1. Introduction
Convergence to the truth is sometimes easy for Bayesians, sometimes difficult, sometimes rare, and sometimes virtually impossible. Speaking somewhat loosely (more careful formulations are given below), we have the following examples.
a) Suppose that the problem is to determine the bias of a coin from knowledge of the outcomes of an infinite sequence of tosses. Typical Bayesian agents are essentially guaranteed to converge to the truth, no matter which hypothesis is true.
b) Suppose that the problem is to determine the propensities underlying a chance process that has a countable infinity of possible outcomes. Then, for typical hypotheses about the propensities, if any of those hypotheses should be true, typical Bayesian agents would be (essentially) guaranteed not to converge to the truth.
c) Suppose that the problem is to determine whether a given binary sequence, revealed bit by bit, encodes a binary expansion of a rational number. For any Bayesian agent, there is a rich infinite family of sequences the agent could be shown that would prevent convergence to the truth. Bayesian agents who are open-minded in a certain appealing sense fail to converge to the truth for typical sequences.
d) Suppose that the problem is to determine, from knowledge of the outcomes of an infinite sequence of tosses, whether a given coin has a bias of two-thirds in favor of heads or in favor of tails—and also whether it is a nickel, a dime, or a quarter. Because the evidence does not distinguish between each pair of fine-grained hypotheses under consideration, convergence to the truth is possible only in special cases.
Bayesian agents tend toward supreme self-confidence: facing a problem of type a, b, or c, they will assign probability zero to the possibility of failing to converge to the truth. While from the outside it is clear that problems of type b, c, and d are far more intractable for Bayesians than are problems of type a, this fact is not reflected in the judgments of Bayesian agents: they rate their chances of success for problems of type b or c just as highly as for problems of type a, even while being willing to admit that success is not inevitable for problems of type d.
In section 5 below, I argue that there is a real problem here for Bayesians—their account of rationality renders a certain sort of arrogance rationally mandatory, requiring agents to be certain that they will be successful at certain tasks, even in cases where the task is so-contrived as to make failure the typical outcome.Footnote 1 The first order of business, in section 2, is to get on the table the notion of typicality that will be in play in the following sections. In section 3, by way of stage setting and because I suspect that the results in question are not as widely known among philosophical Bayesians as they might be, I survey some classic results concerning problems of type a and b. In section 4, I establish the picture sketched above for problems of type c.
2. Typical!
One often says that a property is typical of a given class of objects if the set of objects exemplifying that property is overwhelmingly large in some appropriate sense—or if the set of objects lacking the property is so small as to be negligible in some appropriate sense.
Example: The Unit Interval
Let us begin with a concrete example: the unit interval 
of the real line. We can think of this in a number of ways—notably, as a set, as a topological space (a set together with a notion of convergence of sequences), or as a metric space (a set together with a notion of distance and hence also a notion of convergence).Footnote 2 Each of these brings with it a corresponding notion of smallness and largeness. If we consider 
as a set, then it is natural to think of its countable subsets as being small and to think of the complements of such subsets as being large.
If we consider 
as a equipped with its usual notion of distance, then it is natural to consider a subset 
to be small if and only if it is a (Lebesgue) null set—if and only if, that is, for every 
we can find a collection of subintervals of 
such that each point in 
is in one of these subintervals, while the total length of these subintervals is less than 
. Correlatively, one thinks of complements of null sets as being large.
In the intermediate case, in which we consider 
as a topological space, the relevant notion of smallness is perhaps less familiar. We call a subset 
meagre in 
if 
, where each 
is a nowhere dense subset of 
(in the current case, this means that even if we add to 
those points that arise as limits of sequences of points in 
, the resulting set still does not contain any subintervals of 
as subsets).
To motivate this notion, note that the families of countable subsets of 
and the null sets of 
are closed under the operation of taking countable unions. So one way to arrive at a topological notion of small subset of 
is to isolate a family of undeniably small sets and then look at the family of sets that arise as countable unions of these. Now, it is natural to consider any open dense 
large: no subinterval is so small that it avoids such a 
; every point in such a 
lies in some subinterval wholly contained in that 
. The complement of such a 
is a nowhere dense set—so it is natural to regard such sets as small. The meagre sets are those that result from beginning with the nowhere dense sets and closing under the operation taking countable unions. No subinterval of 
is meagre, and the complement of a meagre set is uncountable and dense in 
(see, e.g., Oxtoby Reference Oxtoby1980, theorem 1.3).
The relations between these three notions of smallness are complex. Every countable set is both a null set and a meagre set. But there exist uncountable subsets of 
that are both meagre and null. The Cantor set, which will play a role in section 4 below, is the standard example. Let 
be the result of deleting the (open) subinterval that is the middle third of 
. Let 
be the result of deleting (open) middle thirds of each of the components of 
. And so on. Then the Cantor set is 
. Equivalently, we can think of 
as the set of those numbers in 
that can be written in base three without using the numeral one. The set 
is uncountable.Footnote 3 But it is a null set.Footnote 4 And 
is itself nowhere dense—and hence is meagre as a subset of 
.Footnote 5
Since there are meagre null sets, there are also sets whose complements are meagre and null. Further, there are null sets with meagre complements—and hence also meagre sets with complements that are null sets (see n. 9 below). In practice, the two notions live side by side, providing complementary, sometimes clashing, notions of smallness.Footnote 6
One of the strengths of the notion of a meagre set is that it is purely topological—and so is invariant under homeomorphisms from the interval to itself (i.e., continuous deformations of the scale on the interval that leave the endpoints fixed). The notion of a null set, though, depends on the metric structure of the interval (or on some other structure that goes beyond its topology): there are homeomorphisms of 
that map the Cantor set to nonnull sets.Footnote 7 Nonnull sets that arise in this way are known as fat Cantor sets.Footnote 8 Each fat Cantor set is meagre—being the image of a meagre set under a homeomorphism—and by taking a countable union of well-chosen fat Cantor sets, we can construct a meagre set whose complement is a null set.Footnote 9
Meagre and Residual
We have been focusing on the interval and its subsets. But in any topological space, we can define the meagre subsets to be those that can be decomposed as countable unions of nowhere dense sets.Footnote 10 The complement of a meagre set is called residual or comeagre. Meagre subsets are also known as sets of first category; nonmeagre subsets are also known as sets of second category.
