生态学中的贝叶斯推断：理论与应用

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"《贝叶斯推断——生态学应用》是专为环境科学家、生态学家和野生动物生物学家设计的一本介绍贝叶斯推断的书籍。书中以数学严谨但易于理解的方式阐述了贝叶斯方法在生态学中的强大功能和实用性。作者包括美国地质调查局帕图森特野生动物研究中心的William A. Link和新西兰奥塔哥大学数学与统计学系的Richard J. Barker。该书由Elsevier出版社出版，首次发行于2010年。" 贝叶斯推断是统计学中的一种重要方法，它允许我们根据现有证据对未知参数的概率分布进行更新。在生态学和环境科学中，贝叶斯方法特别有用，因为它能够处理不确定性，并且可以集成多种来源的数据来构建更全面的模型。生态学家通常面对的是复杂且含有大量不确定性的系统，例如物种分布、种群动态和生态系统功能。贝叶斯框架提供了处理这些不确定性并整合不同数据源的工具。本书《贝叶斯推断——生态学应用》将深入探讨以下关键概念： 1. **贝叶斯定理**：贝叶斯定理是贝叶斯推断的基础，它描述了在给定观察数据的情况下，先验概率如何通过后验概率进行更新。在生态学中，这可能涉及估计物种濒危状态、预测种群数量变化或评估保护措施的效果。 2. **先验知识**：在贝叶斯分析中，先验分布反映了在观察数据出现之前对参数的信念。在生态学研究中，这可能包括基于理论模型或以前研究的经验知识。 3. **后验分布**：通过结合先验分布和似然函数（基于观测数据的概率），我们可以得到参数的后验分布，这代表了在考虑所有现有证据后对参数最可能值的理解。 4. **马尔科夫链蒙特卡洛（MCMC）**：为了实际计算复杂的后验分布，经常使用MCMC方法，如Metropolis-Hastings算法。这种方法允许在高维空间中有效地探索参数空间，这对于生态学中的多参数模型至关重要。 5. **模型选择与比较**：书中可能会讨论贝叶斯信息准则（BIC）和 Widely Applicable Information Criterion（WAIC）等方法，用于比较不同模型的优劣，从而选择最能解释数据的模型。 6. **不确定性分析**：贝叶斯方法提供了一种量化不确定性的方法，通过后验分布的形状和宽度，可以了解参数估计的精度和不确定性。 7. **生态案例研究**：作者很可能会通过具体的生态学案例，如物种分布建模、种群动态模拟或环境影响评估，来展示如何应用贝叶斯方法解决实际问题。这本书对于那些希望在他们的研究中利用贝叶斯统计方法的生态学家和环境科学家来说，是一本宝贵的资源。它不仅提供了理论基础，还强调了贝叶斯方法在解决实际生态问题时的实用性和价值。

PART I

PROBABILITY AND INFERENCE

4 1. INTRODUCTION TO BAYESIAN INFERENCE

should tend toward a skeptical view of Bayesian statistics. Doesn’t embracing the “new”

philosophy mean discarding the old and familiar tools, painstakingly acquired, and which

(after all) work well enough? Why bother with Bayesianism? Isn’t the approach philosophi-

cally ﬂawed? Isn’t it true that “Bayesian methods constitute a radically different way of doing

science ... [and that] Bayesians categorically reject various tenets of statistics and the scientiﬁc

method that are currently widely accepted in ecology and other sciences” (Dennis, 1996). Isn’t

the ﬁeld of Bayesian inference an esoteric quagmire?

“No one after drinking old wine desires new; for he says, ‘The old is good.’ ”

(Luke 5:39, RSV)

Our answer is that the Bayesian approach is attractive because it is useful. Its usefulness

derives in large measure from its simplicity. Its simplicity allows the investigation of far more

complex models than can be handled by the tools in the classical toolbox.

There are interesting philosophical issues to discuss in comparing frequentist and Bayesian

approaches; some of these require more than a superﬁcial understanding of mathematics and

probability theory. It is unfortunate that the philosophical discussion is often pushed to the fore,

its complexity scaring the classically trained researcher back into the fold, without

perhaps

afair exposure to Bayesian methods. The issues are worth addressing and are addressed in

this text. But our priority inwriting this text isutilitarian. The authors were ﬁrst attracted to

Bayesian inference because of its usefulness, and gradually became convinced of its beauty, as

well; we hope that thiswill be the experience of our readers.

1.2 THOMAS BAYES

Little is known of the man who ﬁrst described the mathematical form of reasoning eponymi-

cally known as “Bayesian” inference.

