] means a sequence of positive integers a_1\geq a_2\geq\ldots\geq a_k such that a_1+\ldots+a_k=n. Let p(n) be the number of partitions of [n].

(* there are other definitions, in terms of a partition of the set [n] into k disjoint but unlabelled sets. Be careful about definitions, but the methods often extend to whatever framework is required. *)

Exercise: Show that the generating function of p(n) is:

\left(\frac{1}{1-x}\right)\left(\frac{1}{1-x^2}\right)\left(\frac{1}{1-x^3}\right)\ldots

Note that if we are interested only in partitions of [n], then we don’t need to consider any terms with exponent greater than n, so if we wanted we could take a finite product instead.

Example: the mint group will remember this problem from the first session in Cambridge:

Show that the number of partitions of [n] with distinct parts is equal to the number of partitions of [n] with odd parts.

Rather than the fiddly bijection argument found in the session, we can now treat this as a simple calculation. The generating function for distinct parts is given by:

(1+x)(1+x^2)(1+x^3)\ldots,

while the generating function for odd parts is given by:

\left(\frac{1}{1-x}\right)\left(\frac{1}{1-x^3}\right)\left(\frac{1}{1-x^5}\right)\ldots.

Writing the former as

\left(\frac{1-x^2}{1-x}\right)\left(\frac{1-x^4}{1-x^2}\right)\left(\frac{1-x^6}{1-x^3}\right)\ldots

shows that these are equal and the result follows.

Other things – Multivariate Generating Functions

If you want to track a sequence in two variables, say a_{m,n}, then you can encode this with the bivariate generating function

f(x,y):=\sum_{m,n\geq 0}a_{m,n}x^my^n.

The coefficients are then extracted by [x^ay^b] and so on. There’s some interesting stuff on counting lattice paths with this method.

Sums over arithmetic progressions via roots of unity

Note that we can extract both \sum a_n and \sum (-1)^na_n by judicious choice of x in f(x). By taking half the sum or half the difference, we can obtain

a_0+a_2+a_4+\ldots=\frac12(f(1)+f(-1)),\quad a_1+a_3+a_5+\ldots=\frac12(f(1)-f(-1)).

Can we do this in general? Yes actually. If you want a_0+a_k+a_{2k}+\ldots, this is given by:

a_0+a_k+a_{2k}+\ldots+\frac{1}{k}\left(f(1)+f(w)+\ldots+f(w^{k-1})\right),

where w=e^{2\pi i/k} is a $k$th root of unity. Exercise: Prove this.

For greater clarity, first try the case k=4, and consider the complex part of the power series evaluated at +i and -1.