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### 马尔科夫链蒙特卡洛方法处理计数型相关数据 #### 引言《马尔科夫链蒙特卡洛方法处理计数型相关数据》一书详细探讨了如何利用马尔可夫链蒙特卡洛（Markov Chain Monte Carlo, MCMC）方法来处理和分析具有相关性的计数型数据。这类数据广泛存在于社会科学研究、经济学、生物学以及工程学等多个领域中。传统的方法往往难以有效应对大规模或复杂的相关性结构，而MCMC方法提供了一种灵活且强大的工具。 #### 计数型数据与相关性计数型数据通常指的是非负整数值的数据，例如网站点击次数、电话通话次数、患者就诊次数等。这些数据的特点是分布往往是离散的，并且可能受到各种因素的影响而表现出不同的统计特性。当这些数据之间存在相关性时，即多个计数变量之间相互依赖，传统的统计方法可能不再适用。 #### 相关计数数据模型针对具有相关性的计数型数据，作者提出了一类基于相关潜在效应的模型。在这个模型框架下，每项观察数据都关联着一个潜在效应，这些潜在效应之间存在着相关性。这样的设置能够有效地捕捉到不同计数变量之间的相互作用。 - **特殊案例**：该模型包括多种特殊情况，如Poisson-lognormal分布模型等。 - **估计方法**：为了估计模型参数，作者开发了一个高效的MCMC算法，可以处理多变量正态分布或多变量-t分布假设下的潜在效应。 #### MCMC算法 MCMC算法是一种随机模拟技术，通过构造一个随机过程（马尔科夫链）来逼近目标分布的概率密度函数。这种技术特别适合于处理高维空间中的积分问题，对于复杂的计数型数据模型尤其有用。 - **Metropolis-Hastings算法**：这是MCMC中最常用的算法之一，用于从已知概率分布中抽样。它通过定义一个转移概率来决定是否接受当前样本点，从而构建出一个马尔科夫链。 - **效率优化**：为了提高算法的收敛速度和估计精度，作者对MCMC算法进行了调整和优化。 #### 应用示例书中通过两个真实数据集的例子展示了所提出的模型和算法的有效性： - 第一个例子涉及六维相关计数数据。 - 第二个例子则是一个更为复杂的十六维相关计数数据案例。 #### 关键词解析 - **潜在效应**：指未直接观测到但对计数数据有影响的因素或变量。 - **Metropolis-Hastings算法**：一种重要的MCMC算法，用于从复杂的概率分布中进行抽样。 - **多变量计数数据**：包含多个相互关联的计数变量的数据集。 - **Poisson-lognormal分布**：一种特殊的计数型数据分布模型，其中计数数据服从泊松分布，而泊松分布的均值参数又服从对数正态分布。 #### 结论通过对相关计数型数据的深入分析，《马尔科夫链蒙特卡洛方法处理计数型相关数据》为处理这类数据提供了新的思路和方法。通过引入相关潜在效应并结合高效的MCMC算法，该书不仅解决了传统方法难以处理的问题，也为后续研究提供了宝贵的参考。这对于任何需要分析相关计数数据的领域来说都是极其有价值的贡献。

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Markov Chain Monte Carlo Analysis of

Correlated Count Data

Siddhartha Chib

John M. Olin School of Business, Washington University, St. Louis, MO 63130 (chib@olin.wustl.edu)

Rainer Winkelmann

IZA Bonn, 53072 Bonn, Germany (winkelmann@iza.org)

This article is concerned with the analysis of correlated count data. A class of models is proposed in

which the correlation among the counts is represented by correlated latent effects. Special cases of the

model are discussed and a tuned and ef cient Markov chain Monte Carlo algorithm is developed to

estimate the model under both multivariate normal an d multivariate-

assumptions on the latent effects.

The methods are illustrated with two real data examples of six and sixteen variate correlated counts.

KEY WORDS: Latent effects; Metropolis–Hastings algorithm; Multivariate count data; Poisson–

lognormal distribution.

A large literature on the an alysis of count data is now

available (Cameron and Trivedi 1998, Winkelmann 2000),

but only a small portion of it deals with correlated counts.

