We provide a lay-language primer on the counterfactual model of causal inference and the logic of causal models. Topics include the representation of causal models with causal graphs and using causal graphs to read off relations of conditional independence among variables in a causal domain.

Earlier chapters introduced modeling approaches for a continuous, normally distributed response. Biological data are often not so neat, and the common practice was to transform continuous response variables until the assumption of normality was met. Other kinds of data, particularly presence–absence and behavioral responses and counts, are discrete rather than continuous and require a different approach. In this chapter, we introduce generalized linear models and their extension to generalized linear mixed models to analyze these response variables. We show how common techniques such as contingency tables, loglinear models, and logistic and Poisson regression can be viewed as generalized linear models, using link functions to create the appropriate relationship between response and predictors. The models described in earlier chapters can be reinterpreted as a version of generalized linear models with the identity link function. We finish by introducing generalized additive models for where a linear model may be unsuitable.

Spatial Data Aggregation is defined in our book as the integration of heterogeneous spatial or spatio-temporal data sources, with the aim of predicting the occurrence of hazards and resources. Much of the transition to net zero carbon relies on changing from fossil fuels to materials for renewable energy and batteries. Mineral exploration is therefore key to achieve this goal. Readers will engage in an active mineral exploration for battery metals in Cape Smith, Canada. Readers will learn how data science can be an effective guide in geological field work. To achieve this, several spatial information sources will be used, such as remote sensing, geophysical, and geochemical data. These sources need to be aggregated to guide field geologists in locating areas of interest with the aim of collecting samples. In that context, we introduce Bayes’ rule and Bayesian reasoning about knowledge and information. We emphasize the counterintuitive results of Bayesian reasoning: rare events are very difficult to predict even with very accurate data. Next, we cover the alternative to Bayes’: logistic regression. We emphasize the advantage and disadvantage of these opposite approaches.

Social inquiry needs representations of social reality, and such representations involve ontological choices of how much to freeze history and geography. The freezing metaphor relates to how much social inquiry relaxes conditional independence and unit homogeneity, the two key assumptions of VBA. Unfreezing history and geography brings to light historical and geographic particularities, makes possible new inductive insights into updating theories, and generates new research questions. Comparisons help turn such explorations into discoveries by identifying patterns. CHA employs four comparisons: cross-sectional, serial, contextual, and historical. It thus differs from VBA, which uses comparison for the strictly methodological purpose of controlling for confounders. CHA goes back to the late nineteenth century and has explored four key themes: the evolution of capitalism, regime changes, the administrative transformation of states, and the changing nature of war. CHA does not have a monopoly on historical thinking. VBA employs historical thinking when exploring confounders and historical boundary conditions that might bias their test results.

Data assimilation is a procedure for combining observations and forecasts of a system into a single, improved description of the system state. Because observations and forecasts are uncertain, they are each best described by probability distributions. The problem of combining these two distributions into a new, updated distribution that summarizes all our knowledge is solved by Bayes theorem. If the distributions are Gaussian, then the parameters of the updated distribution can be written as an explicit function of the parameters of the observation and forecast distributions. The assumption of Gaussian distributions is tantamount to assuming linear models for observations and state dynamics. The purpose of this chapter is to provide an introduction to the essence of data assimilation. Accordingly, this chapter discusses the data assimilation problem for Gaussian distributions in which the solution from Bayes theorem can be derived analytically. Practical data assimilation usually requires modifications of this assimilation procedure, a special case of which is discussed in the next chapter.