Comments on "Betting interpretations of probability" by Glenn Shafer
Philip Dawid
University of Cambridge
This is an interesting and informative account of the history and
current status of the betting approach to probability, which has been
sorely undervalued for far too long. I hope that it will help to
revive interest in the topic.
Glenn mentions de Finetti's use of betting ideas to develop his
criterion of coherence: if You are to avoid sure loss to a sceptic who
makes a collection of bets at odds You offer, You should ensure that
they are based on probabilities obeying the usual axioms. This use of
betting arguments to concentrate the mind, before any outcomes are
observed, can be described as "ex ante", as opposed to "ex post" uses,
in which reality decides on the outcomes and the parties are actually
subjected to the appropriate losses or gains. Only in the latter do
we need to take into account the actual behaviour of the world,
determining our empirical preformance. This important distinction is
similar to the dual purpose of penal punishments: ex ante, to keep you
and me on the straight and narrow, for fear of what would happen to us
if we strayed; ex post, to rain down on malefactors their due
punishment. However, I think Glenn is wrong in suggesting that de
Finetti's interest was confined to the ex ante argument. He has a
body of work (see for example de Finetti, 1962, and Dawid and
Galavotti, 2009) which takes ex post empirical assessment seriously;
and he famously ran a competition for his students requiring
probabilistic forecasting of the results of Italian football games,
with rewards and penalties based on the quadratic proper scoring rule.
One interesting aspect of betting and game-theoretic approaches to
probability is the role of strategies for the various players. For
example, a typical theorem of game-theoretic probability is that
Sceptic has a strategy that can force some outcome (generally a
disjunction of (a) a property of Sceptic's capital, and (b) a property
relating Forecaster's forcasts to Reality's outomes), no matter what
the other players do. Here it is important that Sceptic be operating
a strategy, since this means (i) that we are explicit as to what
information he is allowed to access and use, and (ii) we know that he
will be able to succeed in all circumstances. It is equally important
that the other players are not being required to operate strategies,
but can play essentially arbirarily at any point (for example, they
are not precluded from taking additional side-information into
account, or just making something up on the spur of the moment): by
varying these choices, the other players might be able to affect which
element of the disjunction will hold, but they can not avoid the
disjunction. This universality means that game-theoretic results are
actually very much stronger than their counterparts in standard
probability theory, which typically follow from considering the
special case that both Forecaster and Reality are operating
strategies.
"Defensive forecasting" involves a somewhat weaker set-up, since now
Forecaster is seeking a strategy that will ensure some outcome, but
not now for anything that the other players (Sceptic and Reality) may
do, since Sceptic is assumed to operate a known strategy.
Glenn's talk was about Probability, but there are also some
fundamental uses of betting arguments in Statistics, and I wonder how
these can be related? For example, Fisher coined the concept of a
"relevant subset", which, when it exists, can be used to discredit,
(say) a 95% confidence interval, by allowing a system that bets, at
the declared odds of 19:1, against the event that the interval
contains the target parameter, and has positive expected gain no
matter what value that parameter may have. There would appear to be
many similarities, but also many differences, between such ideas and
the game-theoretic interpretation of probability assignments. If a
close connexion can indeed be forged, we might perhaps be able to use
ideas from defensive forecasting to construct confidence intervals
that are immune from such criticism.
Finally, Glenn's analysis of Bayesian conditioning shows the
additional clarity that can be gained from applying betting arguments:
in particular, we can then more readily interpret what it means to
have learned the event A but nothing else. But I am a little puzzled
by his assertion that this "has a meaning without a prior protocol".
Perhaps he can help clarify the distinction in relation to the
following currently discussed problem (see
).
"I have two children. One is a boy born on a Tuesday. What is the
probability I have two boys?"
According to the BBC website the answer is 13/27. However, the
question is ill-posed, since we must take into account, not just the
content of the information received, but how and why that information
came to be supplied. Let us suppose it known that I have two
children, and let us assume that, in the absence of further
information, each is equally likely to be a girl or a boy,
independently, and that the birth day of each is uniformly distributed
across the week, independently of each other and of which sexes they
are. The figure 13/27 then arises if I am asked "Do you have a boy
born on a Tuesday?", and answer "yes". But, if I have two boys, I
might have tossed a coin to choose one of them, and announced his
birth day; or, always announce the birth day of the first born. In
these cases the probability of two boys is 1/3 --- exactly as when no
birth day information is supplied. Yet another alternative is that,
if I have two boys, I will announce the earliest day in the week
(starting from Sunday) that one of them was born. If I announce
Tuesday, the probability I have two boys then turns out to be 18/25.
To me such problems illustrate the necessity of carefully specifying
the protocol governing the supply of information. I should appreciate
Glenn's description of the appropriate game-theoretic formulation(s).
REFERENCES
Dawid, A. P. and Galavotti, M. C. (2009). de Finetti's subjectivism,
objective probability, and the empirical validation of probability
assessments. In Bruno de Finetti, Radical Probabilist (Galavotti,
M. C., ed.), London: College Publications, pp 97-114.
de Finetti B (1962) Does it make sense to speak of "good probability
appraisers?". In The scientist speculates (Good I. J., ed), Basic
Books, New York, pp 357-364.