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Calculation of Marginal Likelihood

Marginal likelihood is vital in comparing models in Bayesian econometrics. Here is the introduction to Chib method with Gibbs sampling.

Begins with the identity of Bayes Theorem
\[\pi(\theta^*|y) = \frac{f(y|\theta^*)\pi(\theta^*)}{f(y)}\,,\]
where$\theta^*$ is a particular value of $\theta$ and $f(y)$ is the marginal likelihood. For numerical accuracy, $\theta^*$ is usually chosen to be the mean of the sample values. The identity can be written as
\[f(y) = \frac{f(y|\theta^*)\pi(\theta^*)}{\pi(\theta^*|y)}\,,\]
The Chib method computes the right-hand side from the output of a Gibbs sampler. The terms in the numerator of the right-hand side are alreadily computed; they are the likelihood function and prior distribution, respectively, evaluated at $\theta^*$. The main problem is to compute $\pi(\theta^*|y)$.

By applying Gibbs sampling, we can the numerical evaluation of this posterior density.

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They say Rome wasn't built in a day, and yet what a difference a day makes.
有人说罗马不是一天建成的，但一天却能改变很多事。


Corporate Finance: A Practical Approach is a solid foundation for those looking to  achieve lasting business growth. In today's competitive business environment,  companies must find innovative ways to enable rapid and sustainable growth. This  text equips readers with the foundational knowledge and tools for making smart  business decisions and formulating strategies to maximize company value. It covers  everything from managing relationships between stakeholders to evaluating merger  and acquisition bids, as well as the companies behind them. Through extensive use  of real-world examples, readers will gain critical perspective into interpreting  corporate financial data, evaluating projects, and allocating funds in ways that  increase corporate value. Readers will gain insights into the tools and strategies  used in modern corporate financial management.  Equity Asset Valuation is a particularly cogent and important resource for anyone  involved in estimating the value of securities and understanding security pricing. A  well-informed professional knows that the common forms of equity valuation—  dividend discount modeling, free cash flow modeling, price/earnings modeling, and  residual income modeling—can all be reconciled with one another under certain  assumptions. With a deep understanding of the underlying assumptions, the  professional investor can better understand what other investors assume when  calculating their valuation estimates. This text has a global orientation, including  emerging markets.  All books in the CFA Institute Investment Series are available through all major  booksellers. Fixed Income Analysis has been at the forefront of new concepts in recent years, and  this particular text offers some of the most recent material for the seasoned  professional who is not a fixed-income specialist. The application of option and  derivative technology to the once staid province of fixed income has helped  contribute to an explosion of thought in this area. Professionals have been  challenged to stay up to speed with credit derivatives, swaptions, collateralized  mortgage securities, mortgage-backed securities, and other vehicles, and this  explosion of products has strained the world's financial markets and tested central  banks to provide sufficient oversight. Armed with a thorough grasp of the new  exposures, the professional investor is much better able to anticipate and understand  the challenges our central bankers and markets face.  International Financial Statement Analysis is designed to address the ever-  increasing need for investment professionals and students to think about financial  statement analysis from a global perspective. The text is a practically oriented  introduction to financial statement analysis that is distinguished by its combination  of a true international orientation, a structured presentation style, and abundant  illustrations and tools covering concepts as they are introduced in the text. The  authors cover this discipline comprehensively and with an eye to ensuring the  reader's success at all levels in the complex world of financial statement analysis.  Investments: Principles of Portfolio and Equity Analysis provides an accessible yet  rigorous introduction to portfolio and equity analysis. Portfolio planning and  portfolio management are presented within a context of up-to-date, global coverage  of security markets, trading, and market-related concepts and products. The  essentials of equity analysis and valuation are explained in detail and profusely  illustrated. The book includes coverage of practitionerimportant but often neglected  topics, such as industry analysis. Throughout, the focus is on the practical  application of key concepts with examples drawn from both emerging and  developed markets. Each chapter affords the reader many opportunities to selfcheck his or her understanding of topics.  One of the most prominent texts over the years in the investment management  industry has been Maginn and Tuttle's Managing Investment Portfolios: A Dynamic  Process. The third edition updates key concepts from the 1990 second edition. Some  of the more experienced members of our community own the prior two editions and  will add the third edition to their libraries. Not only does this seminal work take the  concepts from the other readings and put them in a portfolio context, but it also  updates the concepts of alternative investments, performance presentation standards,  portfolio execution, and, very importantly, individual investor portfolio  management. Focusing attention away from institutional portfolios and toward the  individual investor makes this edition an important and timely work.  The New Wealth Management: The Financial Advisor's Guide to Managing and  Investing Client Assets is an updated version of Harold Evensky's mainstay  reference guide for wealth managers. Harold Evensky, Stephen Horan, and Thomas  Robinson have updated the core text of the 1997 first edition and added an  abundance of new material to fully reflect today's investment challenges. The text  provides authoritative coverage across the full spectrum of wealth management and  serves as a comprehensive guide for financial advisors. The book expertly blends  investment theory and real-world applications and is written in the same thorough  but highly accessible style as the first edition.  

