The Dictionary of Human Geography

Wolff (1987) and Gibson-Graham (2006b
[1996]) have gone further in dissolving the
very notion of the economy as a separate space
with deterministic effects, replacing base and
superstructure with the notion of a decentred,
over-determined totality with no essential,
determining structure (cf.essentialism).
Harvey (1999 [1982]), on the other hand,
continues to emphasize the classical role of the
economy and the dynamics of capitalaccumu-
lationin shaping social (and, crucially,class)
structures under capitalism, but avoids simple
base/superstructure distinctions by conceptu-
alizing economic and superstructural elements
as ‘moments’ in the total circulation process of
capital.
In summary, although the base/superstructure
distinction is too crude to provide an answer, it
doespointtowardsthekeyquestionofthenature
of ‘theeconomy’ in capitalist systems and its
influence on, and interaction with, wider social,
cultural and political structures. kb

Bayesian analysis A type of statistical mod-
elling and estimation deriving from the early
ideas of the Reverend Thomas Bayes, who
developed his ‘doctrine of chances’ in 1763
(Bayes, 1763 [1958]). The Bayesian perspec-
tive differs from traditional or ‘orthodox’ stat-
istical inference in giving explicit recognition
to the role of prior ideas and probabilities and
so is sometimes labelled as a ‘subjective’ ap-
proach to probability and statistics. Much of
the probability theory was developed by 1939,
when Jeffreys wrote his classic text (Jeffreys,
1998 [1939]), but the implementation of
Bayesian methods as a practical statistical
technique is much more recent, and had to
await modern computer technology and the
invention of some very clever new devices.
Bayes’ central idea is that prior probabilities
are updated by confrontation with data to pro-
vide posterior probabilities. For example, sup-
pose we want to make inferences about a
parameteru(which might be a mean or a
regressioncoefficient). Our prior probability
distribution foruisp(u). The observed data
are represented by the likelihood functionp
(yju). Using Bayes’ rule on conditional prob-
abilities gives us the posterior density or distri-
butionp(ujy) as follows:

p(ujy)¼p(u)p(yju)=p(y),

wherep(y)¼

P
up(u)p(yju), the sum over all
possible values ofu, which acts as a normalizing

constant. This term may be ignored in many

instances (though not inmodelcomparison)

to give the unnormalized posterior density:

p(ujy)/p(u)p(yju):

This expression defines the core of Bayesian

inference. Note that this method derives a

posterior probability distribution foru,whereas

classical (or standard) inference uses the

sample data to make inferences about the

unknown, but assumed fixed, parameter

value ofu. Where there are several parameters

in question, such asu 1 andu 2 , thenp(ujy)isa

joint distribution, and the Bayesian statistician

converts this to two marginal posterior distri-

butions by integrating across the range of the

otheru:

p(u 1 jy)/

ð

p(u)p(yju)du 2 :

In this framework, inferences aboutu 1 are

made taking account of the full distribution

ofu 2 , whereas classical inference is based just

on the optimal point estimates and local

curvature around that location.

Opinions about the potential of Bayesian

methods have differed sharply. Some have

seen them as a way of broadening the scope

of quantitative analysis, whilst others have

rejected the notion of bringingsubjectivity

into statistical inference. In practice, Bayesian

methods were little used except for circum-

stances under which they were equivalent to

classical results and so there was no computa-

tional difference, only one of interpretation.

More direct implementation depended on

the facility to do the numerical integrations

required to get the marginal distributions,

and modern computing provided this. In the

social sciences, the work of the Chicago

econometrician Arnold Zellner was very

important in this process (Zellner, 1971).

Modern Bayesian analysis is usually based on

‘uninformative’ or ‘diffuse’ prior information,

reflecting prior ignorance or a determination

not to introduce subjective prior inform-

ation into the analysis; Bayesian estimation is

then used very much as a technical device to

estimate posterior distributions.

Bayesian methods have taken a further leap

forward in the past decade with the construc-

tion of Markov Chain Monte Carlo, or

‘MCMC’, techniques (Gilks, Richardson and

Spiegelhalter, 1996). It has been shown that

Gregory / The Dictionary of Human Geography 9781405132879_4_B Final Proof page 43 31.3.2009 11:01am

BAYESIAN ANALYSIS