Improper priors and finitely additive probabilities (FAP) are the two main alternatives to the standard Bayesian Paradigm using proper priors, i.e. countably additive probabilities in the Kolmogorov axiomatic. Both alternatives are involved in paradoxical phenomena such as non-conglomerability, marginalization paradox, etc. As an heuristic argument, some authors such as Stone (1982) or Kadane et al. (1986, p.218), consider proper prior sequences, for example sequence of uniform prior, and derive under two different topologies two kind of limits: FAPs limits and improper limits. The choice between FAP distribution and improper distribution has been largely debated in the Bayesian literature (see Hartigan, 1983, p.15).
The aim of FAP limits is to preserve the total mass equal to 1, while sacrificing the countable additivity. This point of view has been mainly defended by de Finetti (1972). On the other hand, improper distributions aim to preserve the countable additivity, while sacrificing a total mass equal to 1. Improper distribution appears naturally in the framework of conditional probability, see Rényi (1955) and more recently Taraldsen and Lindqvist (2010, 2016) and Lindqvist and Taraldsen (2018). Conditional probability spaces are also related to projective spaces of measures (Rényi, 1970) which have a natural quotient space topology and a natural convergence mode, named -vague convergence by Bioche and Druilhet (2016).
In Bayesian inference some paradoxes such as non-conglomerability(Stone, 1976, 1982) or the marginalization paradox (Dawid et al., 1973) occur with improper or diffuse FAP priors (Dubins, 1975), but not with proper priors. This lead some authors to include the likelihood in the definition of a convergence mode for the priors, by for instance considering the convergence of the posterior distribution w.r.t. to an entropy criterion (Akaike, 1980) or an integrated version of this criterion (Berger et al., 2009) when the posterior is proper. Bayesian inference with improper posterior is justified by Taraldsen et al. (2019) from a theoretical point of view. Bord et al. (2018)
consider the convergence of proper to an improper posterior for Bayesian estimation of abundance by removal sampling.Tufto et al. (2018) propose to adapt MCMC for the estimation of improper posterior.
In this paper, we only consider convergence of distributions regardless to any statistical model. The implication of our result in Bayesian inference, especially the construction of specific sequences of distribution used in the proof of Theorem 3 is left for future works. In Section 2, we define several concept of convergence in the setting of improper and FAP distributions. In Section 3 we revisit the notion of uniform distribution on integer. In Section 4, we illustrate the fundamental difference between convergence to an improper prior and to an FAP.
2 Convergence of probability sequences
We denote by the set of continuous real-valued bounded functions, by the set of continuous real-valued functions with compact support. For a -finite measure , we denote . Consider a sequence of proper priors . The usual converge mode of to a proper prior is the narrow convergence, also called weak convergence or convergence in law, defined by:
When it exists, the narrow limit of is necessarily unique. In this section, we consider two alternative convergence modes when there is no narrow limit, and especially when the total mass tends to concentrate around the boundary on the domain, more precisely when for all in . The idea is to consider a proper prior as a special case of FAP or as a special case of a Radon measure, and for each case to define in a formalized way, a convergence mode. In both cases the limit is not unique in general but, as a requirement, must coincide with the narrow convergence when Eq. (1) holds.
In the following, we consider that is a metric, locally compact, second countable Hausdorff space
. This is the case, for example, for usual topological finite-dimensional vector spaces or denumerable sets with the discrete topology. In the latter case, any function is continuous and a compact set is a finite set.
2.1 Convergence to an improper distribution
To extend the notion of the narrow limit, we consider here proper distributions within the set of projective space of positive Radon measures as follows: we denote by the set of non-null Radon measures, that is regular countably additive measures with finite mass on each compact set. Note that, in the discrete case any -finite measure is a Radon measure.
We define an improper distribution as an unbounded Radon measure which appear in parametric Bayesian statistics (see, e.g. Jeffreys, 1970). The projective space associated to is the quotient space for the equivalence relation defined by iff for some positive scalar factor . To a Radon measure , it can be associated a unique equivalence class . Therefore, a projective space is a space where objects are defined up to a positive scalar factor. It is natural in Bayesian statistics to consider such projective space since two equivalent priors give the same posterior. The projective space is also naturally linked with conditional probability spaces (Rényi, 1955). All the results presented below on the convergence mode w.r.t. to the projective space can be found in Bioche and Druilhet (2016). The usual topology on is the vague topology defined by
where is the set of all real-valued functions on with compact support.
