26 Likelihood

26.1 Motivation

So far, we have described point processes through quantities such as:

the intensity function \(\lambda(u)\)
the second-order product density \(\lambda^{(2)}(u,v)\)
the pair correlation function \(g(u,v)\)
the \(K\)-function

These summaries are extremely useful, since they describe the first- and second-order structure of a point process.

However, when we come to statistical inference, a more basic question arises:

Given an observed point pattern \(X\), how do we write down the probability of observing that pattern under a model with parameter \(\theta\)?

This leads to the idea of a likelihood function.

In ordinary parametric statistics, likelihood-based inference is often the default approach.
For point processes, the same idea applies, but the form of the likelihood is slightly different, because the data consist of a random configuration of points in space, rather than a finite-dimensional vector.

Note

A likelihood function tells us how plausible the observed data are under different parameter values.

In a spatial point process setting, the observed data are typically the point locations \[ X = \{x_1,\dots,x_n\} \subset W, \] where \(W \subset \mathbb{R}^d\) is the observation window.

The aim of this chapter is to introduce the likelihood for point processes, beginning with the Poisson process, and then extending the discussion to Cox processes.

26.2 What is the data?

Suppose we observe a point process in a bounded window \(W \subset \mathbb{R}^d\).

The data consist of:

the number of observed points, say \(n\)
the locations of those points, say \(x_1,\dots,x_n\)

So the observed realization is

\[ X = \{x_1,\dots,x_n\}. \]

Unlike ordinary data analysis, the sample size \(n\) is itself random.

This is one of the first conceptual differences between point process likelihoods and more familiar likelihoods from regression or classical parametric models.

26.3 Likelihood in a point process setting

Roughly speaking, the likelihood is the density of the observed configuration \(X\) under the model.

For a point process observed on \(W\), we write

\[ L(\theta; X) \]

for the likelihood of parameter \(\theta\) given the observed point pattern \(X\).

In general, one must be a little careful about exactly what this density is taken with respect to.
For now, we will focus on the standard heuristic form of the likelihood, which is the one most commonly used in spatial statistics.

The key idea is simple:

we want the probability of observing points at \(x_1,\dots,x_n\)
and no points elsewhere in the window

For the Poisson process, this can be derived explicitly.

26.4 Poisson process likelihood

We begin with the most important benchmark case.

Let \(X\) be an inhomogeneous Poisson process on \(W\) with intensity function \(\lambda(u)\).

We now derive its likelihood.

26.5 Heuristic derivation via a fine partition

Partition the window \(W\) into many small cells

\[ B_1,\dots,B_m \]

with volumes \(|B_j|\).

Assume the partition is so fine that each cell contains at most one point with high probability.

For a Poisson process:

counts in disjoint cells are independent
for a small cell \(B_j\), \[ N(B_j) \sim \text{Poisson}\!\left(\int_{B_j}\lambda(u)\,du\right) \]

If the cell is very small, then

\[ \int_{B_j}\lambda(u)\,du \approx \lambda(u_j)|B_j| \]

for some representative point \(u_j \in B_j\).

So for a tiny cell \(B_j\):

probability of exactly one point: \[ \mathbb{P}(N(B_j)=1) \approx \lambda(u_j)|B_j|\,\exp(-\lambda(u_j)|B_j|) \]
probability of zero points: \[ \mathbb{P}(N(B_j)=0) \approx \exp(-\lambda(u_j)|B_j|) \]

Now suppose the observed pattern has points in exactly \(n\) cells, say those containing \(x_1,\dots,x_n\), and zero points in all other cells.

By independence of cell counts, the probability of this configuration is approximately

\[ \prod_{i=1}^n \Big[\lambda(x_i)|B_i|\,e^{-\lambda(x_i)|B_i|}\Big] \prod_{j \notin \{i_1,\dots,i_n\}} e^{-\lambda(u_j)|B_j|}. \]

Combining the exponential terms gives

\[ \left(\prod_{i=1}^n \lambda(x_i)|B_i|\right) \exp\left( -\sum_{j=1}^m \lambda(u_j)|B_j| \right). \]

As the partition becomes finer,

\[ \sum_{j=1}^m \lambda(u_j)|B_j| \to \int_W \lambda(u)\,du. \]

Ignoring the small cell-volume factors, which belong to the underlying reference measure, we obtain the likelihood

\[ L(\theta;X) \propto \exp\left(-\int_W \lambda(u)\,du\right) \prod_{i=1}^n \lambda(x_i). \]

This gives the standard Poisson process likelihood:

Important

For an inhomogeneous Poisson process with intensity \(\lambda(u)\), the likelihood is

\[ L(\theta;X) = \exp\left(-\int_W \lambda(u)\,du\right) \prod_{i=1}^n \lambda(x_i). \]

26.6 Interpreting the Poisson likelihood

The Poisson likelihood has two parts:

\[ L(\theta;X) = \exp\left(-\int_W \lambda(u)\,du\right) \prod_{i=1}^n \lambda(x_i). \]

26.6.1 The product term

The factor

\[ \prod_{i=1}^n \lambda(x_i) \]

rewards parameter values that assign high intensity to the observed point locations.

So if many observed points lie in regions where \(\lambda(u)\) is large, this term becomes larger.

26.6.2 The exponential term

The factor

\[ \exp\left(-\int_W \lambda(u)\,du\right) \]

penalises models with large total expected count over the window.

Recall that for a Poisson process,

\[ \mathbb{E}[N(W)] = \int_W \lambda(u)\,du. \]

So this term accounts for the fact that we observed no additional points elsewhere in the window.

