Victor David

Raphaël Fournier-S'niehotta

Nicolas Travers

Parameterisation of Reasoning on Temporal Markov Logic Networks

Keywords: Temporal Markov Logic Networks, Temporal Maximum A-Posteriori Inference, Temporal Parametric Semantics

We aim at improving reasoning on inconsistent and uncertain data. We focus on knowledge-graph data, extended with time intervals to specify their validity, as regularly found in historical sciences. We propose principles on semantics for efficient Maximum A-Posteriori inference on the new Temporal Markov Logic Networks (TMLN) which extend the Markov Logic Networks (MLN) by uncertain temporal facts and rules. We examine total and partial temporal (in)consistency relations between sets of temporal formulae. Then we propose a new Temporal Parametric Semantics, which may combine several sub-functions, allowing to use different assessment strategies. Finally, we expose the constraints that semantics must respect to satisfy our principles.

Introduction

Reasoning on large data sets to extend knowledge is an open challenge [START_REF] Banko | Open information extraction from the web[END_REF][START_REF] Etzioni | Open information extraction from the web[END_REF][START_REF] Pujara | Knowledge graph identification[END_REF][START_REF] Jaradeh | Plumber: A Modular Framework to Create Information Extraction Pipelines[END_REF]]. Most approaches model information with Knowledge Graphs (KGs) [START_REF] Dong | Knowledge vault: A web-scale approach to probabilistic knowledge fusion[END_REF], and rely on Ontologies [START_REF] Rashid | Leveraging semantics for large-scale knowledge graph evaluation[END_REF], Machine Learning [START_REF] Zalmout | All You Need to Know to Build a Product Knowledge Graph[END_REF] or Neural Networks [START_REF] Macdonald | Neural Relation Extraction on Wikipedia Tables for Augmenting Knowledge Graphs[END_REF] representations. Then, Description Logic [START_REF] Ho | Introducing fuzzy temporal description logic[END_REF] and Temporal Logic [START_REF] Rodionova | Temporal logic as filtering[END_REF] are used for rules satisfiability.

For historians, reasoning under uncertainty and inconsistency is at the basis of methodology: the validity of any fact remains questionable. Temporal information is crucial: outside of a temporal interval, a fact becomes false. Using facts, temporal and uncertainty information, a historian may obtain several consistent levels of knowledge.

Recent works on KG reasoning do not focus on both time and uncertainty. Several models with Markov Logic Networks (MLN) have been proposed [START_REF] Chekol | Markov logic networks with numerical constraints[END_REF][START_REF] Rincé | Complex event processing under uncertainty using markov chains, constraints, and sampling[END_REF]. One focused on reasoning on Uncertain Temporal Knowledge Graphs (UTKG) with specific temporal inference rules [START_REF] Chekol | Marrying uncertainty and time in knowledge graphs[END_REF]. However, those representations cannot handle a fully uncertain universe where any fact or rule may be uncertain.

We propose to generalize this approach, by i) introducing a new representation of knowledge, called temporal MLN (TMLN), built on (a temporal) many-sorted logic, for reasoning on both time and uncertainty, ii) enlarging the notion of uncertainty to rules, and iii) adapting reasoning, to deal with both uncertain facts and rules. We define a new temporal semantics and a temporal extension to Maximum A-Posteriori (MAP) inference [START_REF] Riedel | Improving the accuracy and efficiency of map inference for markov logic[END_REF]. This MAP inference produces instantiations, i.e., extended sets of facts maximizing the score w.r.t. a temporal semantics. Finally, we generalise the temporal semantics by a Parametric Semantics, which may combine several sub-functions for various consistency validations.

Background

In a seminal work [START_REF] Chekol | Marrying uncertainty and time in knowledge graphs[END_REF], Chekol et al. formalise the UTKG approach, which integrates both time and uncertainty in Knowledge Graphs to reach a certain world maximisation. However, they do not take into account the possibility to have uncertain rules. We put time at the heart of reasoning, and enlarge their vision. Firstly, we formalise the notion of temporal uncertainty, by combining certain and uncertain formulae. Secondly, our novel representation of TMLN allows for easier manipulations and better analyses.

Many-Sorted First Order Logic

We start by presenting what is a Many-Sorted First-Order Logic (introduced in [START_REF] Wang | Logic of many-sorted theories[END_REF]).

