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.

The previous chapter, as well as Chapter 4, have focused on supervised learning problems, which assume the availability of a labeled training set $𝒟$. A labeled data set consists of examples in the form of pairs (𝑥, 𝑡) of input 𝑥 and desired output 𝑡.

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.

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.

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 looks at the reality of religious plurality – more particularly, the relationship between the Christian church and other religious traditions – including the common typology of exclusivism, inclusivism, and pluralism. It also explores the significance of the fact that the Christian church’s numerical center has shifted from the West to the South and East, with significant consequences for the mission and theology of the church.

This chapter explores how the life, death, and resurrection of Jesus Christ make reconciliation possible. It surveys the diverse biblical motifs surrounding reconciliation/atonement, the variety of traditional “atonement theories” that flowed out of those motifs, and a way of coherently integrating a variety of atonement approaches.

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 analyzes the context and theology of the Protestant Reformation, focusing on key theologians such as Luther, Melanchthon, Zwingli, and Calvin. It also examines responses to the Protestant Reformation, such as the radical reformers and forms of Catholic reformation, concluding with an exploration of the connection between Reformation theology and the rise of Protestant orthodoxy, pietism, and new forms of rationalism.

This chapter elaborates four pillars of an “ecumenical eschatology”: (1) the second coming of Christ (vis-à-vis “going to” heaven); (2) resurrection of the body (vis-à-vis an exclusive emphasis on an immortal soul); (3) final judgment/justice that is historical and public (vis-à-vis one that is heavenly and private); and (4) new heaven and earth (vis-à-vis an exclusive emphasis on heaven). It contends for a creational understanding of this doctrine, which implies that eschatology, rather than being merely a theological afterthought, is a central and mobilizing theme.