Note

The Poisson likelihood balances two things:

fitting the observed points well
not predicting too many unobserved points

26.7 Homogeneous Poisson process as a special case

If the process is homogeneous, so that

\[ \lambda(u) \equiv \lambda, \]

then

\[ \int_W \lambda(u)\,du = \lambda |W|, \]

and the likelihood becomes

\[ L(\lambda;X) = e^{-\lambda |W|}\lambda^n. \]

This is exactly what we would expect, since in the homogeneous case the point locations are conditionally uniform given \(N(W)=n\), and the main contribution to the likelihood comes from the random count.

26.8 Log-likelihood

In practice, one usually works with the log-likelihood rather than the likelihood itself.

For the Poisson process,

\[ \ell(\theta;X) = \log L(\theta;X) = \sum_{i=1}^n \log \lambda(x_i) - \int_W \lambda(u)\,du. \]

This is much easier to manipulate analytically and numerically.

For example, if \(\lambda(u)=\lambda\) is constant, then

\[ \ell(\lambda;X)= n\log\lambda - \lambda |W|. \]

26.9 Why the Poisson likelihood is tractable

The Poisson process is special because its likelihood depends only on the intensity function \(\lambda(u)\).

There is no interaction term involving pairs or higher-order groups of points.

This tractability comes from the defining properties of the Poisson process:

conditional independence of counts in disjoint sets
absence of interaction between points

This is one reason why Poisson processes play such a central role in spatial statistics: they are both mathematically simple and inferentially convenient.

26.10 Moving to Cox processes

A Cox process is more complicated.

Recall that a Cox process is a Poisson process with a random intensity field \(\Lambda(u)\).

That is,

\[ X \mid \Lambda \sim \text{Poisson process with intensity } \Lambda(u). \]

So conditional on the latent field \(\Lambda\), the likelihood is just the Poisson likelihood with \(\lambda(u)\) replaced by \(\Lambda(u)\):

\[ L(\theta;X \mid \Lambda) = \exp\left(-\int_W \Lambda(u)\,du\right) \prod_{i=1}^n \Lambda(x_i). \]

This is called the conditional likelihood.

26.11 Conditional versus marginal likelihood

The important point is that the random field \(\Lambda(u)\) is not observed.

So the conditional likelihood is not yet the likelihood we can use directly for inference.

To obtain the actual likelihood of the observed point pattern, we must average over all possible realizations of the latent field.

This gives the marginal likelihood:

\[ L(\theta;X) = \mathbb{E}_\Lambda\left[ \exp\left(-\int_W \Lambda(u)\,du\right) \prod_{i=1}^n \Lambda(x_i) \right]. \]

This expectation is taken with respect to the distribution of the random intensity field \(\Lambda\).

Important

For a Cox process, the likelihood is obtained by integrating the Poisson likelihood over the distribution of the latent random field.

This is the key structural fact behind likelihood-based inference for Cox processes.

26.12 Why this is already much harder

Compare the two likelihoods.

For a Poisson process:

\[ L(\theta;X) = \exp\left(-\int_W \lambda(u)\,du\right) \prod_{i=1}^n \lambda(x_i), \]

where \(\lambda(u)\) is deterministic.

For a Cox process:

\[ L(\theta;X) = \mathbb{E}_\Lambda\left[ \exp\left(-\int_W \Lambda(u)\,du\right) \prod_{i=1}^n \Lambda(x_i) \right], \]

where \(\Lambda(u)\) is random.

The difficulty is that we are no longer evaluating a known function.
Instead, we must compute an expectation over an entire random field.

This is a fundamentally harder problem.

26.13 Example: likelihood of an LGCP

For a log-Gaussian Cox process,

\[ \Lambda(u)=\exp(Z(u)), \]

where \(Z(u)\) is a Gaussian random field.

Substituting into the Cox process likelihood gives

\[ L(\theta;X) = \mathbb{E}_Z\left[ \exp\left(-\int_W e^{Z(u)}\,du\right) \prod_{i=1}^n e^{Z(x_i)} \right]. \]

Equivalently,

\[ L(\theta;X) = \mathbb{E}_Z\left[ \exp\left( -\int_W e^{Z(u)}\,du + \sum_{i=1}^n Z(x_i) \right) \right]. \]

Even though \(Z(u)\) is Gaussian, this expression is not easy to evaluate, because of the nonlinear term

\[ \int_W e^{Z(u)}\,du. \]

26.14 Example: likelihood of a CSCP

For a single-component chi-square Cox process,

\[ \Lambda(u)=\mu + Z(u)^2, \]

where \(Z(u)\) is a Gaussian random field.

Then the likelihood becomes

\[ L(\theta;X) = \mathbb{E}_Z\left[ \exp\left(-\int_W (\mu + Z(u)^2)\,du\right) \prod_{i=1}^n (\mu + Z(x_i)^2) \right]. \]

Again, the structure is clear, but explicit evaluation is difficult.

26.15 What this chapter has established

The main message of this chapter is not yet that Cox process likelihoods are impossible to use, but rather that they take a very different form from ordinary Poisson likelihoods.

We now have the following picture:

For a Poisson process, the likelihood is explicit and tractable.
For a Cox process, conditional on the latent field, the likelihood is still Poisson.
But the latent field is unobserved, so the marginal likelihood requires integrating over all possible realizations of that field.

This immediately suggests the central question for the next chapter:

Can this likelihood actually be evaluated in closed form, or even computed efficiently?

In most interesting Cox process models, the answer is no.

That is the subject of the next chapter.