Intuitively, in well-behaved spaces meagre subsets are negligibly small, while residual subsets are overwhelmingly large.Footnote 11 Indeed, in the spaces to be considered below, countable sets are always meagre, and meagre sets are always small in a reasonably intuitive sense (i.e., meagre subsets of such spaces have dense complements, so in particular, no nonempty open sets are meagre).Footnote 12
The so-called Banach-Mazur game provides an alternative characterization of meagre subsets that bolsters the intuitive force of their claim to be negligibly small. Let 
be a topological space and let 
be a basis for the topology of 
: that is, 
is a collection of open sets of 
such that every open set can be written as a union of sets in 
. A subset 
of 
is identified, and a two-player game is to be played. Player 1 and player 2 take turns choosing elements 
of 
, with player 1 choosing first and with each 
constrained to be contained in 
(for 
). Player 1 wins if 
contains a point in 
; otherwise, player 2 wins. Intuitively, the smaller 
is, the harder it is for player 1 to win—certainly, if 
is all of 
, then player 1 cannot lose, while if 
is countable (and 
uncountable and well behaved), it is easy for player 2 to win, no matter what player 1 does. It turns out that player 2 has a winning strategy (a rule for selecting an interval at each stage, given what has gone on before, that is guaranteed to result in victory no matter what player 1 does) if and only if 
is meagre.Footnote 13 For applications, see footnotes 41, 45, and 46 below.
Below, our concern will be with the Bayesian framework, in which each agent has in view a space of live hypotheses over which prior and posterior probability measures are defined. In the usual case, spaces of hypotheses and the spaces of probability measures over them have a topological structure (one knows which sequences of points in these spaces converge and which do not), but they are not equipped with natural notions of distance or of measure.Footnote 14 So we will count meagre subsets of spaces of hypotheses or of priors as small and residual subsets as large. A property of hypotheses or of priors will count as typical, just in case the set of objects exemplifying that property is residual in the relevant space.
For the sort of cases we will be concerned with, the various notions of largeness and smallness mesh in interesting ways.
Remark 1. On the one hand, if 
is a reasonable topological space with uncountably many points, then every Borel probability measure on 
assigns measure zero to some residual subset of 
.Footnote 15 So there is a sense in which each Bayesian prior embodies some very strong a priori bias that is immune to empirical refutation.
Remark 2. On the other hand, if 
is a (measurable) meagre subset of a reasonable space of hypotheses, then the probability measures according to which 
has measure zero form a residual subset of the space of probability measures over that space of hypotheses.Footnote 16
3. Good News, Bad News
Turn we now to a family of problems from the heartland of Bayesian statistics, by way of getting more clear concerning what it means for a Bayesian agent to be guaranteed to converge to the truth—and to have a quick look at some representative facts about when this is possible. Suppose that we are interested in a repeatable chance process with a countable number of possible outcomes—tossing a coin, spinning a roulette wheel, drawing a ball from an urn, and so on. We assume that each outcome has a stable chance of occurring from repetition to repetition, with the outcomes of various repetitions being independent of one another. The propensities of the various outcomes are unknown. So there is a space 
of chance hypotheses whose points correspond to the various ways these propensities might be: in the case of a coin, 
will be the set of ordered pairs, 
, of nonnegative numbers that sum to one; in the case of a die, 
will be the set of sextuples, 
, of nonnegative numbers that sum to one; in the case of a process with a countably infinite number of possible outcomes, 
will be the set of sequences, 
, of nonnegative numbers that sum to one. In each case, 
gives the chance of getting outcome 
on any given trial.
A Bayesian agent facing this sort of setup will begin with a prior probability measure 
over 
, assigning various degrees of credence to various (sets of) chance hypotheses. After the first trial, 
will be updated by conditionalization to yield a posterior probability measure 
on 
; after the second trial, 
will be updated by conditionalization to yield a new posterior 
on 
, and so on.
We are interested in the conditions under which the sequence of posteriors 
generated by 
and an infinite sequence of outcomes converges to the truth. The idea is as follows. Suppose that 
is the true chance hypothesis. Look at a sequence of outcomes generated by the process (given that 
encodes the true propensities), and consider the corresponding sequence of posteriors 
determined by these outcomes and by the given prior 
. We say that this sequence of probability measures converges to the truth if as time goes on its terms becomes more and more tightly peaked around the true hypothesis 
. Here and below, to say that a sequence of probability measures 
becomes more and more tightly peaked about a given hypothesis 
is to say that the 
converges (in the weak topology; see n. 14 above) to the probability measure 
that puts unit weight on 
.Footnote 17
Clearly, there can be no general guarantee of success, even for Bayesian agents with eminently reasonable priors. Faced with an infinite sequence of tosses in which the limiting frequency of heads is one-third, any such agent will, as time goes on, become more and more certain that the bias of the coin is very close to one-third. But of course such a sequence of outcomes is possible even if the coin is in truth fair. So there can be no prior that is absolutely guaranteed to lead us to the right answer for this sort of problem.