From the related quotient topology, we can derive a convergence mode, called q-vague convergence: a sequence in converge to a (non-null) improper distribution in if converges to w.r.t. the quotient topology where is the equivalence class associated to and similarly for . The limit is unique whereas is unique only up to a positive scalar factor. It is not always tractable to check a convergence in the quotient space. However, there is an equivalent definition in the initial space : converges to if there exists some scalar factors such that converges vaguely to :
The -vague convergence can be considered as an extension of the narrow convergence in the sense that if and are proper distributions and converge narrowly to then converge q-vaguely to . Note that the converse part holds if and only if is tight (see Bioche and Druilhet, 2016, Proposition 2.8).
When a sequence of proper distributions converges q-vaguely to an improper distribution, then for any compact (Bioche and Druilhet, 2016, Prop. 2.11). The following lemma gives an apparently stronger, but in fact equivalent, result. It will be useful to establish the main result and to construct examples in Section 4.3.
Let be a sequence of proper distributions such that for any compact . Then there exists a non-decreasing sequence of compact sets such that and . Moreover, may be chosen such that, for any compact , there exists an integer such that .
Let , , be a increasing sequence of compact sets with . For each , , so there exists an integer such that and for . Consider now such a sequence of integers , . For any there exists a unique integer such that . We define by . So, . Since increases with , . Futhermore, the sequence can be chosen such that, for any compact , is a subset of all but finitely many , (see e.g. Bauer, 2001, Lemma 29.8). By construction, the same property holds for the sequence . ∎
Note that, for any compact set does not imply that converge q-vaguely. See Section 4 for some examples of such sequences.
2.2 Convergence to a FAP
In this section, we consider a proper distribution as special cases of FAPs. When a sequence of proper distribution does not converge narrowly to a proper prior, we need to define a weaker convergence mode so that the sequence can converge to some FAPs. This convergence mode will be named FAP convergence, and the corresponding limits, the FAP limits.
Denote by the set of bounded real-valued measurable functions on . A FAP can be defined as a linear functional on which is positive, i.e. if and which satisfies . Therefore, the set of FAPs belongs to the topological dual of equipped with the sup-norm. For any measurable set , we define , where if and 0 otherwise. We also denote . Since proper distributions are required to be special cases of FAPs, we do impose here measurability conditions in the definition of , contrary to many authors (see, e.g. Heath and Sudderth, 1978). In the case where is a denumerable set equipped with the usual discrete topology, any functions or sets are measurable and therefore this distinction is not relevant.
Consider a sequence of FAPs. The minimal requirement for to be a FAP limit is that for any , where is the set of such that exists. The existence of a limit that satisfies this requirement is guaranteed by the Hahn-Banach theorem (see Rudin, 1991; Huisman, 2016) as follows: define the linear function on by and the sublinear functional . Then, there exists a linear functional on that coincides with on and that satisfies on . The functional will be called a FAP limit. The condition implies that is a FAP. Conversely, a FAP necessarily satisfies . Replacing by gives . Therefore, a FAP limit can be characterized by the following lemma:
A FAP is a FAP-limit of the sequence if and only if
or equivalently if and only if
When , the FAP limit is unique and corresponds to the convergence associated to the weak-topology. In the general case, the FAP limit is not unique and its existence relies on the axiom of choice. Unlike the q-vague convergence, the FAP convergence cannot be considered as an extension of the narrow convergence. For example, consider the proper distributions , where is the Dirac measure. The sequence converges narrowly to but is not a FAP limit of . To show this, consider , with the set rational numbers, we have . However, in the special case where is a denumerable set, any real-valued function on is continuous and therefore if a sequence of proper distribution converges narrowly to a proper distribution , then is a FAP limit.
Another way to define a FAP limit in a formalized way has been proposed by Stone (1982) for denumerable sets but can be extended to more general sets considered in this paper. The existence of a limit relies on the Banach-Alaoglu-Bourbaki theorem (see, e.g. Rudin, 1991), since a FAP belongs to the unit ball in the dual of which is compact for the weak-topology. Hence, for any sequence of FAPs, there exists at least one accumulation point which is defined as a FAP limit. We recall that is an accumulation point of for the weak-topology if and only if for any integer , any in and any , there exists an infinite number of such that , . Note that, since is not in general first-countable, there does not necessarily exist a subsequence that converges to . We can only say that, for any , there exists a subsequence such that converges to .
