A couple of years ago I
showed how to construct hyperreal finitely additive probabilities on
infinite sets that satisfy certain symmetry constraints and have the
Bayesian regularity property that every possible outcome has non-zero
probability. In this post, I want to show a result that allows one to
construct such probabilities for an infinite sequence of independent
random variables.

Suppose first we have a group *G* of symmetries acting on a space
*Ω*. What I previously showed
was that there is a hyperreal *G*-invariant finitely additive
probability assignment on all the subsets of *Ω* that satisfies Bayesian regularity
(i.e., *P*(*A*) > 0
for every non-empty *A*) if and
only if the action of *G* on
*Ω* is “locally finite”,
i.e.:

- For any finitely generated subgroup
*H* of *G* and any point *x* in *G*, the orbit *H**x* is finite.

Here is today’s main result (unless there is a mistake in the
proof):

**Theorem.** For each *i* in an index set, suppose we have a
group *G*_{i}
acting on a space *Ω*_{i}. Let *Ω* = ∏_{i}*Ω*_{i}
and *G* = ∏_{i}*G*_{i},
and consider *G* acting
componentwise on *Ω*. Then the
following are equivalent:

there is a hyperreal *G*-invariant finitely additive
probability assignment on all the subsets of *Ω* that satisfies Bayesian regularity
and the independence condition that if *A*_{1}, ..., *A*_{n}
are subsets of *Ω* such that
*A*_{i} depends
only on coordinates from *J*_{i} ⊆ *I*
with *J*_{1}, ..., *J*_{n}
pairwise disjoint if and only if the action of *G* on *Ω* is locally finite

there is a hyperreal *G*-invariant finitely additive
probability assignment on all the subsets of *Ω* that satisfies Bayesian
regularity

the action of *G* on
*Ω* is locally finite.

Here, an event *A* depends
only on coordinates from a set *J* just in case there is a subset
*A*′ of ∏_{j ∈ J}*Ω*_{j}
such that *A* = {*ω* ∈ *Ω* : *ω*|_{J} ∈ *A*′}
(I am thinking of the members of a product of sets as functions from the
index set to the union of the *Ω*_{i}). For
brevity, I will omit “finitely additive” from now on.

The equivalence of (b) and (c) is from my old result, and the
implication from (a) to (b) is trivial, so the only thing to be shown is
that (c) implies (a).

**Example:** If each group *G*_{i} is finite and
of size at most *N* for a fixed
*N*, then the local finiteness
condition is met. (Each such group can be embedded into the symmetric
group *S*_{N},
and any power
of a finite group is locally finite, so *a fortiori* its
action is locally finite.) In particular, if all of the groups *G*_{i} are the same
and finite, the condition is met. An example like that is where we have
an infinite sequence of coin tosses, and the symmetry on each coin toss
is the reversal of the coin.

**Philosophical note:** The above gives us the kind of
symmetry we want for each individual independent experiment. But
intuitively, if the experiments are identically distributed, we will
want invariance with respect to a shuffling of the experiments. We are
unlikely to get that, because the shuffling is unlikely to satisfy the
local finiteness condition. For instance, for a doubly infinite sequence
of coin tosses, we would want invariance with respect to shifting the
sequence, and that doesn’t satisfy local finiteness.

Now, on to a sketch of the proof from (c) to (a). The proof uses a
sequence of three reductions using an ultraproduct construction to cases
exhibiting more and more finiteness.

First, note that without loss of generality, the index set *I* can be taken to be finite. For if
it’s infinite, for any finite partition *K* of *I*, and any *J* ∈ *K*, let *G*_{J} = ∏_{i ∈ J}*G*_{i},
let *Ω*_{J} = ∏_{i ∈ J}*Ω*_{i},
with the obvious action of *G*_{J} on *Ω*_{J}. Then *G* is isomorphic to ∏_{J ∈ K}*G*_{J}
and *Ω* to ∏_{J ∈ K}*Ω*_{J}.
Then if we have the result for finite index sets, we will get a regular
hyperreal *G*-invariant
probability on *Ω* that
satisfies the independence condition in the special case where *J*_{1}, ..., *J*_{n}
are such that *J*_{i} and *J*_{j} for distinct
*i* and *j* are such that at least one of
*J*_{i} ∩ *J*
and *J*_{j} ∩ *J*
is empty for every *J* ∈ *K*. We then take an
ultraproduct of these probability measures with respect to *K* and an ultrafilter on the
partially ordered set of finite partitions of *I* ordered by fineness, and then we
get the independence condition in full generality.

Second, without loss of generality, the groups *G*_{i} can be taken
as finitely generated. For suppose we can construct a regular
probability that is invariant under *H* = ∏_{i}*H*_{i}
where *H*_{i}
is a finitely generated subgroup of *G*_{i} and satisfies
the independence condition. Then we take an ultraproduct with respect to
an ultrafilter on the partially ordered set of sequences of finitely
generated groups (*H*_{i})_{i ∈ I}
where *H*_{i}
is a subgroup of *G*_{i} and where the
set is ordered by componentwise inclusion.

Third, also without loss of generality, the sets *Ω*_{i} can be taken
to be finite, by replacing each *Ω*_{i} with an orbit
of some finite collection of elements under the action of the finitely
generated *G*_{i}, since such
orbits will be finite by local finiteness, and once again taking an
appropriate ultraproduct with respect to an ultrafilter on the partially
ordered set of sequences of finite subsets of *Ω*_{i} closed under
*G*_{i} ordered
by componentwise inclusion. The Bayesian regularity condition will hold
for the ultraproduct if it holds for each factor in the
ultraproduct.

We have thus reduced everything to the case where *I* is finite and each *Ω*_{i} is finite.
The existence of the hyperreal *G*-invariant finitely additive
regular probability measure is now trivial: just let *P*(*A*) = |*A*|/|*Ω*|
for every *A* ⊆ *Ω*. (In
fact, the measure is countably additive and not merely finitely
additive, real and not merely hyperreal, and invariant not just under
the action of *G* but under all
permutations.)