Definition 1 (Many-Sorted FOL). Let So = {s 1 , . . . , s n } be a set of sorts. A Many-Sorted First-Order Logic MS-FOL, is a set of formulae built up by induction from: a set C = {a 1 , . . . , a l } of constants, a set V = {x s , y s , z s , . . . | s ∈ So} of variables, a set P = {P 1 , . . . , P m } of predicates, a function ar : P → N which tells the arity of any predicate, a function sort s.t. for P ∈ P, sort(P ) ∈ So ar(P ) , and for c ∈ C, sort(c) ∈ So, the usual connectives (¬, ∨, ∧, →, ↔), Boolean constants ( and ⊥) and quantifier symbols (∀, ∃). A ground formula is a formula without any variable.

Lowercase (resp. uppercase) greek letters like φ, ψ (resp. Φ, Ψ ) denote formulae (resp. sets of formulae). MS-FOL is the set of all formulae.

Example 1. For instance let So = {s 1 , s 2 }, let P 1 ∈ P such that sort(P 1 ) = s 2 × s 1 × s 1 , let a 1 , a 2 , t 1 , t 2 ∈ C such that sort(a 1 ) = sort(a 2 ) = s 2 , sort(t 1 ) = sort(t 2 ) = s 1 and let x s2 ∈ V. We can build then the following MS-FOL formulae: P 1 (a 1 , t 1 , t 2 ), ∀x s2 P 1 (x s2 , t 1 , t 2 ). However, P 1 (t 1 , t 2 , a 1 ) or ∀x s2 P 1 (a 1 , a 2 , x s2 ) cannot be built because they do not respect the sorts.

MS-FOL formulae are evaluated via a notion of structure called n-sorted structures [START_REF] Gallier | Logic for computer science: foundations of automatic theorem proving[END_REF]. Classical first-order logic formulae are captured as 1-sorted structures.

Definition 2 (Structure

). A n-sorted structure is St = ({D 1 , . . . , D n }, {R 1 , . . . , R m }, {c 1 , . . . , c l })
, where D 1 , . . . , D n are the (non-empty) domains, R 1 , . . . , R m are relations between domains' elements, and c 1 , . . . , c l are distinct constants in the domains.

Our running example is presented in Example 2. These sentences gather biographical elements about a French philosopher from the 14th century, Nicole Oresme. In MS-FOL, though without uncertainty, we may then define a suitable structure. Then, satisfying formulae is recursively defined by: - Definition 3 does not target the satisfaction of implications and equivalences, while they can be defined by: (φ → ψ) ≡ (¬φ ∨ ψ), and

I St |= P i (a 1 , . . . , a k ) iff (I St (a 1 ), . . . , I St (a k )) ∈ R i , -I St |= ∃x si φ iff I St,x s i ←v |= φ for some v ∈ D i , -I St |= ∀x si φ iff I St,x s i ←v |= φ for each v ∈ D i , -I St |= φ ∧ ψ iff I St |=
(φ ↔ ψ) ≡ (φ → ψ) ∧ (ψ → φ).

Temporal and Uncertain Knowledge Representation

Markov Logic Networks (MLNs) combine Markov Networks and First-Order Logic (FOL) by attaching weights to first-order formulae and treating them as feature templates for Markov Networks [START_REF] Richardson | Markov Logic Networks[END_REF]. We extend this framework to temporal information by resorting to Many-Sorted First-Order Logic (MS-FOL).

Temporal Markov Logic Networks

Let start by defining what we call a Temporal Many-Sorted First-Order Logic TS-FOL.

Definition 6 (Temporal Many-Sorted FOL).

A TS-FOL evaluated by a structure St is a constrained MS-FOL where |So| ≥ 2, for any interpretation I St (s 1 ) = T ime, any predicate P i ∈ TS-FOL has ar(P i ) ≥ 3 with the sort of the last two parameters belonging to s 1 and t min and t max are time constants indicating the minimum and maximum time points for any pre-order between the time constants.

Using this constrained MS-FOL accompanied with a temporal domain (named T ime) and temporal predicates (the last two parameters indicate the validity temporal bounds), we may represent temporal facts and rules. Finally, Temporal Markov Logic Networks (TMLN) extend TS-FOL (resp. Markov Logic Networks) by associating a degree of certainty to each formula (resp. by adding a temporal validity to the predicates).