Therefore, the set of FAP limits of obtained by Stone’s approach is included in the set of FAP limits obtained by using the Hahn-Banach theorem as above and (4) or (5) still hold but are not sufficient conditions. The converse inclusion is false in general. It is easy to see that the closed convex hull of the set of FAP limits defined by Stone is included in the set of FAP limits defined in this paper. We conjecture that, conversely, the set of FAP limits defined by (4) is the convex hull for the limits defined by Stone. Consider for example and . There are only two FAP limits and with Stone’s construction, whereas any , is a FAP limit with our construction. In Section 4.1, we illustrate the difference between the two convergence modes with another example.
2.3 FAP limits vs q-vague convergence
The fact that a sequence of proper distributions have both improper and FAP limits may suggest a connection between the two notions as proposed heuristically by many authors. The following results show that this is not the case. Roughly speaking, it is shown that any FAP which is a FAP limit of some proper distribution sequence can be connected to any improper prior by this mean.
Let be a sequence of proper distributions such that for any compact set . Then, for any improper distribution , it can be constructed a sequence which converges q-vaguely to and which has the same set of FAP limits as .
For any FAP or any proper or improper distribution we define the distribution by . From Lemma 1, it can be constructed an exhaustive increasing sequence of compact sets such that . Put and define the sequence of proper distributions , with the complement of . By Lemma 8 and 7 in Appendix A, has the same FAP limits as . By Lemma 9, converges q-vaguely to .
Let be a sequence of proper distributions that converges q-vaguely to an improper distribution . Then, for any other improper distribution , it can be constructed a sequence that converges q-vaguely to and that has the same FAP limits as .
The only link that can be established between improper q-vague limits and FAP limits of the same proper distribution sequence is that the FAP limits are necessarily diffuse, i.e. they assign a probability 0 to any compact set.
3 Uniform distribution on integers
In this section, we compare different notions of uniform distributions on the set of integers . by using several considerations such as limit of proper uniform distributions.
We illustrate the fact that FAP uniform distributions are not well defined objects (de Finetti, 1972, pp.122,224). Contrary to uniform improper distributions, FAP limits of uniform distributions on an exhaustive sequence of compact sets are highly dependent on the choice of that sequence.
3.1 Uniform improper distribution
There are several equivalent ways to define a uniform improper prior on integers. These definitions lead to a unique, up to a scalar factor, distribution. The uniform distribution can be defined directly as a flat distribution, i.e. for integer. It is also the unique (up to a scalar factor) measure that is shift invariant, i.e. such that for any integer and any set of integers . The uniform distribution is also the q-vague limit of the sequence of uniform proper distributions on . More generally and equivalently, the uniform distribution is the q-vague limit of any sequence of proper uniform priors on an exhaustive increasing sequence of finite subsets of integers.
3.2 Finitely additive uniform distribution
The notion of uniform finitely additive probabilities is more complex. Contrary to the improper case, there is no explicit definition since for any integer . We present here several non equivalent approaches to define a uniform FAP. The first two ones can be found in Kadane and O’Hagan (1995) and Schirokauer and Kadane (2007).
3.2.1 Shift invariant (SI) uniform distribution
As for the improper case, a uniform FAP distribution can be defined as been any shift invariant FAP, i.e. FAPs satisfying for any subset of integers and any integer . Such a distribution will be called SI-uniform. In that case, one necessarily has : , for any . Kadane and O’Hagan (1995) also investigate the properties of FAPs satisfying only , where are called residue classes.
3.2.2 Limiting relative frequency (LRF) uniform distributions.
Kadane and O’Hagan (1995) consider a stronger condition to define uniformity. For a subset , define its limiting relative frequency LRF by
when this limit exists. A FAP on is said to be LRF-uniform if when .
Let be the uniform proper distribution on , then . Therefore a FAP is LRF uniform if and only if it is a FAP limit of . It is worth noting that, unlike the improper case, the FAP limits are highly dependent on the choice of the increasing exhaustive sequence of finite sets . Changing the sequence changes the notion of uniformity. For example, if is the uniform distribution on , then , whereas .
