key: cord-0044896-2w512dvt
authors: Pelessoni, Renato; Vicig, Paolo
title: Conditioning and Dilation with Coherent Nearly-Linear Models
date: 2020-05-15
journal: Information Processing and Management of Uncertainty in Knowledge-Based Systems
DOI: 10.1007/978-3-030-50143-3_11
sha: 4ba7921178940b8436799b609f0c554df2bd5828
doc_id: 44896
cord_uid: 2w512dvt

In previous work [1] we introduced Nearly-Linear (NL) models, a class of neighbourhood models obtaining upper/lower probabilities by means of a linear affine transformation (with barriers) of a given probability. NL models are partitioned into more subfamilies, some of which are coherent. One, that of the Vertical Barrier Models (VBM), includes known models, such as the Pari-Mutuel, the [Formula: see text]-contamination or the Total Variation model as special instances. In this paper we study conditioning of coherent NL models, obtaining formulae for their natural extension. We show that VBMs are stable after conditioning, i.e. return a conditional model that is still a VBM, and that this is true also for the special instances mentioned above but not in general for NL models. We then analyse dilation for coherent NL models, a phenomenon that makes our ex-post opinion on an event A, after conditioning it on any event in a partition of hypotheses, vaguer than our ex-ante opinion on A.

Among special imprecise probability models, neighbourhood models [10, Sec. 4.6.5] obtain an upper/lower probability from a given (precise) probability P 0 . One reason for doing this may be that P 0 is not considered fully reliable. Even when it is, P 0 (A) represents a fair price for selling event A, meaning that the buyer is entitled to receive 1 if A is true, 0 otherwise. A seller typically requires a higher price than P 0 (A), P (A) ≥ P 0 (A), for selling A. The upper probability P , to be interpreted as an infimum selling price in the behavioural approach to imprecise probabilities [10] , is often obtained as a function of P 0 .

Recently, we investigated Nearly-Linear (NL) models [1] , a relatively simple class of neighbourhood models. In fact, they derive upper and lower probabilities, P and P respectively, as linear affine transformations of P 0 with barriers, to prevent reaching values outside the [0, 1] interval. As proven in [1] , some NL models are coherent, while other ones ensure weaker properties. The most important coherent NL models are Vertical Barrier Models (VBM), including several well-known models as special cases, such as the Pari-Mutuel, the Total Variation, the ε-contamination model, and others.

In this paper we explore the behaviour of coherent NL models when conditioning. Precisely, after recalling essential preliminary notions in Sect. 2, in Sect. 3 the events in the (unconditional) domain of P , P are conditioned on an event B, and the lower/upper natural extensions E, E of P ,P on the new (conditional) environment are computed. The natural extension is a standard inferential procedure in the theory of imprecise probabilities [10] , which gives the least-committal coherent extension of a lower (upper) probability. Since P (P ) is 2-monotone (2-alternating) in a coherent NL model, E (E) is given by easy-to-apply formulae and is 2-monotone (2-alternating) too. An interesting result is that VMBs are stable after conditioning: the conditional model after applying the natural extension is still a VBM. We show that this property extends also to the mentioned special VBMs: conditioning each special VBM model returns a special VBM model of the same kind. By contrast, the property does not hold for other NL models. In Sect. 4 we explore the phenomenon of dilation, where, given a partition of events B, it happens for some event A that E(A|B) ≤ P (A) ≤ P (A) ≤ E(A|B), for all B ∈ B\{∅}. This means that our a posteriori evaluations are vaguer than the a priori ones. We derive a number of conditions for dilation to occur or not to occur. Section 5 concludes the paper.

In this paper we shall be concerned with coherent lower and upper probabilities, both conditional and unconditional. Coherent means in both cases Williamscoherent [11] , in the structure-free version studied in [6] : Definition 1. Let D = ∅ be an arbitrary set of conditional events. A conditional lower probability P :

A similar definition applies to upper probabilities. However, when considering simultaneously lower and upper probabilities, they will be conjugate, i.e.

Equation (1) lets us refer to lower (alternatively upper) probabilities only. When D is made of unconditional events only, Definition 1 coincides with Walley's coherence [10] . In general, a (Williams-)coherent P on D has a coherent extension, not necessarily unique, on any set of conditional events D ⊃ D.