But we can have something almost as good (although we have to put up with the unfortunate name that statisticians have given this good thing).
We call a prior 
on 
consistent at 
if, according to the chance hypothesis 
, the chance of a sequence of outcomes arising that, together with 
, would generate a sequence 
of posteriors that did not become more and more tightly peaked around 
is zero. And we call a prior 
consistent if it is consistent at every 
.
In the chance setting, a consistent prior is the best one can ask for: such a prior is essentially guaranteed to lead to the truth, in the sense that no matter which chance hypothesis is true, any nonpathological stream of data generated by that hypothesis would lead an agent with that prior to pile up more and more credence on smaller and smaller neighborhoods of the true hypothesis.
For problems of the kind under consideration, we have the following sort of result about subjective certainty:
For any prior 
, the set of hypotheses at which 
is not consistent is a set of 
-measure zero. (Freedman Reference Freedman1963, sec. 1)
Now, the fact that a given prior assigns measure zero to the set of hypotheses at which it is inconsistent is compatible with that set being very large indeed. If I start out life certain that the coin has a one-third chance of coming up heads on any toss, then my prior is consistent for just one point in 
.
A natural hope is that this sort of blatant a priori bias is the only obstruction to consistency—that any prior that was in some suitable sense spread over the entire space of hypotheses would turn out to be consistent. We say that a hypothesis 
is in the support of prior 
, if 
assigns positive measure to every open set containing 
. We say that a prior has full support, if it assigns positive measure to every nonempty open subset of 
.
When the number of possible outcomes of our chance process is finite, a prior 
is consistent at a hypothesis 
if and only if 
is in the support of 
.Footnote 18 (Freedman Reference Freedman1963, sec. 3)
A prior is consistent at a hypothesis, if it assigns that hypothesis a positive probability. But a prior can also be consistent at a hypothesis that it assigns zero probability, so long as it puts a finite amount of probability on each open set containing that hypothesis. The consistent priors are those that are nondogmatic in an intuitive sense: in this setting, the consistent priors are just the priors of full support—those that put positive measure on every open neighborhood of every hypothesis. And the priors of full support form a residual family in the space of priors on 
.Footnote 19
The situation changes markedly when we consider chance processes with a countable infinity of possible outcomes.
When the number of possible outcomes is infinite, for a prior 
to be consistent at a hypothesis 
it is necessary (but not sufficient) that 
be in the support of 
.Footnote 20 (Freedman Reference Freedman1963, sec. 5)
In this setting, for any two hypotheses 
, it is possible to find a prior 
with 
in its support, such that conditionalizing 
on any data sequence that is nonpathological according to 
leads to a sequence of posteriors 
that become more and more tightly peaked around 
as more and more data are revealed (Freedman Reference Freedman1963, theorem 5). So an agent with prior 
can be expected to be misled into thinking that the true chance propensities are given by 
when they are in fact given by 
.
Further, there are priors with full support that are consistent only at a meagre subset of 
(Freedman Reference Freedman1963, sec. 5, remark 6). Indeed, the priors of full support that are consistent only at a meagre set of hypotheses form a residual set in the space of priors defined on 
.Footnote 21 So there is a sense in which the typical prior is of full support but fails to converge to the truth in typical situations. But this is not because the problem is so-contrived as to put convergence to the truth beyond the pale of possibility: there exist consistent priors for this problem.Footnote 22
Where does this leave us? In ideally well-behaved settings, the posteriors of Bayesian agents whose priors have full support are essentially guaranteed to converge to the truth, and such priors are typical. But in the last case considered, typical priors fail to converge to the truth in typical situations, although there still exist special priors guaranteed to converge to the truth in every situation under consideration. For some problems, convergence to the truth is all but automatic for Bayesians; for others, it is very difficult (but not impossible) to come by.
Remark 3. We have been concerned with the most straightforward sort of statistical inference problem, in which the hypotheses under consideration are very closely connected with the chance distributions of observations. In many problems, this connection is more subtle. In some problems, convergence to the truth is in general impossible—and is not expected by typical Bayesian agents—because multiple hypotheses induce the same chance distribution on the space of sequences of measurement outcomes, so that observation of outcomes provides no way to discriminate between hypotheses. Statisticians call this phenomenon nonidentifiability. It is easy to construct silly examples of nonidentifiability. Example: suppose that one is told the outcomes of a sequence of tosses of a coin and asked to determine whether the coin is biased two-thirds in favor of heads or of tails and also whether it is a nickel, a dime, or a quarter. One expects that this evidence will allow one to correctly identify the bias of the coin but not its denomination.Footnote 23
Remark 4. Nonidentifiable models aside, statisticians have a rule of thumb that being of full support should suffice for a prior to be consistent when the space of hypotheses is finite dimensional (parametric models) but not when the space of hypotheses is infinite dimensional (nonparametric models).Footnote 24 Even in the nonparametric setting, sufficient conditions are known for a prior to be consistent—often they involve requiring that the problem be identifiable, that the prior avoid certain sorts of jaggedness, and that it spread credence over the space of hypotheses in some sense more demanding than having full support (see Wasserman Reference Wasserman, Dey, Müller and Sinha1998; Ghosh and Ramamoorthi Reference Ghosh and Ramamoorthi2003, chap. 4).