Correlated counts typically arise in three varieties—as gen-

uine “multivariate” data on several related counted outcomes,

as longitudinal measurements o n a large number of su bjects

over a sh ort period of time, or as measurements on a small

set of subjects over a long period of time (the seemingly

unrelated case). Although the longitudinal situation has been

actively studied (e.g., see Hausman, Hall, and Griliches 1 984;

Blundell, Grif th, and Van Reenen 1995; Wooldridge 1997;

Chib, Greenberg, and Winkelmann 1998, henceforth CGW)

and a number of useful models and approaches are avail-

able, the other cases have been analyzed only under simpli-

fying assumptions (King 1989; Jung and Winkelmann 1993;

Gurmu and Elder 1998; Munkin and Trivedi 1999). The latter

approaches either do not allow a general correlation structure

or are dif cult to extend beyond the case of a few outcomes.

This article is an effort to deal with both problems. To

model the correlation among a large number of counts in a

 exible fashion, we introduce a set of correlated latent effects,

one for each subject and outcome. Conditioned on the latent

effects, t he counts are assumed to be independent Poisson with

a conditional mean function that depends on the latent effects

and a set of covariates. To complete the model we assume

that the latent effects follow a multivariate Gaussian distribu-

tion with a zero mean vector and full unrestricted covariance

matrix. As an extension of this model, we also consider the

case in which the latent effects follow a multivariate-

distri-

bution. To estimate this model, we develop a Markov ch ain

Monte Carlo (MCMC) simulation method that is based on the

work of CGW. Under t his framework, we are able to sample

the posterior distribution of the parameters and latent effects

without computing the likelihood function of the model.

The methods that we develop in this article can be applied to

datasets with large numbers of correlated counts. We demon-

strate this feature by  tting our model to a problem with 16

response variables. In our view this is an important illustration

that highlights what is possible from a Bayesian simulation-

based perspective.

The rest of the article is organized as follows. In Section 1

we present the basic model and some special cases and exten-

sions. The  tting algo rithm is developed in Section 2, while

Section 3 gives two real data examples. Section 4 concludes.

1. MODEL

Following the usual notation for multivariate d ata, let

1 : : : 1 y

denote the collection of

counts on the

th sub-

ject in the sample,

i µ n

. Let

1 : : : 1 b

denote a set

subject and outcome-speci c latent effects, and suppose

that, conditioned on

and parameters

‚

the coun ts

j µ J

, are independent Poisson:

—

1 ‚

Poisson

4Œ

exp

‚

for

j µ J

and

i µ n1

(1)

where

are covariates. To model the correlation among the

counts, let

—

1 D51 i µ n1

(2)

where

is an unrestricted covariance ma trix.

To understand some of the features of this model, let

exp

and

1 : : : 1 v

. Then

4Œ1 è5

, a multi-

variate lognormal distribution with mean

exp

5 diag

4D55

and dispersion matrix

diag

4Œ556

exp

4D5

diag

4Œ55

Hence,

—

‹

1 v

Poisson

4‹

, where

‹

exp

‚

This is, therefore, in the form of a Poisson–lognormal distribu-

tion as discussed by Aitchison and Ho (1989).

In this setup, the expectation and va riance of t he marginal

joint distribution of

can be derived without integration. Let

‹

(i.e.,

‹

and

‹

differ only b y a constant fac-

tor),

‹

1 : : : 1

‹

, and

diag

‹

. Applying the

Journal of Business & Economic Statistics

October 2001, Vol. 19, No. 4

428

Chib and Winkelmann: Markov Chain Monte Carlo Analysis of Correlated Count Data 429