Definition 7 (TMLN).

A Temporal Markov Logic Network M = (F, R), based on a TS-FOL, is a set of weighted temporal facts and rules where F and R are sets of pairs such that:

-F = {(φ 1 , w 1 ),. . . , (φ n , w n )} with ∀i ∈ {1,. . . , n}, φ i ∈ TS-FOL such that it is a ground formula and w i ∈ [0, 1], -R = {(φ 1 , w 1 ),. . . , (φ k , w k )} with ∀i ∈ {1,. . . , k}, φ i ∈ TS-FOL such that
it is not a ground formula and in the form (premises,conclusion), i.e., (ψ 1 ∧. . . ∧ψ l ) → ψ l+1 where ∀j ∈ {1,. . . , l + 1}, ψ j ∈ TS-FOL, and

w i ∈ [0, 1].
The universe of all TMLNs is denoted by TMLN.

In the rest of the paper, we simplify the example by directly using the structure defined in Example 3 (c.f. Section 2.1). Table 2: Ground Rules Instantiating R1 and R2 (from Table 1) for Nicole Oresme.

Example 2 (Continued). The TMLN representation of our running example can be found in Table 1. We may identify 6 independent facts and 2 rules, each one with temporal validity and certainty weights (arbitrary extracted from Example 2).

In order to select what is the most probable and consistent set of ground formulae with a MAP inference, we need first to have our data (facts and rules) represented in a TMLN. Then, we must obtain the ground rules (if possible), by replacing the variables in the rules by constants (according to our TMLN). We call this second step the instantiation.

TMLN Instantiation

Let M be a TMLN, we denote by MI(M) the Maximal TMLN Instantiation of M. MI(M) contains the set of M's facts and all ground rules that can be constructed by instantiating all its predicates containing variables by other deductible ground predicates (with logical consequence, from Definitions 4 and 5). A ground rule's weight is the minimum of the weights of the formulae in M used to construct the instantiated rule.

Formally, to define the set of instantiations, we have to define two useful notions. Firstly, we denote by TF(M) = (φ,w)∈M φ the set of temporal formulae of M ∈ TMLN.

Secondly, we define the function W : TS-FOL × TMLN → [0, 1], returning the maximal weight of a temporal formula deductible from a TMLN: W(φ, M) = max(min w (M 1 ), . . . ,

min w (M m )) s.t. {M 1 , . . . , M m } = {M i ⊆ M | TF(M i ) φ and M i ⊂ M i s.t.
TF(M i ) φ} and min w (M i = {(ψ 1 , w 1 ), . . . , (ψ l , w l )}) = min(w 1 , . . . , w l ). Definition 8 (TMLN Instantiation). Given M = (F, R) ∈ TMLN, the set of instantiations MI of M is defined as follows: c n is a vector of constants replacing each occurrence of the variables, X i ⊆ X is the set of variables in φ i , C i ⊆ C is the vector of constants replaced in φ i and the instantiated rule satisfies the 2 following conditions:

MI(M) = F ∪ { (φ 1 ∧ . . . ∧ φ k → φ k+1 ) X←C , w | (φ 1 ∧ . . . ∧ φ k → φ k+1 , w) ∈ R, X = {x 1 , . . . , x n } is the set of variables in φ 1 ∧. . .∧φ k → φ k+1 , C = c 1 , . . . ,
1.

∀φ i ∈ {φ 1 , . . . , φ k }, φ iX i ←C i ∈ Cn(TF(M)) 2. w = min(w, W(φ 1X 1 ←C 1 , M), . . . , W(φ kX k ←C k , M)}
Where φ X←C is the formula φ s.t. all the occurrences of the variable x i ∈ X are replaced by the constant c i ∈ C.

Currently, we only deal with universal (i.e., ∀) rules and no existential one (i.e., ∃), to simplify the maximal TMLN instantiation. Indeed, with existential rules, we would have to deal with a set of sets of instantiations, given that we would not know which set of instantiations would be true. We keep this question for future works.