The natural extension E of P on D is the least-committal coherent extension of P to D , meaning that if Q is a coherent extension of P , then E ≤ Q on D .

Further, E = P on D iff P is coherent [6, 10] .

In this paper, we shall initially be concerned with unconditional lower probabilities (P (·)) and their conjugates (P (·)).

Coherence implies that [10, Sec. 2.7.4] if A ⇒ B, P (A) ≤ P (B), P (A) ≤ P (B) (monotonicity)

The domain D of P (·), P (·) will often be A(IP ), the set of events logically dependent on a given partition IP (the powerset of IP , in set theoretic language). A lower probability P , coherent on

2-monotone and 2-alternating coherent imprecise probabilities have some special properties [8] [9] [10] . In particular,

). If P is a coherent 2-monotone lower probability on A(IP ) and P is its conjugate, given B ∈ A(IP ) such that P (B) > 0, then, ∀A ∈ A(IP ),

Nearly-Linear models have been defined in [1] , where their basic properties have been investigated. 

and P (∅) = P (∅) = 0, P (Ω) = P (Ω) = 1. In Eqs. (5), (6), P 0 is an assigned (precise) probability on A(IP ), while

It has been shown in [1, Sec. 3.1] that NL models are partitioned into three subfamilies, with varying consistency properties. Here we focus on the coherent NL models, which are all the models in the VBM subfamily and some of the HBM (to be recalled next), while, within the third subfamily, P and P are coherent iff the cardinality of IP is 2 (therefore we neglect these latter models). 

and c is given by (7) (hence c ≥ 0). In a Horizontal Barrier Model (HBM) P , P are given by Definition 2, hence by (5), (6), (7) ∀A ∈ A(IP ) \ {∅, Ω}, where a, b satisfy the constraints Thus, VBMs and (partly) HBMs ensure very good consistency properties, while being relatively simple transformations of an assigned probability P 0 . Further, a VBM generalises a number of well-known models. Among them we find:

• if a + b = 0, the vacuous lower/upper probability model [10, Sec. 2.9.1]: [4, 7] , [10, Sec. 2.9.3]:

(12) 

Given a coherent NL model (P , P ) on A(IP ) and an event B ∈ A(IP ) \ {∅}, assumed or known to be true, we look for the natural extensions E(A|B), E(A|B) of P , P respectively, for any A ∈ A(IP ). In other words, P , P are extended on A(IP )|B. When P (B) = 0, we determine E, E quickly thanks to Proposition 3, which follows after a preliminary Lemma, stated without proof in a finite setting in [7] . Lemma 1. Let P : D → R be a coherent lower probability on D, non-empty set of unconditional events, and B ∈ D,

Then, the lower probability P defined by

To prove coherence of P , take E j ∈ D (j = 1, . . . , n), A k |B ∈ S (k = 1, . . . , m) and n + m real coefficients s j (j = 1, . . . , n), t k (k = 1, . . . , m), such that at most one of them is negative, and define

According to Definition 1, we have to prove that max{G|H} ≥ 0, where H = Ω if n > 0, H = B otherwise. We distinguish two cases. 

Proposition 3 ensures that: (14) and apply conjugacy).

Let us now assume P (B) > 0.

Then, E, E are given by the next

Proof. We derive first the expression (15) for E(A|B).

Since P is 2-alternating, we may apply Eq. (3). There, E depends on P (A∧B) and P (A c ∧ B), which cannot be simultaneously 0: by (2) , this would imply

using in both derivations conjugacy first, monotonicity then. Thus, only the following exhaustive alternatives may occur: 

Turning now to E(A|B), its value in Eq. (16) may be obtained simply by conjugacy, using E(A|B) = 1 − E(A c |B) and (15).

For the VBM, it is productive to write E(A|B), E(A|B) as follows:

Proposition 5. Let (P , P ) be a VBM on A(IP ). For a given B ∈ A(IP ), with P (B) > 0, we have that

Proof. Preliminarily, note that the denominator in (20) is positive. In fact, by assumption P (B) > 0, meaning by (8) that bP 0 (B) + a > 0. Using (7) and recalling that c ≥ 0 in a VBM, it holds also that

Given this, let us prove (18) (the argument for (19) is analogous or could be also derived using conjugacy and will be omitted). For this, recalling (16), it is sufficient to establish that

(i) Using (20) and the product rule at the first equality, non-negativity of b, c in the VBM at the inequality, (7) at the second equality, we obtain:

= 0 by (9) (and since bP 0 (A c ∧ B) + c ≥ 0). (ii) Taking account of the first equality in the proof of (i) above and since bP 0 (B)

(iii) Immediate from (i), (ii) and recalling (17) (exchanging there A with A c ).