4. Good News, Rotten News
Let us now take as our space of hypotheses the space 
of infinite binary sequences. We will need a few facts about this space, known as the Cantor space. Space 
is the result of taking the product of 
with itself infinitely many times and as such carries the natural product space topology—the topology of pointwise convergence of sequences.Footnote 25 We can characterize the open sets in 
as follows: let 
be a string of bits; then, the set 
consisting of those sequences whose first 
bits are given by 
is an open set, and every open set arises as a union of sets of this form. It is straightforward to set up a correspondence between the points in the Cantor space and the points of the standard Cantor set: the mapping 
is a homeomorphism.
Think of each sequence in 
as corresponding to a possible sequence of records of outcomes of some family of observations. We might be looking at the outcomes of a sequence of coin tosses (perhaps with each toss involving the same coin, perhaps with different tosses involving coins of different biases). Or we might be looking at successive bits in a binary expansion of a constant of nature or determining whether there is more gold in China or in India, minute by minute.
We will be interested in Bayesian agents who begin life with priors over 
and then update by conditionalization as the bits of data making up some particular sequence are revealed one by one. In the current context, conditionalization works as follows. Consider the first bit learned. Using the notation introduced above, 
and 
(the set of sequences that begin with 0 and the set of sequences that begin with 1) are open sets. When you learn the first bit of data, you set your prior to zero on one of these sets and, up to normalization, leave it as it is on the other (i.e., from now on, you assign measure zero to any set contained in one of 
and 
but leave unchanged the ratio between the measures of sets contained in the other). And similarly, for each new bit you learn—if the first bit was 1, then after seeing that bit your posterior is concentrated on 
. After seeing the second bit, you have a new posterior that vanishes either on 
or on 
(depending on what the second bit was). And so on.Footnote 26
It is easy to construct priors of full support in this setting. For example, the fair coin measure, which assigns measure 
to any open set of the form 
, is a measure of full support. So is any measure that assigns nonzero weight to each point in a countable dense subset of 
(since then any open set includes points assigned positive measure).
Consistency in the sense of section 3 is easy to come by for this problem: every prior of full support is consistent. Given the way that conditionalization works for this problem, whatever the true sequence 
is, the result of conditionalizing a prior 
of full support on longer and longer initial segments of 
is to deliver a sequence of posterior probability measures 
whose support is limited to a sequence of smaller and smaller open sets that shrink toward 
.Footnote 27 So as time goes on, these posteriors become as tightly peaked as you like around the true hypothesis.Footnote 28 So convergence to the truth is guaranteed for priors of full support, and priors of full support are typical in the space of priors.Footnote 29
Nonetheless, there is a sense in which convergence to the truth can be formidably difficult in this setting. As we will see, there are questions we could ask our Bayesian agents about the sequences they are seeing such that then any prior will fail to converge to the truth for infinitely many sequences in 
—and such that priors that display a certain sort of attractive open-mindedness fail to converge to the truth for typical hypotheses in 
.
Let 
be any countable dense subset of 
.Footnote 30 Now, to say that 
is a dense subset of 
is to say that every point in 
is a limit of a sequence of points in 
—or, equivalently, that for any finite binary string 
, the set 
contains an element of 
. Further, each 
is an uncountable set, and so it must also have a nonempty intersection with the complement of 
. So the complement of our countable dense 
is also dense in 
. Examples of countable dense subsets of 
are the set of binary sequences that correspond to binary expansions of rational numbers, the set of binary sequences that are eventually constant, and the set of binary sequences that are periodic.Footnote 31
For the remainder of this section, we will give our Bayesian agents the following problem: determine whether the true sequence 
lies in 
.Footnote 32 Since the prior 
and the sequence 
of posteriors sum up the Bayesian’s opinion at each stage, the natural method for a Bayesian facing this problem is to be guided at each stage by 
.
What would count as getting at the truth here? Well, a rather undemanding criterion would be that either 
and, for sufficiently large 
, 
is always greater than or equal to one-half or 
and, for sufficiently large 
, 
is always less than one-half. So let us say that for the problem currently under consideration, the failure set for a prior is the set of hypotheses for which this condition does not hold. As one would expect, there is the usual sort of guarantee of subjective certainty of convergence to the truth:
Let 
be any prior on 
. Then, the corresponding failure set has 
-measure zero.Footnote 33
The question, of course, is what the failure sets of various priors look like. Some priors correspond to agents who begin life certain that the true sequence lies in 
and who remain thus no matter what data they see—for these priors, the failure set is the uncountable dense complement of 
in 
.Footnote 34 Other priors correspond to agents who begin life certain that the true sequence 
lies outside of 
and who maintain this certainty no matter what data they see—for such priors, the failure set is the countable dense set 
.Footnote 35
Either sort of prior can have full support.Footnote 36 So in the current context, being of full support does not suffice to guarantee convergence to the truth—indeed, it is compatible with being entirely closed minded about whether the true sequence belongs to 
.
What would it take to truly leave the door open concerning the question whether the true sequence belongs to 
? Since this problem has the feature that any finite amount of data is consistent both with the true sequence being in 
and with it not being in 
, it is natural to consider a condition along the following lines.