3.2.3 Bernoulli Scheme (BS) uniform distribution
We propose here another notion of uniformity that is not dependent of the choice a particular increasing sequence of as for the LRF uniformity. Consider a Bernoulli Scheme, that is a sequence. Define the random set . A FAP is said to be BS-uniform if, for any ,
, almost surely. Note that the strong law of large numbers, LRF, almost surely.
Let be an increasing sequence of finite subsets of , with being infinite. Then any FAP-limit of the sequence of uniform distributions on is BS-uniform.
When , this proposition shows that any FAP limit of uniform distribution is BS-uniform. In particular, a LRF uniform FAP is also BS uniform. However, if for example is the set of even numbers less or equal to , then any FAP-limit of the sequence of uniform distributions on will be BS-uniform, although it is intuitively, certainly not uniform on but on . Therefore, BS uniformity looks much more like a necessary condition for a FAP to be uniform, than like a complete definition.
4 Comparison of convergence modes on examples
We consider here some examples that illustrate the difference between convergence of proper distributions to an improper distribution or a to FAP.
4.1 FAP limits on .
For a sequence of proper distribution on , it is known that there does not necessarily exist a q-vague limit, but if it exists, it is unique in the projective space of Radon measures, i.e. unique up to a scalar factor. At the opposite, we have seen that a FAP limit always exists but is not necessarily unique.
We illustrate the non-uniqueness of FAP limits with an extreme case. Consider the sequence of proper distributions , where is the Dirac measure on . This sequence has no q-vague limit since for . Moreover, for any subset of so that and are both infinite:
whereas, for any finite set , and for any cofinite set , . Therefore, from (5), the set of FAP limits of is the set of all diffuse FAPs on . This shows that all the diffuse FAPs are connected through the same sequence .
Let’s examine the set of FAP limits of obtained with Stone’s definition of FAP convergence (see Section 2.2). For any subset , there exits a subsequence such that convergences to . So, Therefore the FAP limits of in Stone’s sense are all remote FAPs, that is diffuse FAPs such that , as defined by Dubins (1975, p.92). This also proves the existence of remote FAPs. Note that a remote distribution is neither BS uniform nor SI and therefore cannot be LRF uniform.
4.2 Convergence of sequence of Poisson distributions
We consider the sequence
of Poisson distributions with mean. For any finite set , we have . However this sequence of proper priors does not converge -vaguely to any improper distribution (Bioche and Druilhet, 2016, §5.2). As a remark, let be shifted measures on , defined by , where can be seen as a measure on the set of all integers with for
. Then, using the approximation of the Poisson distribution by a the normal distribution, it can be shown that the sequenceconverges vaguely to the improper uniform measure on .
We consider now the FAP limits of the sequence . The next result shows that the limit have some properties of uniformity described in Section 3 but not all of them. The proof is given in Appendix B.
Any FAP limit of the sequence of Poisson distribution with mean is shift invariant and BS-uniform but not necessarily LRF uniform.
Therefore, the FAP limits of the Poisson distribution sequence are examples of SI- and BS-uniform distributions that are not LRF uniform. Kadane and Jin (2014) give another example of SI but not LFR uniform FAPs using paths of random walks. Even if they consider FAP on a subset of bounded function, it can be extended to using Hahn-Banach theorem similarly to Section 2.2.
4.3 FAP vs q-vague convergence of uniform proper distributions
To illustrate the fact that any FAP limit can be related with any improper distribution, consider again the sequence of Poisson distributions with mean and any given improper distribution on the integers. Since for any finite set, the proof of Lemma 1 shows how to construct an exhaustive sequence of finite set such that . For example, choose and define the sequence of proper distributions by :
for any set . From Theorem 3, converge -vaguely to and has the same FAP limit as the Poisson distribution sequence.
As another example, consider the sequence of uniform distribution on and choose . We have . Therefore, for any improper distribution on the set of integers, the sequence constructed as in (6) have the same FAP limits as that of sequence of uniform distributions and converge q-vaguely to . This shows again the difficulty to connect improper and FAP uniform distributions by limits of proper distribution.
4.4 Convergence of beta distributions
In this section, we consider the limit of the sequence of beta distributiondefined on when and go to 0. We see that the FAP limits depend on the way and goes to , which is not the case for the q-vague improper limit. This illustrates the difference between FAP limits and q-vague limits of proper distribution sequences.