From Example 2, the instantiation of R 1 (resp. R 2 ) are GR 11 (from F 1 , F 3 and F 4 ) and GR 12 (from F 1 , F 3 and F 5 ) (resp. GR 2 from F 2 and F 3 ). Hence GR 11 has a weight of 0.4, GR 12 of 0.5 and GR 2 of 0.8, see Table 2.

A TMLN instantiation I ⊆ MI(M) is a TMLN only composed of ground formulae, I is also called a state of the TMLN M. The universe of all TMLN instantiations is denoted by TMLN * . Note that an instantiation can be inconsistent. In our example, GR 11 , F 1 , F 3 , F 4 imply P easantF amily(N O, t min , t max ) while GR 2 , F 2 , F 3 imply ¬P easantF amily(N O, t min , t max ), i.e., they are inconsistent together. Thus, to obtain the most consistent and informative set of instantiations, we have to compute the Maximum A-Posteriori (MAP) inference [START_REF] Chekol | Marrying uncertainty and time in knowledge graphs[END_REF][START_REF] Riedel | Improving the accuracy and efficiency of map inference for markov logic[END_REF].

Temporal and Uncertain Knowledge Reasoning

Computing the MAP inference means "finding the most probable state of the world" [START_REF] Chekol | Marrying uncertainty and time in knowledge graphs[END_REF]. In order to do that, we integrate semantics on TMLN, before examining principles governing those semantics and (in)consistency relations.

Temporal MAP Inference

Semantics are methods which compute the strength of a TMLN state. We denote the universe of all semantics by Sem, such that for any S ∈ Sem, S : TMLN * → [0, +∞[. We compute a strength above 0, instead of a probability between 0 and 1. One semantics may maximise the amount of information, while another may maximise the quality. Temporal Maximum A-Posteriori (MAP) Inference in TMLN returns the most probable, temporally consistent, and expanded state w.r.t. a given semantics. Given M ∈ TMLN and S ∈ Sem, a method solving a MAP problem is denoted by: map : TMLN × Sem → P(TMLN * ), where P(X) denote the powerset of X, such that:

map(M, S) = {I | I ∈ argmax I ⊆ MI(M)

S(I) and I ∈ argmax

I ⊆ MI(M) S(I ) s.t. I ⊂ I }.
To determine a MAP inference we need to define semantics, however not all methods are desirable. Below we present some principles that a semantics should satisfy.

Principles for semantics in Temporal MAP Inference

Our first principle states that adding new weightless information does not change the strength of the MAPs of a TMLN, whatever the temporality is. Given that the same set of predicates or formulae instantiated on different temporalities are not equivalent, we homogenises TS-FOL with the maximal time interval to define the information novelty.

For time homogenization, we denote by τ : P(TS-FOL) → P(TS-FOL), the function transforming any temporal predicate into the maximal time interval (t min and t max ). The last principle states that if we add a new temporal fact to a TMLN such that it is consistent with each instantiation, then the fact will be present in each new instantiation.

Temporal Consistency and Inconsistency

We study here new temporal consistency interactions required to define our Temporal MAP inference. Temporal Consistency relations need to be refined according to the temporal validity of the predicates.For a predicate and its negation, no clear definition exists to express the temporal consistencies based on their time intervals. We propose a temporal consistency with a general case (partial) and a special case (total).

To establish these different temporal consistency relations, we introduce a function TI to create pre-orders between the temporal constants in the domain T ime of a TS-FOL and which extracts the interval of time points from two constants.

Definition 9 (Temporal (in)consistency). Let a set of formulae Φ ⊆ TS-FOL. Temporal consistency:

-Φ has a partial temporal consistency denoted by pCon(Φ) iff: ∀φ, ψ ∈ Cn(Φ) s. -Φ has a partial temporal inconsistency denoted by pInc(Φ) iff: ∃φ, ψ ∈ Cn(Φ) s.t. φ=P (x 1 ,. . . , x k , t 1 , t 1 ), ψ=¬P (x 1 ,. . . , x k , t 2 , t 2 ) and TI(t 1 , t 1 ) ∩ TI(t 2 , t 2 ) = ∅. Otherwise ¬pInc(Φ) is true.

