While phase transitions have been discussed at length, no phase transition has been rigorously established up to this point. This gap is filled in this final chapter, where comparison with oriented percolation is used to prove that contact processes with sufficiently large infection rates survive. This result depends on a comparison between k-dependent and independent random variables due to Ligett, Schonmann, and Stacey, which is proved in detail. In the final exercises, the reader is asked to apply the method to other interacting particle systems, such as a model with cooperative branching.

This chapter introduces neural networks as flexible function approximators built by composing layers of simple processing units. A network with no hidden layers performs linear regression if its output layer is linear and logistic regression if its output layer uses softmax. Hidden layers increase expressivity: a network with one hidden layer and ReLU activations can approximate any continuous function on a closed and bounded input domain, though complex functions may require many units. Deep networks, with multiple hidden layers, are more efficient and scalable than shallow ones, especially for learning hierarchical structure. Neural networks are trained using gradient-based optimisation, with gradients computed via backpropagation. Training adjusts weights to minimise a loss function, using small batches of data. Techniques like early stopping and small batches act as implicit regularisers, while weight decay provides explicit regularisation. Convolutional neural networks use convolution and pooling layers to exploit spatial structure in image data. More broadly, architectural choices often reflect domain-specific assumptions.

Duality is one of the most basic tools in the study of interacting particle systems. There exist two forms: pathwise and distributional duality, of which the first is the stronger. It is shown how pathwise dualities naturally follow from the basic Poisson construction of Chapter 4 by looking backward in time. The classical additive and cancellative dualities, which are pathwise dualities, are discussed in detail. As an example of dualities that are not pathwise, Lloyd–Sudbury duality is discussed. It is demonstrated how duality can be used to prove various results, such as clustering in the voter model, uniqueness of a homogeneous nontrivial invariant law for the contact process, and equality of the critical points for finite survival and for the existence of a nontrivial invariant law in a contact-voter model.

This chapter introduces directed acyclic graphs (DAGs) as a way to represent multivariate probability distributions. DAGs help clarify the structure of probabilistic models and the dependencies among their variables and serve as a central tool in later chapters. Every DAG corresponds to a specific factorisation of a joint mass or density function into a product of conditional distributions. While a DAG encodes how the distribution breaks down into conditionals, it does not fully determine the distribution itself. Instead, it implies certain dependency constraints among variables. These constraints can be examined using the concept of d-separation, which allows us to infer conditional independence relationships directly from the graph.

An introduction of the four classical models: the voter model, the contact process, stochastic Ising models, and the exclusion process, complemented by some variations on the classical models such as the two-stage contact process or biased and rebellious voter models, as well as some other models such as systems of branching, coalescing, and annihilating random walks and an example of a kinetically constrained model. Some well-known major results are mentioned together with many open problems and topics that are from the mathematical side and still poorly understood, such as the critical exponents associated with continuous phase transition and (in the final section) periodic behavior.

This chapter introduces key concepts and methods in Bayesian statistical modelling. The posterior predictive distribution captures both epistemic uncertainty in model parameters and aleatory uncertainty in future outcomes. A Bayesian p-value gives the probability that a statistic computed from data output by a given model will be more extreme than the value of the same statistic computed from observed data. Bayesian p-values close to 0 or 1 suggest the model may be inadequate. Markov chain Monte Carlo is a general-purpose tool for sampling from complex, unnormalised distributions. It produces dependent samples, so the effective sample size is usually smaller than the number of iterations. Informative priors are useful when data leave large uncertainties in parameter values. Empirical Bayes combines information across related datasets by estimating a distribution over parameters using frequentist methods. Hierarchical modelling provides a unified Bayesian framework for handling multiple related datasets, capturing group structure via a hierarchical graph.

This central chapter builds on Chapter 2 to give sufficient conditions for an interacting particle system to be uniquely defined by a Poisson point process called a graphical representation. The generator construction is also discussed. The basic construction result is applied to prove that interacting particle systems with sufficiently weak interactions are ergodic in the sense that they have a unique invariant law that is the long-time limit started from any initial state. The theory is demonstrated on a stochastic Ising model, leading to a proof that the magnetization is zero at sufficiently high temperatures.

In this chapter, we explore an unsupervised learning problem: estimating a distribution function from two-dimensional data. Although there is no response variable, the workflow mirrors that of supervised learning. We select the best-fitting function within a family by maximising the sum of the log of the distribution's values at the observed data points. As in supervised learning, excessive flexibility leads to overfitting, while insufficient flexibility leads to underfitting. We use cross-validation to identify a function family that achieves a happy medium.

This chapter introduces simple and multiple linear regression models – core tools in predictive modelling due to their simplicity and interpretability. These models assume the response variable is a linear function of the predictor(s), plus a noise term. The regression function gives the expected response given the predictors. The coefficient of determination, R2, measures how much of the variance in the response is explained by the model. In simple linear regression, R2 equals the square of the Pearson correlation between response and predictor; in multiple regression, it equals the square of the correlation between response and predicted values. Each coefficient in multiple regression reflects the expected change in the response for a one-unit increase in that predictor, holding others fixed. Standardising predictors lets us compare coefficient sizes. Strong collinearity between predictors increases uncertainty in the fitted coefficients. Models using only a subset of predictors may generalise better than those using all and overfitting. The squared error risk of a modelling procedure – its expected test error – can be broken down into bias, variance and irreducible noise.