Thomas Bayes was born in 1701 (or 1702) in London,

England (or perhaps, Hertfordshire). He was admitted to the University of Edinburgh in

1719, where he studied logic and theology. Like his father, he was a nonconformist minister

(i.e., Protestant but not Anglican); he is included in a 1727 listing of “Approved Ministers of

the Presbyterian Denomination.”

Bayes was elected a Fellow of the Royal Society in 1742, having been recommended as

“a Gentleman of known merit, well skilled in Geometry and all parts of Mathematical and

Philosophical Learning.” It is thought that this nomination came in consequence of his“An

Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections

of the Author of the Analyst” (1736), inwhich he defended Isaac Newton against

the Bishop

Berkeley, who had referred to Newton as an “inﬁdel Mathematician.”

Apart from his defense of Newton and a well-received contribution to a debate on the

purpose of God in creation (“Divine Benevolence, or an Attempt to Prove That the Principal End of

the Divine Providence and Government is the Happiness of His Creatures”, 1731) no other works of

Bayes’ are known to have been published during hislifetime.

1. “Eponymous” is a good word, like supercalifragilisticexpialidocious – if you say it loud enough, you’ll always sound

precocious. Bayesian writers seem unable to resist it.

I. PROBABILITY AND INFERENCE

1.2 THOMAS BAYES 5

1.2.1 The Doctrine of Chances

The work which preserved Bayes’ place inhistory is“An Essay towards solving a Problem

in the Doctrine of Chances” published posthumously (1763) through the good ofﬁces of Bayes’

friend Richard Price. The essay begins as follows:

“Given the number of times inwhich an unknown event has happened and failed: Required the chance that

the probability of its happening inasingle trial lies somewhere between any two degrees of probability that

can be named.”

Bayes’ “unknown event” is what we would today call a Bernoulli trial: a random event with

two possible outcomes, labeled “success” and “failure.” The ﬂipofacoin, an attempt to guess

the correct answer on a multiple choice exam, whether a newly hatched chick survives to

ﬂedging; each is a Bernoulli trial.

The “probability of the event happening inasingle trial” is referred to as the success para-

meter of the Bernoulli trial, and denoted by p. Sometimes, the nature of the trial leads to

a speciﬁcation of p.Acoin is deemed “fair” if “Heads” isaslikely as “Tails”, so that the

probability of getting “Heads” is p =1/2. On a multiple choice exam with

ﬁve options per

question, we might assume that the probability of guessing correctly is p=1/3 if we could rule

out two of the options and were forced to guess among the remaining three. In both cases, the

nature of the event leads to a speciﬁcation for p.

More frequently, however, there isnobasis for knowing p. Indeed, even ifwehadabasis

for an educated guess, we might wish to put aside whatever prejudices we have, and allow

the data to speak for themselves. Suppose that having observed 10 newly hatched chicks, we

note that only four survive to ﬂedging. Bayes’ essay seeks means to translate such data into

conclusions like “The probability that p is between 0.23 and 0.59 is 0.80.” Given “two degrees

of probability,” say a and b, the problem is to determine Pr(a ≤p≤b), i.e.,

the probability that

p is in the interval from a to b.

For those familiar with such things, it may sound as though Bayes were trying to construct

a conﬁdence interval for p. Doubtless, Bayes’ goal was the same as that of the frequentist

statistician who computes a conﬁdence interval: to quantify uncertainty about an unknown

quantity, using data and probabilities. As we shall see, however, Bayes’ way of accounting for

uncertainty about p isquite different from the frequentist approach, which developed later.

The difference lies in the Bayes’ use of the term “probability.” We illustrate withasimple

example.

The

authors of this text have a deformed New Zealand 50 cent piece; it is convex, rather

than ﬂat, with the Queen’s image on the side that bulges outward (see Fig. 1.1). Consider the

event that the coin lands, when ﬂipped, with the Queen’s image facing upward (“Heads”);

the unknown quantity of interest is p = Pr(Heads).

The convexity of the coinraises doubts

about whether p =

2. There isanimplicit ceteris paribus in deﬁning the event, as also in Bayes’ essay: individual ﬂips of the coin are replicates,

in the sense that conditions are assumed to be relatively constant with regard to the outcome. For instance, the authors have

observed that p appears to depend on the hardness of the surface on which the coin lands. Speaking entirely in terms of the

coin, and not in terms of a mathematical model, it seems impossible to explicitly deﬁne what we mean by “replicates,” and

whatever ambiguity remains in that deﬁnition is carried over into our deﬁnition of the probability p.

I. PROBABILITY AND INFERENCE

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