law of the iterated expectation, one obtains E

—

‚1 D5

‹

and var

—

‚1 D5

exp

4D5

where we have

‚

4‚

1 : : : 1 ‚

. Hence, the covariance between the counts

is represented by the terms

cov

1 y

‹

exp

‹

exp

5‹

exp

which can b e positive or negative depending on the sign of

, the

4j1 k5

element of

. Moreover, the model allows for

overdispersion, a variance in excess of the expectation, as long

0. The correlation structure of the counts is thus unre-

stricted. Note, however, that the marginal distribution of the

counts

cannot be obtained by direct computation, re quiring

as it does the evaluation of a

-variate integral of the Poisson

distribution in (1) with resp ect to the distribution of

It is interesting to note that our model is similar to that

of Gurmu and Elder (1998) except that in their model the

distribution of

is left unspeci ed. Under that assumption,

the model becomes computationally intractable for anything

more than a few correlated counts. As we show in this arti-

cle, it is possible to  t higher-dimensional models provided

one is willing to make a parametric distributional assumption

for

, which in turn provides a clean i nterpretation for the

correlation structure. The assumption of normality is not cru-

cial and can be generalized. For example, it is easy to let the

distribution of the latent effects be multivariate-

instead of

the multivariate-normal, as will be discussed, or to model the

distribution by a  nite mixture of normal distributions. More

importantly, it is possible to relax the assumption, implicit in

the preceding formulation, that the

are independent of the

covariates by letting the mean of

be a function of one or

more of the available covariates. The estimation approach that

we will present needs to be modi ed only slightly to incorpo-

rate this feature. Finally, our model can be specialized to the

panel-data setting (where the index

represents time) by let-

ting the conditional mean function be

exp

‚

where

is a set of covariates that are a subset of

. This is

exactly the model of CGW that in turn is a generalization of

the model of Hausman et al. (1984). It should be noted that,

in this specialization of the general model, fewer than

latent

effects appear in the conditional mean function of subject

2. ESTIMATION OF THE MODEL

2.1 Likelihood Function

Let us suppose that the observations

1 : : : 1 y

are

conditionally independent across clusters. Then, the likelihood

function is the product of the contributions

p4y

—

‚1 D5

, where

p4y

—

‚1 D5

is the joint probability of the

counts in cluster

given by

p4y

—

‚1 D5

f 4y

—

‚

1 b

5”

—

1 D5

(3)

where

, as previously, is the Poisson mass function condi-

tioned on

4‚

1 b

and

”

is the

-variate normal density fu nc-

tion. This multiple integral cannot be solved in closed form

for arbitrary

, but some simpli cations are possible if

assumed to be a diagonal matrix. To deal with the general case,

however, it is necessary to turn to simulation-based methods.

2.2 MCMC Implementation

The main idea of the estimation approach is to focus on the

posterior distribution of the parameters and the latent effects

and then to summarize this p osterior distribution by MCMC

methods. Since mu ch has been written about MCMC methods

(e.g., see Tierney 19 94; Chib and Greenberg 1995, 1996), we

can be brief.

With MCMC methods, one designs an ergodic Markov

chain with the property that the limiting invariant distribution

of the chain is the posterior density of interest. Then, draws

furnished by sampling the Markov chain, after an initial tran-

sient o r burn-in stage, can be taken as approximate correlated

draws fro m the posterior distribution. This output forms th e

basis for summarizing the posterior distribution and for com-

puting Bayesian point and interval estimates. Ergodic laws of

large numbers for Markov chains on continuous state spaces

are used to justify that these estimates are simulation consis-

tent, converging to the posterior expectations as the simulation

sample size becomes large.

One standard method for constructing a Markov chain with

the correct limiting distribution, is via a recursive simulation

of the so-called ful l conditional densities—that is, the den-

sity of a set or block of parameters, given the data and the

remaining blocks of parameters. Each of the full conditio nal

densities in the simulation is then sampled either directly (if

the full conditional density belongs to a known family of dis-

tributions) or by utilizing a techniqu e such as the Metropolis–

Hastings (M–H) method. An important and crucial point is

that these methods do not require knowled ge of the intractable

normalizing constant of the posterior distribution.

In th e present case, we apply MCMC methods to simu-

late the augmented posterior distribution of the parameters

and the latent effect. For the prior on the parameters, assume

that

4‚1 D

independently follow the distributions

‚

4‚

1 B

51 D

Wishart

4

1 R

where

4‚

1 B

1 v

1 R

are known hyperparameters and Wishart

4¢1 ¢5

is the Wishart

distribution with



df and scale matrix

. Then, by Bayes

theorem, the posterior density is proportional to

”

4‚

—

‚

1 B

—



1 R

p4y

—

‚1 b

5”

—

1 D51

where

is the Wishart density. We now consider a sampling

procedure to simulate this density.

Following CGW, we co nstruct our Markov chain u sing the

blocks of parameters

‚

, and

and the full conditional

distributions

—

y1 ‚1 D73 6‚

—

y1 b73 6D

—

b71

(4)

where

1 : : : 1 b

. The simulation output is obtained

by recursively simulating these distributions, using the most

recent values of the conditioning variables at each step.

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