Let 
be a measure on 
and 
be a (measurable) subset of 
. Measure 
is open-minded for 
if, for any data set an agent with 
as a prior sees, there are extensions of this data set that would lead the agent to assign a credence of at least one-half to the proposition that the true sequence lies in 
and other extensions of the data that would lead the agent to assign this proposition credence less than one-half.
Agents with open-minded priors never make up their minds irredeemably on the basis of a finite amount of data. It is not hard to see that open-minded priors exist for countable and dense 
of the sort we are concerned with.Footnote 37
We need one more definition before getting down to business.
We say that hypothesis 
flummoxes prior 
if there are infinitely many 
for which 
and infinitely many 
for which 
.
Agents faced with flummoxers flip-flop ad infinitum on the question whether it is at least as likely as not that the true sequence lies in 
.
1. Any prior whose failure set was empty would have to be open-minded. No matter what data an agent has been shown, those data are consistent with both
and . If the prior has an empty failure set, it has to be prepared for either eventuality, so there must be ways of continuing that would lead to credence above one-half and others that would lead to credence below one-half.2. Any open-minded prior is flummoxed by some hypothesis. Given an open-minded prior, we can construct a flummoxing hypothesis as follows.Footnote 38 Start off by showing some data that lead to a credence greater than one-half that the true sequence is in
. There are sequences that begin this way but that are not in (since the complement of is dense in ). Pick one of them, and continue the data stream by showing a block of bits from this sequence long enough to force the agent’s credence that the true sequence is in below one-half—there must be such a continuation since the agent is open-minded. Now, whatever string of bits has been shown to the agent, there is some sequence in that begins this way (since is dense in ). So, since the agent is open-minded, we can choose a continuation of the data stream that will eventually force the agent’s credence that the sequence being revealed lies in above one-half. Continue this bait and switch ad infinitum. In this way, we construct a binary sequence that flummoxes our open-minded agent.3. No prior has an empty failure set. This follows immediately from the preceding two points and the fact that every flummoxer of a prior is in its failure set.
4. Each prior has an infinite failure set that is dense in the space of hypotheses. Let
be any prior and let be any binary sequence in our space of hypotheses. We need to show that there is a sequence of hypotheses in the failure set of that converges to (this will establish that the failure set of is dense in and hence infinite). It suffices to show that for each there is an in the failure set of that has the same first bits as .Footnote 39 So for arbitrary, let be the initial -bit string of . There are two cases: there is some flummoxer of that begins with , or there is not. If there is, then we can take this flummoxer as . If not, then for every , there is some point in the binary sequence after which has made up its mind irredeemably and can no longer be caused to flip-flop on the question whether the sequence being revealed lies in . In particular, this is the case for itself. Suppose that makes up its mind irredeemably after being shown the first bits of . So no matter how we continue the data sequence after (the first bits of ), we always find that gives us the same answer about whether it is more likely than not that the binary sequence being revealed lies in . But this means that for some binary sequences in , settles on the wrong answer since some sequences in lie in and others do not. So in this case we can let be one of these binary sequences that is wrong about.5. The set of flummoxers of an open-minded prior is residual in
. We can think of a prior as a machine that, when fed a binary sequence bit by bit, outputs a sequence of Ys and Ns (telling us, at each stage, whether ). For a given prior and a given data sequence , we say that the prior never switches for that data sequence, if the prior outputs a sequence that is all Ys or all Ns. We say that it switches at least once if the output sequence begins with one or more Ys followed by one or more Ns (or vice versa). We say that it switches at least twice if the output begins with some Ys followed by some Ns followed by some Ys (or vice versa). And so on. Consider an open-minded prior . It is not hard to see that for each , the set of for which the prior switches at least times is open and dense in .Footnote 40 Now, the set of flummoxers of is just the set of points in that get to switch at least times for each (i.e., the set of flummoxers of is just the intersection of all the ). But a countable intersection of open dense sets is residual. So the set of flummoxers of is residual in .Footnote 416. Open-minded priors fail to converge to the truth for typical hypotheses when faced with the problem of deciding whether the sequence they are being shown lies in
. The set of flummoxers of a given prior is a subset of its failure set. So the failure set of an open-minded prior is residual in the space of hypotheses. So, in particular, the failure set is an uncountable set, dense in the space of hypotheses.Footnote 42
and 
. If the prior has an empty failure set, it has to be prepared for either eventuality, so there must be ways of continuing that would lead to credence above one-half and others that would lead to credence below one-half.
. There are sequences that begin this way but that are not in 
(since the complement of 
is dense in 
). Pick one of them, and continue the data stream by showing a block of bits from this sequence long enough to force the agent’s credence that the true sequence is in 
below one-half—there must be such a continuation since the agent is open-minded. Now, whatever string of bits has been shown to the agent, there is some sequence in 
that begins this way (since 
is dense in 
). So, since the agent is open-minded, we can choose a continuation of the data stream that will eventually force the agent’s credence that the sequence being revealed lies in 
above one-half. Continue this bait and switch ad infinitum. In this way, we construct a binary sequence that flummoxes our open-minded agent.
be any prior and let 
be any binary sequence in our space of hypotheses. We need to show that there is a sequence 
of hypotheses in the failure set of 
that converges to 
(this will establish that the failure set of 
is dense in 
and hence infinite). It suffices to show that for each 
there is an 
in the failure set of 
that has the same first 
bits as 
.
arbitrary, let 
be the initial 
-bit string of 
. There are two cases: there is some flummoxer of 
that begins with 
, or there is not. If there is, then we can take this flummoxer as 
. If not, then for every 
, there is some point in the binary sequence 
after which 
has made up its mind irredeemably and can no longer be caused to flip-flop on the question whether the sequence being revealed lies in 
. In particular, this is the case for 
itself. Suppose that 
makes up its mind irredeemably after being shown the first 
bits of 
. So no matter how we continue the data sequence after 
(the first 
bits of 
), we always find that 
gives us the same answer about whether it is more likely than not that the binary sequence being revealed lies in 
. But this means that for some binary sequences in 
, 
settles on the wrong answer since some sequences in 
lie in 
and others do not. So in this case we can let 
be one of these binary sequences that 
is wrong about.