-Φ has a total temporal inconsistency denoted by tInc(Φ) iff: ∃φ, ψ ∈ Cn(Φ) s.t. φ = P (x 1 ,. . . , x k , t 1 , t 1 ), ψ = ¬P (x 1 ,. . . , x k , t 2 , t 2 ) and (TI(t 1 , t 1 ) = TI(t 2 , t 2 ). Otherwise ¬tInc(Φ) is true.

We now examine the interaction properties between pCon, tCon, pInc and tInc, such as complementarity, subsumption and inclusion 1 .

Definition 10 (Complementarity & Subsumption). ∀Φ ⊆ TS-FOL, ∀ relation r 1 , r 2 if:
r 1 (Φ) ↔ ¬r 2 (Φ) then r 1 and r 2 are complementary.

-

r 1 (Φ) → r 2 (Φ) then r 1 subsume r 2 .
The next two propositions show that firstly tCon and pInc are complementary; secondly different subsumption relations exist between the temporal consistencies. In the following, for temporal consistency and inconsistency relations, we denote by {r} = {Φ ⊆ TS-FOL | r(Φ)} their set of sets of formulae respecting their condition, where r ∈ {pCon, tCon, pInc, tInc, ¬pCon, ¬tCon, ¬pInc, ¬tInc}.

Definition 11 (Inclusion).

Let two relations of temporal consistency r 1 , r 2 ∈ {pCon, tCon, pInc, tInc, ¬pCon, ¬tCon, ¬pInc, ¬tInc}, r 1 is considered included in r 2 if:

{r 1 } ⊆ {r 2 } iff ∀Φ ⊆ TS-FOL, r 1 (Φ) → r 2 (Φ).
Some inclusions of temporal consistency relations may be defined between the sets of sets of formulae respecting them. -∆ : TMLN * → {0, 1},

-σ : TMLN * → +∞ k=0 [0, 1] k , -Θ : +∞ k=0 [0, 1] k → [0, +∞[, For any M ∈ TMLN, I ⊆ MI(M), the strength of a temporal parametric semantics TPS = ∆, σ, Θ is computed by: TPS(I) = ∆(I) • Θ σ(I) .
We propose below key properties that must be satisfied by each of the three functions ∆, σ, Θ of a TPS. Those properties constrain the range of functions to be consid- ered, and discard those exhibiting undesired behaviours. Definition 13. A temporal parametric semantics TPS = ∆, σ, Θ is well-behaved according to a temporal consistency relation Con iff the following conditions hold:

∆-(a) ∆(I) = 1 if Con(TF(I)) is true (i.e., I is temporally consistent w.r.t. Con). Θ-(a) Θ() = 0. (b) Θ(w) = w. (c) Θ is symmetric. (d) Θ(w 1 ,. . . , w k ) = Θ(w 1 ,. . . , w k , 0). (e) Θ(w 1 ,. . . , w k , y) ≤ Θ(w 1 ,. . . , w k , z) if y ≤ z. σ-(a) σ() = (). (b) σ({(φ 1 , w 1 ),. . . , (φ k , w k )}) = (w 1 ,. . . , w n ) such that if k ≥ 1 then n ≥ 1. (c) σ({(φ 1 , w 1 ),. . . , (φ k , w k ), (φ k+1 , 0)}) = (σ({(φ 1 , w 1 ),. . . , (φ k , w k )}), 0), if τ (TF({φ 1 ,. . . , φ k })) τ ({φ}). (d) σ({(φ 1 , w 1 ),. . . , (φ k , w k )}) ⊂ σ({(φ 1 , w 1 ),. . . , (φ k , w k ), (φ k+1 , w k+1 )}) if φ k+1 is a ground temporal formula, τ (TF({φ 1 ,. . . , φ k })) τ ({φ k+1 }) and
Con(TF({(φ 1 , w 1 ),. . . , (φ k , w k ), (φ k+1 , w k+1 )})).

(e) Θ σ({(φ 1 , w 1 ),. . . , (φ k , w k )}) ≤ Θ σ({(φ 1 , w 1 ),. . . , (φ k , w k ),

(φ k+1 , w k+1 )}) if τ (TF({φ 1 ,. . . , φ k })) τ ({φ k+1 }) and
Con(TF({(φ 1 , w 1 ),. . . , (φ k , w k ), (φ k+1 , w k+1 )})).

We also say that ∆ is well-behaved according to Con, and Θ, σ are well-behaved.