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Financial analysis is the process of examining a company’s performance in the context of its industry and economic environment in order to arrive at a decision or recommendation. Often, the decisions and recommendations addressed by fi nancial analysts pertain to providing capital to companies—specifi cally, whether to invest in the company’s debt or equity securities and at what price. An investor in debt securities is concerned about the company’s ability to pay interest and to repay the principal lent. An investor in equity securities is an owner with a residual interest in the company and is concerned about the company’s ability to pay dividends and the likelihood that its share price will increase. Overall, a central focus of financial analysis is evaluating the company’s ability to earn a return on its capital that is at least equal to the cost of that capital, to profi tably grow its operations, and to generate enough cash to meet obligations and pursue opportunities. Fundamental financial analysis starts with the information found in a company’s financial reports. These financial reports include audited financial statements, additional disclosures required by regulatory authorities, and any accompanying (unaudited) commentary by management. Basic fi nancial statement analysis—as presented in this chapter—provides a foundation that enables the analyst to better understand information gathered from research beyond the financial reports. This chapter is organized as follows: Section 2 discusses the scope of financial statement analysis. Section 3 describes the sources of information used in financial statement analysis, including the primary financial statements (balance sheet, statement of comprehensive income, statement of changes in equity, and cash flow statement). Section 4 provides a framework for guiding the financial statement analysis process. A summary of the key points and practice problems in the CFA Institute multiple-choice format conclude the chapter. The role of fi nancial reporting by companies is to provide information about a company’s performance, fi nancial position, and changes in fi nancial position that is useful to a wide range of users in making economic decisions.1 The role of fi nancial statement analysis is to use financial reports prepared by companies, combined with other information, to evaluate the past, current, and potential performance and financial position of a company for the purpose of making investment, credit, and other economic decisions. (Managers within a company perform financial analysis to make operating, investing, and financing decisions but do not necessarily rely on analysis of related fi nancial statements. They have access to additional financial information that can be reported in whatever format is most useful to their decision.) In evaluating fi nancial reports, analysts typically have a specifi c economic decision in mind. Examples of these decisions include the following:  

• Evaluating an equity investment for inclusion in a portfolio.
• Evaluating a merger or acquisition candidate.
• Evaluating a subsidiary or operating division of a parent company.
• Deciding whether to make a venture capital or other private equity investment.
• Determining the creditworthiness of a company in order to decide whether to extend a loan
to the company and if so, what terms to offer.
• Extending credit to a customer.
• Examining compliance with debt covenants or other contractual arrangements.
• Assigning a debt rating to a company or bond issue.
• Valuing a security for making an investment recommendation to others.
• Forecasting future net income and cash flow.


These decisions demonstrate certain themes in fi nancial analysis. In general, analysts seek to examine the past and current performance and financial position of a company in order to form expectations about its future performance and fi nancial position. Analysts are also concerned about factors that aff ect risks to a company’s future performance and financial position. An examination of performance can include an assessment of a company’s profitability (the ability to earn a profit from delivering goods and services) and its ability to generate positive cash flows (cash receipts in excess of cash disbursements). Profi t and cash fl ow are not equivalent. Profi t (or loss) represents the diff erence between the prices at which goods or services are provided to customers and the expenses incurred to provide those goods and services. In addition, profi t (or loss) includes other income (such as investing income or income from the sale of items other than goods and services) minus the expenses incurred to earn that income. Overall, profi t (or loss) equals income minus expenses, and its recognition is mostly independent from when cash is received or paid. Example 1 illustrates the distinction between profit and cash flow.  

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