probability-aez-notes

A \(\sigma\)-field, \(\Sigma\), over a set \(\Omega\) is a subset of the power set of \(\Omega\) (ie it is a set of subsets of \(\Omega\)) with three properties:

1. \(\Sigma\) contains \(\Omega\),
2. \(\Sigma\) is closed under complements, and
3. \(\Sigma\) is closed under countable unions.

Given a set, \(\Omega\) and a \(\sigma\)-field, \(\Sigma\), over it, a real valued function, \(\mu\) is a measure if

1. \(\mu\) is non-negative,
2. \(\mu(\emptyset) = 0\), and
3. \(\mu\) has countable additivity, ie the measure of a union of a countable set of disjoint sets is the sum of the measures of them.

A probability measure is a measure where \(\mu(\Omega) = 1\).

For a probability measure if \(P(B) > 0\), then the probability of \(A\) given \(B\) is defined by

\[ P(A\mid B) = \frac{P(A \cap B)}{P(B)} \]

This is a continuous time process which counts the arrivals which occur at exponentially distributed times with rate \(\lambda\).

For a discrete random variable \(X\) taking values from the non-negative integers, the probability generating function is defined as

\[ G(z) = \mathbb{E}(z^{X}) = \sum_{x=0}^{\infty} f_X(x) z^{x} \]

where \(f_X\) is the probability mass function (PMF).

Consider a process where individuals give birth with rate \(\lambda\). In this case the rate at which the population increases in size depends on the current population size. The master equation for this is

\[ \frac{dp_i}{dt} = \lambda (i - 1) p_{i-1} - \lambda i p_i \]

for \(i\geq n\) where the process starts with a population of size \(n\): \(p_{n}(0) = 1\). Multiplying through by \(z^i\) and summing over \(i\) gives the following PDE for the PGF, \(g(z,t)\)

\[ \frac{\partial g}{\partial t} = \lambda (z^2 - z) \frac{\partial g}{\partial z}. \]

Given the initial condition, the boundary condition for the PDE is \(g(z,0) = z^n\). The method of characteristics looks suitable for solving this; the function, \(g\), will be constant along the curves satisfying

\[ \frac{dz}{dt} = \lambda (z - z^2). \]

(Hopefully you remember how to do partial fractions!) This leads us to conclude that the function is constant upon curves where

\[ C = \frac{z \exp\{ -\lambda t\}}{1 - z (1 - \exp\{ -\lambda t\})}. \]

So we know that \(g\) must be a function of the RHS of the previous equation. With the initial condition \(n=1\) — which due to independence is usually sufficient to consider — that function is the identity function so

\[ g(z,t) = \frac{z \exp\{ -\lambda t\}}{1 - z (1 - \exp\{ -\lambda t\})}. \]

From this we can then see that

\[ \lim _{z\to 1} g_{z}(z,t) = \exp\{\lambda t\} \]

which agrees with the mean-field approximation. To check that this is actually a solution to the PDE, we can use the following little bit of Maxima

define(g(z,t), z * exp(-l * t) / (1 - z * ( 1- exp(-l * t))));

is(equal(l * z * (z - 1) * diff(g(z,t), z), diff(g(z,t), t)));
is(equal(g(z,0), z));

Consider a process where individuals give birth at a rate \(\lambda\) and die at a rate \(\mu\). The master equation for this is

\[ \frac{dp_i}{dt} = \lambda (i - 1) p_{i-1} - \lambda i p_{i} + \mu (i + 1) p_{i+1} - \mu i p_{i}. \]

with the condition that \(p_i=0\) for \(i<0\). Multiplying through by \(z^i\) and summing over \(i\) gives the following PDE for the PGF, \(g(z,t)\)

\[ \frac{\partial g}{\partial t} = (1 - z) ( \mu - \lambda z) \frac{\partial g}{\partial z}. \]

Given the initial condition, the boundary condition for the PDE is \(g(z,0) = z^n\). The method of characteristics looks suitable for solving this; the function, \(g\), will be constant along the curves satisfying

\[ \frac{dz}{dt} = (1 - z) ( \mu - \lambda z). \]

(Hopefully you remember how to do partial fractions!) This leads us to conclude that the function is constant upon curves where

\[ C = \frac{(\lambda z - \mu) \exp\{ - (\lambda - \mu) t \} - \mu (z-1)}{(\lambda z - \mu) \exp\{ - (\lambda - \mu) t \} - \lambda (z-1)}. \]

So we know that the solution can only be a function of the RHS of the previous equation. Since this simplifies to \(z\) at \(t=0\) and assuming the initial condition \(n=1\) we get the solution

\[ g(z,t) = \frac{(\lambda z - \mu) \exp\{ - (\lambda - \mu) t \} - \mu (z-1)}{(\lambda z - \mu) \exp\{ - (\lambda - \mu) t \} - \lambda (z-1)}. \]

We can then test this in maxima with the following

define(g(z,t), ((b * z - a) * exp(- (b - a) * t) - a * ( z- 1)) / ((b * z - a) * exp(- (b - a) * t) - b * ( z- 1)));

is(equal((1-z)*(b*z-a) * diff(g(z,t), z), -diff(g(z,t), t)));
is(equal(g(z,0), z));

Taking the derivative and the limit we get that the expected population size is \(exp\{(λ-μ)t\} which again agrees with the mean field approximation.

For notes on moment closure, see my statistics notes.

This is a particularly nifty trick when dealing with tricky MCMC problems, see my statistics notes.

\[ \sum_{k=0}^{\infty} r^{k} = \frac{1}{1 - r} \]

for \(|r| < 1\).

Let \(\{a_n\}\) and \(\{c_n\}\) be two sequences which converge to \(l\). If there is an \(N\) such that for all \(n\geq N\) we have \(a_n \leq b_n \leq c_n\), then \(\{b_n\}\) also converges to \(l\).

• The Dirichlet process is not from the Dirichlet distribution.
• There are three main ways to think about the DP:
  • the Chinese restraunt process,
  • the stick-breaking process,
  • and the Polya urn scheme
• The DP has "rich-get-richer" behaviour.
• The DP is defined by a base distribution, \(H\), and a scaling parameter, \(\alpha\).
  • For larger values of \(\alpha\) the distribution looks more like \(H\), for smaller values it looks like a resampled sample from \(H\) where previously sampled values become more likely to be resampled.
• This has found substantial use in Bayesian nonparameteric inference where it is used as a prior.

This distribution is constrained to have a non-negative correlation. See Karlis and Ntzoufras (2003) for an example.

See Famoye (2010) for a bivariate negative binomial distribution constructed using the multiplicative method described above.

This distribution comes from Edwards and Gurland (1961), who call it the compound correlated bivariate Poisson distribution. The PGF of this distribution is

\[ g_{X_1, X_2}(z_1, z_2) = \left( A + B_1 z_1 +B_2 z_2 + B_{12} z_1 z_2 \right)^{-k} \]

where \(A = 1 - B_1 - B_2 - B_{12}\). The PGF can then be used to get the PGFs of the marginal distributions and from this we can get the MGF and hence the moments.

Although Edwards and Gurland (1961) provide expressions for the PMF, they suggest that the recurrence relations that they provide can provide a computationally more convenient approach.