The density of a beta distribution Beta is given by
where is the beta function.
From Bioche and Druilhet (2016), the unique (up to a scalar factor) q-vague limit of when and go to 0 is the Haldane improper distribution:
The q-vague limit gives no information on the relative concentration of the mass around and : for , . To explore this concentration, we temporally replace the space by . This has no consequence on the Beta distribution but will change radically the q-vague limit. Put and assume that converges to some . The sequence converges narrowly, and hence q-vaguely, to the proper distribution . Contrary to the Haldane prior, shows how the mass concentrate on the boundary of the domain, but gives no information on the behavior of the sequence inside the domain. Note that the Haldane distribution is not a Radon measure on since where is a compact set. Therefore cannot be a candidate for the q-vague limit on .
We now consider the FAP limits on of and we show that they give an information similar to that given by on the way the mass concentrate on the boundary of the domain. Again, we assume that converge to some . Easy calculations show that for any and . Therefore, for any FAP-limit and for any , we have and , whereas for .
5 Conclusion and perspectives
In this paper, we have shown that improper distributions and FAPs limits give quite different information on the behaviour of the sequence of proper distribution and are therefore complementary.
In a Bayesian context, under some mild regularity condition on the likelihood
, these results can be applied to the joint distributionand to the marginal distribution when it is well defined, both converging to an improper prior if the sequence of prior converge to an improper distribution, similarly for FAP limits. As limits of proper priors, FAP limits and improper prior limits represent two different features of the sequence : the first one describes the behavior of the sequence inside the domain, whereas the second one describes how the mass concentrate on the boundary of the domain. We hope to use these differences in future works to understand some paradoxical phenomenon that occurs in Bayesian statistics.
We establish some lemmas useful to prove Theorem 3. The first one is straightforward.
Let and be two sequences of proper distributions and be a sequence of scalars that converges to . Then, the sequence defined by has the same FAP limits as .
For any , and . The result follows from Lemma 2. ∎
Let be a sequence of proper priors and be a non-decreasing sequence of compact sets such that , then the sequence defined by has the same FAP limits than .
First, note that is not defined when , but this cannot occur more than a finite number of times. For any , . Since is bounded, . Moreover, . Therefore, and have the same limit-sup and limit-inf and thus they have the same FAP limits by Lemma 2.
At the opposite of Lemma 8, the following lemma shows that if we consider the restriction of a sequence of a proper or improper distribution on a exhaustive increasing sequence of compact sets, we preserve the q-vague limits.
Let be a non-decreasing sequence of compact sets such that and such that, for any compact , there exists such that . A sequence of Radon measures converges q-vaguely to the Radon measure if and only if converges q-vaguely to .
Assume that converges q-vaguely to , then there exists some positive scalars such that for any in , . Put and denote by a compact set that includes the support of . Then, there exists an integer such that for . Therefore, for , . The result and its reciprocal follow. ∎
In order to show that is SI, we consider as distribution on , extending them by on the non-positive integers. Define the by , for any subset of . It is known that that , where is the total variation norm. Therefore, for any subset of , . Letting go to infinity, we deduce that, for any FAP limit of , and any integer : .
The fact that is uniform in BS sense comes from an easy adaptation of the Hoeffding inequality in that context. Let be a Bernoulli scheme, of parameter , and denote by being the associated probability. Hoeffding inequality gives, that, for any :
for some positive constant . The expected conclusion is then obtained thanks to the Borel-Cantelli lemma.
The fact that some of the FAP limits are not LRF uniform is a direct consequence of the following lemma.
For any , there exists a set and some FAP limits of such that and .
First note that , for any set , if, and only if, . Therefore, for any set with and for any set such that , one has both and . Take now for set the following :
For that , one has :
and thus . However, converges to . Indeed, if is some random variable with law , one has :
The rigth-hand side term above converges to
thanks to the central limit theorem. Hencewhile converges to , and converges to . Now, for any , choose two numbers , so that . Take the set to be :
then again and converges to , still thanks to the central limit theorem. Let . Then and . Now, any FAP-limit of subsequence is also a FAP-limit of : is therefore shift uniform and BS uniform, but one has . ∎
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