. We can think of a prior as a machine that, when fed a binary sequence bit by bit, outputs a sequence of Ys and Ns (telling us, at each stage, whether 
). For a given prior 
and a given data sequence 
, we say that the prior never switches for that data sequence, if the prior outputs a sequence that is all Ys or all Ns. We say that it switches at least once if the output sequence begins with one or more Ys followed by one or more Ns (or vice versa). We say that it switches at least twice if the output begins with some Ys followed by some Ns followed by some Ys (or vice versa). And so on. Consider an open-minded prior 
. It is not hard to see that for each 
, the set 
of 
for which the prior switches at least 
times is open and dense in 
.
is just the set of points in 
that get 
to switch at least 
times for each 
(i.e., the set of flummoxers of 
is just the intersection of all the 
). But a countable intersection of open dense sets is residual. So the set of flummoxers of 
is residual in 
.
. The set of flummoxers of a given prior is a subset of its failure set. So the failure set of an open-minded prior is residual in the space of hypotheses. So, in particular, the failure set is an uncountable set, dense in the space of hypotheses.5. Orgulity
As we have just seen, if a Bayesian agent is shown a binary sequence bit by bit and asked after each bit is revealed for the probability that the sequence as a whole has a property of a certain type—such as being a binary expansion of a rational number or being periodic or being eventually constant—then there will be a rich family of sequences that the agent could be shown that would frustrate convergence to the truth on this question. Agents who are open-minded, in the sense that they never make up their minds irredeemably on the basis of a finite amount of evidence, are frustrated in this way by typical binary sequences.
This in itself does not represent a shortcoming of Bayesianism: define a method to be a function from finite binary strings to guesses about whether the sequence being revealed has the target property, and define an open-minded method to be one that never makes up its mind irredeemably on the basis of a finite amount of evidence, and so on. Then analogues 1–6 of section 4 carry over to this more general setting. What sets Bayesian agents apart is that, despite the patent intractability of the problem at hand, they assign probability zero to the set of sequences that they cannot handle successfully.Footnote 43
Suppose that I face a problem of the type under discussion. I am inclined to be a Bayesian and have extraordinary powers of voluntary belief. So I sit down to look at some priors to decide which to adopt. One factor among many that I will weigh up in evaluating a prior is the nature of its failure set—the set of binary sequences with the feature that if I am shown them, I will fail to latch on to the truth about whether the sequence has the property of interest. The failure set of each prior I consider is nontrivial—always infinite and dense, in some cases even residual. And each prior considers its own failure set to be negligible, assigning it measure zero. I may find a prior that I like on other grounds but worry that its failure set is rather large, but tinkering with it will never get me very far. For example, if a prior 
has a countable failure set, I can construct a new prior 
arbitrarily similar to 
that converges to the truth for every sequence in the failure set of 
, but now 
will fail to converge to the truth for infinitely many sequences that 
was able to handle.Footnote 44
Reflecting on this I may well, even while endorsing my favorite method as well suited to this problem, also think that there is some chance that nature will be unkind and frustrate my desire to reach the truth. According to Bayesianism, this combination of thoughts is incoherent: to commit to a method is to commit to a prior, and (for this problem) no prior licenses the belief that there is some chance that it will fail to lead to the truth in the long run.
Bayesian convergence-to-the-truth theorems tell us that Bayesian agents are forbidden to think that there is any chance that they will be fooled in the long run, even when they know that their credence function is defined on a space that includes many hypotheses that would frustrate their desire to reach the truth. This is a bizarre limitation—and one that ought to be unwelcome to those who are attracted to Bayesianism because of its apparent flexibility.
Let me wind up by heading off two lines of response that may seem appealing.
i) Some Bayesians may be attracted to a view on which an agent with a given prior is entitled to consider all hypotheses in its failure set to be mere skeptical hypotheses. Collectively, the set of skeptical scenarios ought to be assigned measure zero. So there is no embarrassment here, no matter how large the failure set should turn out to be.
But that would be a strange way to think of the current problematic. We are interested in the behavior of a Bayesian agent’s posteriors in the infinite long run, in a setting in which every false hypothesis is eventually ruled out definitively by observation and in which these posteriors converge to the delta function measure concentrated on the true hypothesis. By ordinary standards, the hypotheses in the failure set of a prior are not skeptical scenarios.
ii) Bayesians who countenance substantive restrictions on rational priors may be tempted to forbid open-minded priors for problems of the type under consideration. This is in itself a strange idea. Suppose, for instance, that I am interested in some physical system whose dynamics may well be chaotic and that the binary sequence I am to be shown encodes some information about the history of the system (imperfect example: it encodes which of two points on the surface of Hyperion is closer to the center of Saturn, second by second). Suppose, further, that by examining this sequence bit by bit, I aim to determine whether the sequence as a whole is periodic. I begin by thinking that the system’s dynamics is likely chaotic but that there is some chance that it is instead periodic. If it is chaotic, then I expect that the sequence I am shown will not be periodic but that it will mimic periodic sequences for long stretches of time.Footnote 45 If the system is periodic, the sequence will be periodic, but whatever reasons I have to think that the system is likely chaotic are also reasons to think that if the sequence is periodic, it will be very complicated. Is it really plausible that rationality requires me to think that there is some possible finite evidence set that would decide the question one way or the other, no matter what evidence should later come in?