Theorem 1. Any temporal parametric semantics well-behaved w.r.t a temporal consistency relation Con satisfies the principles from Section 4.2: Temporal Neutrality, Consistency Monotony and Invariant Consistent Facts (the last two according to Con).

Once temporal consistency relations are defined, we may enhance semantics for MAP inference with temporal validation functions. One TMLN instantiation can be valid or not according to different criteria (i.e., accept an instantiation).

Definition 14 (Temporal Consistency Validation Function). Let M ∈ TMLN, an instantiation I ⊆ MI(M) and x ∈ {pCon, tCon, pInc, tInc}. We define ∆ x : TMLN * → {0, 1}, a temporal consistency validation function according to x, such that:

∆pCon(I) = 1 if pCon(TF(I)) 0 if ¬pCon(TF(I)) ∆tCon(I) = 1 if tCon(TF(I)) 0 if ¬tCon(TF(I)) ∆pInc(I) = 0 if pInc(TF(I)) 1 if ¬pInc(TF(I)) ∆tInc(I) = 0 if tInc(TF(I)) 1 if ¬tInc(TF(I))
Corollary 1. For any I ⊆ TMLN * , ∆ tCon (I) = ∆ pInc (I).

Corollary 2. Let x ∈ {pCon, tCon, pInc, tInc}, each ∆ x is well-behaved.

We show next that we can order the value of the ∆ x for any instantiation. Theorem 2. Let M ∈ TMLN, for any σ and Θ, we denote by: -TPS tCon = ∆ tCon , σ, Θ , TPS pInc = ∆ pInc , σ, Θ , -TPS pCon = ∆ pCon , σ, Θ , TPS tInc = ∆ tInc , σ, Θ .

Hence: ∀I tCon ∈ map(M, TPS tCon ), ∀I pInc ∈ map(M, TPS pInc ), ∀I pCon ∈ map(M, TPS pCon ), ∀I tInc ∈ map(M, TPS tInc ), TPS tCon (I tCon ) = TPS pInc (I pInc ) ≤ TPS pCon (I pCon ) ≤ TPS tInc (I tInc ).

We study different instances of the aggregate function, using different sums. This type of parameters will determine the strength of an instantiation, in various ways.

Definition 15 (Aggregate Functions

). Let {w 1 ,. . . , w n } such that n ∈ [0, +∞[ and ∀i ∈ [0, n], w i ∈ [0, 1]. -Θ sum (w 1 ,. . . , w n ) = n i=1 w i , if n = 0 then Θ sum () = 0. -Θ sum,α (w 1 ,. . . , w n ) = n i=1 (w i ) α 1 α s.t. α ≥ 1, if n = 0 then Θ sum,α () = 0. -Θ psum (w 1 ,. . . , w n ) = w 1 . . . w n , where w 1 w 2 = w 1 + w 2 -w 1 • w 2 , if n = 0 then Θ psum () = 0 and if n = 1 then Θ psum (w) = w.
Those aggregate functions target different kinds of semantics. For instance, Θ sum,α emphasises strong weights for inference, while Θ psum targets more controversial instantiations (mixed weights). Proposition 5. The three functions Θ sum , Θ sum,α and Θ psum are well-behaved.

We propose below a selective function σ id which returns all the weights, more complex functions selecting weights with a threshold (σ thresh,α ) and another one, σ rule , setting the weight of ground rules to 0 when one of their premises is not deductible (i.e., the rule is unusable). The latter is interesting to say if unground rules (no deductible premisesprem & imp) must be kept or not in any instantiation.

Definition 16 (Selective Functions). Let M ∈ TMLN, {(φ 1 , w 1 ),. . . , (φ n , w n )} ⊆ MI(M):

σ id ({(φ 1 , w 1 ),. . . , (φ n , w n )}) = (w 1 ,. . . , w n ) -σ thresh,α ({(φ 1 , w 1 ),. . . , (φ n , w n )}) = (max(w 1 -α, 0),. . . , max(w n -α, 0)) s.t.

