Starting from the biblical depiction of Christ’s incarnate personhood, this chapter traces the early church’s discernment of the deity and humanity of Christ that culminated in the Chalcedonian definition. Exploring the range of christological options, the chapter emphasizes the promise of a kenotic christology for making sense of Chalcedon’s intentions and parameters, including a fuller appreciation of Christ’s humanity.

This chapter articulates the identity of the church by way of key biblical metaphors and the Nicene Creed. It then examines the church’s mission, the role of proclamation and the sacraments, and the missional shape of the doctrine of election.

In this chapter, we consider inference and learning problems that can be formulated as the optimization of an information-theoretic measure (ITM).

As introduced in Chapter 4, setting up a learning problem requires the selection of an inductive bias, which consists of a model class $\mathscr{H}$ and a training algorithm. By the no-free-lunch theorem, this first step is essential in order to make generalization possible. A trained model generalizes if it performs well outside the training set, on average with respect to the unknown population distribution.

As discussed in Chapter 2, learning is needed when a “physics”-based mathematical model for the data generation mechanism is not available or is too complex to use for design purposes. As an essential benchmark setting, this chapter discusses the ideal case in which an accurate mathematical model is known, and hence learning is not necessary. As in large part of machine learning, we specifically focus on the problem of prediction. The goal is to predict a target variable given the observation of an input variable based on a mathematical model that describes the joint generation of both variables. Model-based prediction is also known as inference.

In brisk and engaging prose, this comprehensive introductory textbook traverses the broad sweep of US history since 1945. Winds of Hope, Storms of Discord explores how Americans of all walks of life –a political leaders, businesspeople, public intellectuals, workers, students, activists, migrants, and others – struggled to define the nation’s political, economic, geopolitical, demographic, and social character. It chronicles the nation’s ceaseless ferment, from the rocky conversion to peacetime in the early aftermath of World War II; to the frightening emergence of the Cold War and repeated US military adventures abroad; to the struggles of African Americans and other minorities to claim a share of the American Dream; to the striking transformations in social attitudes catalysed by the women’s movement and struggles for gay and lesbian liberation; to the dynamic force of political, economic, and social conservatism. Carrying the story to the spring of 2022, Winds of Hope also shows how dizzying technological changes at times threatened to upend the nation’s civic and political life.

This chapter considers the meaning of the claim that human beings are created in God’s image (imago Dei), offering a representational-relational conception of the doctrine. It also examines how the corruptive effects of sin deface humanity’s image-bearing, pointing to the need for the image to be restored through the person of Jesus Christ, the true image of God.

In brisk and engaging prose, this comprehensive introductory textbook traverses the broad sweep of US history since 1945. Winds of Hope, Storms of Discord explores how Americans of all walks of life –a political leaders, businesspeople, public intellectuals, workers, students, activists, migrants, and others – struggled to define the nation’s political, economic, geopolitical, demographic, and social character. It chronicles the nation’s ceaseless ferment, from the rocky conversion to peacetime in the early aftermath of World War II; to the frightening emergence of the Cold War and repeated US military adventures abroad; to the struggles of African Americans and other minorities to claim a share of the American Dream; to the striking transformations in social attitudes catalysed by the women’s movement and struggles for gay and lesbian liberation; to the dynamic force of political, economic, and social conservatism. Carrying the story to the spring of 2022, Winds of Hope also shows how dizzying technological changes at times threatened to upend the nation’s civic and political life.

Previous chapters have formulated learning problems within a frequentist framework. Frequentist learning aims to determine a value of the model parameter $\theta$ that approximately minimizes the population loss. Since the population loss is not known, this is in practice done by minimizing an estimate of the population loss $L_p(\theta )$ based on training data – the training loss $L_{𝒟}(\theta )$.

This chapter takes up the doctrine of God generally, concentrating on the traditional understanding of divine being as a complex of attributes derived via positiva and via negative (i.e., “classical theism”). Given classical theism’s tensions with the biblical narrative, as well as its role in helping to foster the rise of modern atheism, the chapter argues for a rigorous trinitarian rethinking of key traditional divine attributes.

As seen in the preceding chapter, when a reliable model $p(x,t)$ is available to describe the probabilistic relationship between input variable x and target variable t, one is faced with a model-based prediction problem, also known as inference. Inference can in principle be optimally addressed by evaluating functions of the posterior distribution $p(t|x)=p(x,t)/p(x)$ of the output t given the input x.

This chapter provides a refresher on probability and linear algebra with the aim of reviewing the necessary background for the rest of the book. Readers not familiar with probability and linear algebra are invited to first consult one of the standard textbooks mentioned in Recommended Resources, Sec. 2.14. Readers well versed on these topics may briefly skim through this chapter to get a sense of the notation used in the book.

In the examples studied in Chapter 4, the exact optimization of the (regularized) training loss was feasible through simple numerical procedures or via closed-form analytical solutions. In practice, exact optimization is often computationally intractable, and scalable implementations must rely on approximate optimization methods that perform local, iterative updates in search of an optimized solution. This chapter provides an introduction to local optimization methods for machine learning.

In brisk and engaging prose, this comprehensive introductory textbook traverses the broad sweep of US history since 1945. Winds of Hope, Storms of Discord explores how Americans of all walks of life –a political leaders, businesspeople, public intellectuals, workers, students, activists, migrants, and others – struggled to define the nation’s political, economic, geopolitical, demographic, and social character. It chronicles the nation’s ceaseless ferment, from the rocky conversion to peacetime in the early aftermath of World War II; to the frightening emergence of the Cold War and repeated US military adventures abroad; to the struggles of African Americans and other minorities to claim a share of the American Dream; to the striking transformations in social attitudes catalysed by the women’s movement and struggles for gay and lesbian liberation; to the dynamic force of political, economic, and social conservatism. Carrying the story to the spring of 2022, Winds of Hope also shows how dizzying technological changes at times threatened to upend the nation’s civic and political life.