Elementary computations ensure that b B > 0, a B ≤ 0, 0 < a B + b B ≤ 1.

are conditioned on the same B, the resulting model is still a VBM. (Note the this holds also when P (B) = 0: here Proposition 5 does not apply, but from Proposition 3 we obtain the vacuous lower/upper probabilities, a special VBM.) We synthesise this property saying that a VBM is stable under conditioning. A natural question now arises: does a HBM ensure an analogous property? Specifically, condition on B, with P (B) > 0, the events of A(IP ), which are initially given a HBM lower probability P . Is it the case that the resulting E(·|B) is determined by the equation

with a B , b B given by (20) and obeying the HBM constraints (11) (and similarly with P , E(·|B))?

The answer is negative: although E(A|B) may occasionally be obtained from (21), for instance-as is easy to check-when P (A ∧ B), P (A c ∧ B) ∈]0, 1[, ∀A ∈ A(IP ) \ {∅, Ω}, this is not true in general, as shown in the next example. Table 1 describes the values of an assigned P 0 , and of P , P obtained by (5), (6) with a = −10, b = 12. Since a, b satisfy (11) and, as can be easily checked, P is subadditive, (P , P ) is a coherent HBM by Definition 3 and Proposition 2. Now, take A = ω 2 , B = ω 1 ∨ ω 2 . From Thus, a coherent HBM differs from a VBM with respect to conditioning on some event B, being not stable.

It is interesting to note that not only the VBM, but also its special submodels listed in Sect. 2.1 are stable: conditioning one of them on B returns a model of the same kind. Let us illustrate this in some detail.

For the linear-vacuous model

Note that this follows also from Proposition 3, since P V (B) = 0. By conjugacy, E V (·|B) is vacuous too.

With the ε-contamination model, its conditional model is again of the same type: from (18) 

At first sight, the conditional model differs from a TVM. However, we may easily write (13) in the form (22):

Comparing (22) and (23) we see that the TVM is stable too under conditioning: P T V M , E T V M may be thought of as normalised on the P 0 -probability of what is currently assumed to be true (Ω first, B then).

, ∀A ∈ A(IP )) still made of all probabilities at a total variation distance 2 from P 0 not larger than −a (> 0). This is like the unconditional TVM, the difference being that any A is replaced by A ∧ B, 

Clearly, a B + b B = 1, and we may conclude that E P MM (·|B) is again a PMM, with δ replaced by δ B . Note that the starting P P MM (·) may be written as

Similarly, we obtain E P MM (A|B) = min{(1 + δ B )P 0 (A|B), 1}.

Note that δ B ≥ δ, with equality holding iff P 0 (B) = 1. As well known [7, 10] , δ (hence δ B ) has the interpretation of a loading factor, which makes a subject 'sell' A (A|B) for a selling price P P MM (A) (E P MM (A|B)) higher than the 'fair price' P 0 (A) (P 0 (A|B)). With respect to this kind of considerations, conditioning increases the loading factor, and the smaller P P MM (B), the higher the increase. Next to this, conditioning makes both the seller and the buyer more cautious, in the sense that they restrict the events they would sell or buy. From the seller's perspective, we can see this noting that P P MM (A) < 1 iff

Here t Ω is the threshold to ensure that selling A may be considered: when P (A) = 1, the seller is practically sure that A will occur. On the other hand, s/he will find no rational buyer for such a price: in fact, the buyer should pay 1 to receive at most 1 if A occurs, 0 otherwise. After conditioning,

When t B < t Ω , the seller may have the chance to negotiate A, but not A|B, for some events A.

Analogously, a subject 'buying' A (A|B) will be unwilling to do so when P P MM (A) = 0 (when E P MM (A|B) = 0), which happens iff

Here t B ≥ t Ω , and again conditioning makes the buyer more cautious, see also Fig. 1 . 