In any case, there are close relatives of our problem that afflict all priors, open-minded or not. Suppose that a Bayesian agent is shown a binary sequence
encoding, for each of a sequence of days, whether it snows at a given location. On day , our agent is required to announce the probability of snow on the next day, given the pattern of snowy and snowless days to date. The discrepancy on day is . A Bayesian agent is high-low calibrated relative to a given binary sequence if the mean discrepancy goes to zero when we specialize to days on which and also to days on which . If a forecaster fails to be high-low calibrated, then a rival can complain that there is a mismatch, for days on which the forecaster predicted (or did not predict) snow, between the actual relative frequency of snowy days and the forecast probabilities of snow. All Bayesian forecasters assign probability zero to the set of sequences for which they are not high-low calibrated (Dawid Reference Dawid1982). And all Bayesian forecasters fail to be calibrated for typical data sequences.Footnote 46
encoding, for each of a sequence of days, whether it snows at a given location. On day 
, our agent is required to announce the probability 
of snow on the next day, given the pattern of snowy and snowless days to date. The discrepancy on day 
is 
. A Bayesian agent is high-low calibrated relative to a given binary sequence if the mean discrepancy goes to zero when we specialize to days on which 
and also to days on which 
. If a forecaster fails to be high-low calibrated, then a rival can complain that there is a mismatch, for days on which the forecaster predicted (or did not predict) snow, between the actual relative frequency of snowy days and the forecast probabilities of snow. All Bayesian forecasters assign probability zero to the set of sequences for which they are not high-low calibrated (Dawid Some have seen in the tendency of Bayesian agents to converge to the truth—and in related results concerning the eventual merger of opinion between Bayesian agents whose initial credences share a certain amount of common ground—the materials for acquitting personalist Bayesianism of the charge of excessive subjectivity.Footnote 47 But recent philosophical commentators (some Bayesians among them) have tended to downplay the significance of these results, pointing out that what they guarantee is that Bayesian agents think that there is no chance that their own future opinions will fail to converge to the truth, which is not the same thing as saying that the opinions of each Bayesian agent are in fact certain to converge to the truth.Footnote 48 The truth concerning Bayesian convergence-to-the-truth results is significantly worse than has been generally allowed—they constitute a real liability for Bayesianism by forbidding a reasonable epistemological modesty.
as follows: if 
is the ternary expansion of 
in which each 
is 0 or 2, then we define 
by 
and let 
be the real number in 
whose binary expansion is 
. This map is nondecreasing and continuous. But it is not one to one: 1/2 can be written in binary either as 
or as 
, so 
maps 1/3, which admits the ternary expansion 
, to the same point that it maps 2/3, which admits the ternary expansion 
. For a characterization of the countable set of pairs of points at which this problem arises, see, e.g., Gelbaum and Olmsted (
sum to one.
in constructing 
is open. So each 
is closed—so 
itself is closed. So in order to see that 
is nowhere dense, it suffices to note that, since each such subinterval contains numbers that require ones in their ternary expansions, 
has an empty interior.
that interchanges the null sets and the meagre sets and that is its own inverse. So from a suitably lofty perspective, the null sets and the meagre sets are functionally equivalent: any sentence that involves just set-theoretic terminology together with ‘meagre’ or ‘null set’ is equivalent to the sentence that arises by swapping the roles of these two terms. See Oxtoby (
. Let 
be a function from 
to itself that maps 
to 
, where 
is the Lebesgue measure on 
. For 
, let 
. This is a strictly increasing continuous map from 
onto itself—and hence a homeomorphism (see, e.g., Dieudonné 
is constant on the complement of 
, the derivative of 
on this set is 1/2, so this set of measure one gets mapped to a set of measure one-half, which means that 
must get mapped onto a set with the same measure. This is a special case of a more general result; see Oxtoby (
and construct 
by deleting the middle open interval of length 
from 
, construct 
by deleting the two middle intervals of length 
from 
, and so on, deleting at each stage intervals of total length 
. The intersection 
of the 
has Lebesgue measure 
and is a fat Cantor set. See Gelbaum and Olmsted (
be a countable union of fat Cantor sets 
with 
. See, e.g., Gelbaum and Olmsted (
of a topological space 
is called nowhere dense if the closure of 
has an empty interior.
