Palgrave Handbook of Econometrics: Applied Econometrics

John DiNardo 127

blood from an experimental rock. In many such cases, we will not be able to

put any proposition to a “severe test.”

For the Bayesian the resolution of the problem is quite different – the data-
generation process (DGP) doesn’t (and shouldn’t) matter. This if often referred to
as “the likelihood principle.” To see how this works, recall the statement of Bayes’
rule in equation (3.5):

P(Ai|B) =

P(B|Ai)P(Ai)

∑k

j= 1 P(B|Aj)P(Aj)

.

Consider two different likelihoods such that:

zP(B|Ai)=P∗(B|Ai) ∀Ai,z>0.

Now use Bayes’ rule to show that one’s inference is unaffected by use ofP∗(B|Ai)
instead ofP(B|Ai):

P(Ai|B)=

P∗(B|Ai)P(Ai)

∑k

j= 1 P

∗(B|Aj)P(Aj)

=

zP(B|Ai)P(Ai)

∑k

j= 1 zP(B|Aj)P(Aj)

=

zP(B|Ai)P(Ai)

z

∑k

j= 1 P(B|Aj)P(Aj)

=

P(B|Ai)P(Ai)

∑k

j= 1 P(B|Aj)P(Aj)

.

Indeed, because of this property, Bayes’ rule is often written as:

P(Ai|B)

︸ ︷︷ ︸

Posterior

∝ P(B|Ai)

︸ ︷︷ ︸

Likelihood

P(Ai)

︸︷︷︸

Prior

. (3.9)

Consequently, how the data was generated does not matter for the typical Bayesian
analysis. It is also why a Bayesian would view the information in the binomial
versus negative binomial experiment as being the “same.”
This property of Bayesian inference has been frequently cited as being one of the
most significant differences between Bayesians and non-Bayesians.^36

3.5.3 If the DGP is irrelevant is the likelihood really everything?

A great deal more follows from the Bayesian approach. Unlike the previous exam-
ple, which might discomfit some non-Bayesians, another implication seems a
bit more problematic. One significant difficulty with the simplest versions of
Bayesian analysis concerns the distinction between “theorizing after the fact” and
“predesignation.”