When Eq. (24) holds, the imprecision of our evaluation on A increases, or at least does not decrease, after assuming B true. Equation (24) is a condition preliminary to dilation, which we shall discuss later on. Next, we investigate when (24) holds. Preliminarily, say that (24) occurs trivially when it holds and its three inequalities are equalities, and that A is an extreme event if either P (A) = P (A) = 0 or P (A) = P (A) = 1. When referring to extreme events in the sequel, we shall rule out ∅, Ω (for which no inference is needed).

Next, we investigate if Eq. (24) obtains when: P (B) = 0 (Lemma 2); P (B) > 0 and A is either an extreme event (Lemma 3) or non-extreme (Proposition 7). Hence,

With some algebraic manipulations on the right-hand side of (25), noting that by assumption and (2), (5), (6), (7) we have that 0

which proves (i).

(ii) It can be obtained in a very similar way or directly by conjugacy.

From Proposition 6, and recalling (15), (16), it follows straightforwardly that The previous results pave the way to considerations on dilation with NL models. When discussing dilation [2, 7] , we consider a partition B of non-impossible events and say that (weak) dilation occurs, with respect to A and B, when

Recall that an event A is logically non-independent of a partition B iff ∃B ∈ B \ {∅} such that either B ⇒ A or B ⇒ A c , logically independent of B otherwise.

We can now introduce several results concerning dilation. Proof. We need not consider those B ∈ B such that P (B) = 0, if any: by Lemma 2, (a 0 ) they ensure (24). For the others apply Proposition 7.

We derive now an interesting sufficient condition for dilation with a VBM, extending an analogous property of a PMM [7, Corollary 2]. (28)

Proof. If P (B) = 0, apply Lemma 2, (a 0 ). Otherwise, by Proposition 7, we have to check that, when P 0 (B c ) > 0, P (A) ≤ P 0 (A|B c ) ≤ P (A) holds. Now, if P (B) > 0, then necessarily by (5) bP 0 (B) + a > 0, hence P 0 (B) > 0, because a ≤ 0 in a VBM. Further, if (28) holds, then P 0 (A ∧ B c ) = P 0 (A) · P 0 (B c ), hence for those B ∈ B such that P 0 (B c ) > 0, also P 0 (A|B c ) = P 0 (A). Thus, the condition to check boils down to P (A) ≤ P 0 (A) ≤ P (A), which always applies for a VBM.

Note that dilation occurs if event A in Proposition 10 is P 0 -non-correlated with any B ∈ B.

Among coherent NL models, VBMs ensure the property of being stable with conditioning, as also do several known submodels of theirs, such as the PMM. This implies also that results found in [5] on natural extensions of (unconditional) VBMs to gambles still apply here to conditional gambles X|B defined on the conditional partition IP |B = {ω|B : ω ∈ IP }. By contrast, those HBMs that are coherent are generally not stable, thus these models confirm their weaker properties, in comparison with VBMs, already pointed out, from other perspectives, in [1] . Concerning dilation of A w.r.t. partition B, we have seen that it may depend on more conditions, such as whether A is extreme or not, or it is logically independent of B or not, and we supplied several results. In future work, we plan to study the regular extension [10, Appendix J] of coherent NL models, determining how its being less conservative than the natural extension may limit extreme evaluations, as well as dilation. Concepts related to dilation and not presented here are also discussed in [2] for the ε-contamination and the Total Variation models, among others. The assumptions in [2] are usually less general than the present framework. The role of these notions within NL models is still to be investigated.

Nearly-Linear uncertainty measures

Divisive Conditioning: Further Results on Dilation

Markov chains and mixing times

Pari-mutuel probabilities as an uncertainty model

Extending Nearly-Linear Models

Williams coherence and beyond

Inference and risk measurement with the parimutuel model

Lower previsions. Wiley series in probability and statistics

Coherent Lower (and upper) Probabilities

Statistical reasoning with imprecise probabilities. No. 42 in Monographs on statistics and applied probability

Notes on conditional previsions

We are grateful to the referees for their stimulating comments and suggestions. We acknowledge partial support by the FRA2018 grant 'Uncertainty Modelling: Mathematical Methods and Applications'.