, the corresponding space of priors will be the space of (countably additive) Borel probability measures over 
equipped with the topology of weak convergence: to say that the sequence of measures 
converges to the measure 
is to say that 
for each bounded continuous function 
on 
.
of n. 9 above has measure one; our result tells us that typical Borel probability measures consider both to be sets of measure zero.
be the number of possible outcomes. For 
and 
, a natural number less than or equal to 
, let 
be the set of 
such that 
, for 
, where 
and 
are the weights assigned to the 
th possible outcome by 
and by 
, respectively. Thus, 
weakly if and only if for all 
and 
, 
.
and 
, let 
be the set of 
such that 
. In the current context, a prior has 
in its support if and only if it assigns a positive measure to each 
.
currently under consideration are compact and metrizable (Freedman 
corresponds to an infinite sequence 
of weights assigned to each possible outcome (relative to some fixed enumeration of the outcomes). For each 
and 
, let 
be the set of 
in 
such that 
. A prior on 
includes 
in its support if and only if it assigns positive measure to 
for each 
.
be a point in 
. And for each 
, let 
be a point in 
. Then 
in this topology if and only if, for each 
, there is a 
such that 
for all 
.
converge weakly to 
. This follows from Billingsley (
of some topological space 
is closed under finite intersections and that every open subset of 
is a countable union of sets in 
; then, if 
are measures on 
such that if 
for every 
, then the 
converge weakly to 
. Now, in our case, the class of 
satisfies these hypotheses. So we need only show that for any binary string 
, 
. To see this, note that if 
is in 
, then 
, and for sufficiently large 
, the support of 
is contained in 
so that 
, while if 
, then 
, and for sufficiently large 
, 
is outside the support of 
, so we have 
.
has the topology of the Cantor set, which as a closed subset of a compact metrizable space is compact and metrizable. So the results quoted in n. 19 above apply to probability measures on 
.
is, of course, (Borel) measurable, being a countable union of singleton sets (each of which is closed).
arises as a limit of a sequence 
drawn from either set—for each 
, let 
be an eventually constant or periodic binary sequence that begins with the same first 
bits as 
.
with dense complement. Everything below would still go through, except for the argument of n. 37.
such that neither 
and 
nor 
and 
is a set of measure zero. Their corollary 5 justifies us in treating every subset of this set as a set of measure zero, and the failure set is itself such a subset.
, and then put 9/10 unit of credence on the first, 9/100 on the second, etc.
as a whole.
that is a nontrivial mixture of the fair coin measure with a measure of the sort discussed in n. 34 is open-minded. As usual, we write 
for the posterior generated from 
and 
bits of data. Let us also write 
for 
and 
for 
. In order to show that our 
is open-minded, it suffices to show that after seeing 
bits, (i) if 
, then there is a way of continuing that data set to include 
bits such that 
and, (ii) if 
, then there is a way of continuing that data set to include 
bits such that 
. Since all that matters is the ratio of 
to 
, let us ignore normalization—update 
by just killing off the part of it that lives on the region inconsistent with the bit revealed at stage 
. In order to see i, note that at each stage, no matter what bit turns up next, the measure of the complement of 
will go down by a factor of two. Meanwhile, the measure of 
decreases by a factor of nine whenever a new bit is shown that rules out the member of 
of maximum probability among those consistent with the data thus far. So by choosing at each stage to show a bit of this kind, we can ensure that after a finite number of new bits, 
. In order to see ii, let 
be a point in 
consistent with the data through stage 
. Then, for some 
, 
. So if we extend the data set by showing bits from 
, we can ensure that 
, even while 
gets halved every time a new bit is revealed. So after a finite number of bits are revealed on this scheme, 
, as desired.
, 
for 
, so we have 
pointwise.
causes 
to switch at least 
times. Then there must be some 
such that the first 
bits of 
already suffice to make 
switch 
times. So 
must switch at least 
times for every 
(where 
is the 
-bit initial segment of 
). So 
is open. And if 
is any point in 
, then we can construct a sequence 
of points in 
that converge to 
: each 
starts with the first 
bits of 
and then continues in some way that makes 
change its mind 
times (possible since 
is open-minded). The 
converge pointwise to 
—for each 
, we know that 
for 
. So 
is dense in 
.
to be class of (nonempty) sets of form 
, so that we can think of player 1 and player 2 as taking turns specifying finite binary strings that are then concatenated to yield a binary sequence, with player 1 winning if and only if the resulting sequence lies in the target set 
. Let 
be the set of nonflummoxers of prior 
. Player 2 has a winning strategy: no matter what player 1 does, player 2 plays a string of bits that causes 
to output Y followed by a string that causes it to output N. This always results in an infinite sequence that flummoxes 
. Since player 2 has a winning strategy, the set of nonflummoxers must be meagre in 
.
by rescaling 
to make available an arbitrarily small amount of credence and then parcelling it out among the sequences in the failure set of 
. Prior 
will have an infinite failure set, disjoint from that of 
. The failure set of 
may even be residual, whereas the failure set of 
was meagre—e.g., 
may be the fair coin measure, and 
may be a measure of the type considered in n. 37 above.
consist of those sequences that do not start with some string 
repeated 
times, for each 
. Consider a Banach-Mazur game of the type described in n. 41 above, with 
as the designated set. Player 2 has a winning strategy: on the 
th turn, play 
repetitions of whatever has been played so far. So 
is meagre. This means that even though typical sequences in 
are not periodic (the set of periodic sequences is countable and, so, meagre), typical sequences begin by repeating some string twice, some string three times, some string four times, etc.
consisting of sequences for which a given forecaster is high-low calibrated. Player 2 has a winning strategy: each bit played is designed to make the forecaster look bad (so a 1 is played as the 
th bit if and only if 
), and enough bits are played on each turn to make the mean discrepancy on days for which snow is forecast go above .25 or the mean discrepancy for days on which no snow is forecast go below −.25. For further discussion, see Belot (
