This chapter addresses the formal doctrine of the Trinity, which is the distinctive Christian understanding of God and the linchpin and framework for all other Christian doctrines. After laying out this doctrine’s biblical basis and historical development, it presents a typology of trinitarian models, finding special promise in a social model for its biblical-orthodox soundness, conceptual coherence, and ethical implications.

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 focuses on three key problems that underlie the formulation of many machine learning methods for inference and learning, namely variational inference (VI), amortized VI, and variational expectation maximization (VEM). We have already encountered these problems in simplified forms in previous chapters, and they will be essential in developing the more advanced techniques to be covered in the rest of the book. Notably, VI and amortized VI underpin optimal Bayesian inference, which was used, e.g., in Chapter 6 to design optimal predictors for generative models; and VEM generalizes the EM algorithm that was introduced in Chapter 7 for training directed generative latent-variable models.

This chapter surveys important modern developments connected to the Enlightenment of the 18th century, which put Christianity in a defensive position, having to justify its basic beliefs in light of a new prime criterion (i.e., not revelation but reason) and a new defining context (i.e., history). This context led some thinkers (e.g., Protestant liberalism) to significant departures from traditional forms of theology and other thinkers (e.g., neo-orthodoxy) to significant attempts to rethink Christian orthodoxy in a new mode.

The previous chapters have adopted a limited range of probabilistic models, namely Bernoulli and categorical distributions for discrete rvs and Gaussian distributions for continuous rvs. While these are common modeling choices, they clearly do not represent many important situations of interest for machine learning applications. For instance, discrete data may a priori take arbitrarily large values, making categorical models unsuitable. Continuous data may need to satisfy certain constraints, such as non-negativity, rendering Gaussian models far from ideal.

This chapter examines the concept of revelation, including the traditional Christian distinction between general and special revelation (especially scripture). After exploring the relationship of these two forms of revelation, it examines some of the complexities involved in articulating a knowledge of God.

So far, this book has focused on conventional centralized learning settings in which data are collected at a central server, which carries out training. When data originate at distributed agents, such as personal devices, organizations, or factories run by different companies, this approach has two clear drawbacks: • First, it requires transferring data from the agents to the server, which may incur a prohibitive communication load. • Second, in the process of transferring, storing, and processing the agents’ data, sensitive information may be exposed or exploited.

In full recognition of the issues involved in the intellectually and existentially challenging problem of evil, this chapter seeks to mount a response to the reality of evil given faith in the triune God. After critically examining a number of traditional, philosophically-inclined theodicies, the chapter suggests a number of “moments” in a narratival response to evil, focusing on the pivotal theological realities of the cross, resurrection, and new creation.

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 duration of all express trusts except charitable trusts is limited by the application of the rule against perpetuities. As explained in chapter 15, the rule does not impose a fixed time limit on the duration of a trust. In most cases, it imposes a limit (nowadays usually 80 years) beyond a which an interest arising under the trust cannot vest. In South Australia, the perpetuity rule has been abolished but a court may, upon application, vary the terms of a trust 80 or more years after the date of the instrument creating the trust so that any interests which have not yet vested vest immediately. There are no limits on the duration of a charitable trust beyond the practical limitation of the availability of trust money to be applied for the charity’s objects. If the original objects of the charity become impossible or impracticable to achieve, a cy-près scheme will be approved enabling the trust property to be applied for objects which are as close as possible to the original objects.

The word ‘equity’ is one of the most ambiguous in the law. Its most obvious meaning is fairness and justice. Many would argue that equity is the overriding goal of all law. How could the law ever justify unfair or inequitable outcomes? But a moment’s thought will show that applying, without more, the criterion of ‘fairness’ to solve all legal problems is open to serious objections. Decisions will inevitably reflect the subjective beliefs and values of the decision-maker as to what is fair. In a pluralist democracy, disputes about what is fair or equitable are settled by elected legislators, not by unelected judges, except where legislation has explicitly authorised judges to determine cases by reference to considerations of fairness. Judges do not assess what is equitable without reference to some standard or benchmark.

A trustee owns the trust property and can exercise an owner’s legal right to manage and dispose of the property. But the exercise of these rights is not absolute. It is subject to the beneficiary’s equitable rights to compel the proper performance of the trust and, in accordance with principles described in chapter 24, to bring the trust to an end and become owner of the property. Legal ownership presents opportunities for the trustee to betray and exploit the trust for personal gain. It also exposes the beneficiary to the risk that the trustee may neglect the trust property. Equity therefore imposes strict obligations on the trustee to ensure that the trust is carried out strictly according to its terms. Even when a trustee holds property on a bare trust for a single, adult beneficiary, so that an adult beneficiary can claim the trust property under the rule in Saunders v Vautier, the trustee will have active duties to perform, including a duty of care with respect to the property. The duties are enforceable by the remedies discussed in Part B, all of which are designed not only to provide complete relief for the trust but also to deter wrongdoing.

The fiduciary may not be the only person who is accountable for a breach of fiduciary obligation. Other parties may also be liable. Suppose a solicitor misappropriates client money and then pays it into his wife’s bank account. The solicitor might also have been helped to commit the breach by an accountant who gave the solicitor access to the client account. Will these third parties – the wife and accountant – be liable in equity for their participation in the fiduciary’s breach? If the solicitor is solvent, the answer to this question may not matter much. The solicitor will be ordered to restore the money to the fund, together with compound interest to compensate for the loss of investment opportunity caused by the misappropriation. But if the solicitor is insolvent, the client will look to other participants in the fraud with ‘deep pockets’ to recover her funds.