α ∈ [0, 1[ -• let φ = (ψ 1 ∧. . . ∧ψ k ) → ψ k+1 a rule, prem(φ) = {ψ 1 ,. . . , ψ k }. • imp((φ, w), {(φ 1 , w 1 ),. . . , (φ n , w n )}) =        0 if φ is a ground rule s.t. ∃ψ i ∈ prem(φ) s.t. ψ i / ∈ Cn({φ 1 ,. . . , φ n }) w otherwise • σ rule ({(φ 1 , w 1 ),. . . , (φ n , w n )}) = imp((φ 1 , w 1 ), {(φ 2 , w 2 ),. . . , (φ n , w n )}),.
. . , imp((φ n , w n ), {(φ 1 , w 1 ),. . . , (φ n-1 , w n-1 )}) Proposition 6. The three functions σ id , σ thresh,α and σ rule are well-behaved.

In [START_REF] Chekol | Marrying uncertainty and time in knowledge graphs[END_REF], the MAP inference uses a semantics working on Herbrand models (containing temporal formulae, included in TS-FOL) and built from uncertain (w < 1) and certain (w = 1) temporal facts and with a set of TMLN certain rules. This semantics also determines the temporal inconsistency by a classical consistency (if there is no formula ϕ such that Φ ϕ and Φ ¬ϕ) and by summing all the weights of facts in the instantiation. Therefore, for TMLNs without any uncertain rule, the MAP inference will return the same instantiations as ours, using the temporal parametric semantics ∆ tInc , σ id , Θ sum . Our Temporal MAP inference generalises their work.

Example of Reasoning on TMLN

Example 2 (Continued). Table 3 illustrates various TPS leading to different instantiations and possible inferences from Example 2. The last column focuses on the Peasant-Family information which has not been proven yet by Historians. Another interesting analyse is about "did Nicole Oresme studied at the College of Navarre and when?".

Θ sum,2 keeps highly weighted facts and rules. Instantiations insist on the conflict between GR 11 /GR 12 and GR 2 (PeasantFamily or not), keeping those which maximise the strength. More precisely, these ground rules are temporally inconsistent depending on the set of facts in the instantiation. Indeed, if ground rules' premises are not present in the instantiation, then they have no conflict, as it appears in the TPS using σ id . σ rule gives an interesting vision on the MAP inference result, since it keeps (i.e., w > 0) only useful ground rules, i.e., rules having deductible premises in the instantiation. Hence, σ rule never give importance simultaneously to GR 11 /GR 12 with GR 2 .

Eventually, notice that even if P easantF amily(N O, t min , t max ) has a weight of 0.5 and its negation 0.8, or even if in this example we have more inference on the negation, we cannot conclude that ¬P easantF amily(N O, t min , t max ) is more likely.

For instance, ∆ tInc , σ id , Θ sum corresponding to Chekol's semantics, advocates that Nicole Oresme was a Peasant while the one with ∆ pCon (the partial temporal consistency) infers the opposite. The parametric choice of our approach would allow the historian deciding which inference corresponds to their own reasoning.

Conclusion

Reasoning on Knowledge Graphs have recently been approached with MLNs, to find the most probable state of the world. However, they were limited to a strict temporal inconsistency. We extend UTKGs with a fully-formalized model of TMLN, capable of combining consistency and inconsistency for facts and rules, allowing to reason with the MAP inference using new principles and relations between partial and total aspects of temporal (in-)consistency. Moreover, we introduce temporal parametric semantics which offer flexibility to tailor semantics reasoning to one's needs.

As a perspective, we first wish to extend this work to existential rules in order to capture more real data (see Section 3.2). Secondly, some relationships between formulae in a TMLN are not yet exploited and could be represented with argumentation graphs (e.g., the notion of support [START_REF] Cohen | A survey of different approaches to support in argumentation systems[END_REF] or similarity between pieces of information [START_REF] Amgoud | Measuring similarity between logical arguments[END_REF][START_REF] Amgoud | A General Setting for Gradual Semantics Dealing with Similarity[END_REF]), to enhance the weight of inferred knowledge. Thirdly, as we can see in Table 3, each parameter of the TPS have an impact on the temporal MAP inferences and its possible conclusions. It will be useful to investigate some properties to describe the strategies and specific behaviours of the parametrisation functions. Finally, implementing our results and confronting them with real data will likely prove very useful, at least to generate new research hypotheses in historical science.

Example 2 .

 2 Nicole Oresme was a person and a philosopher born in the Middle Ages between 1320-1382. It would appear that Nicole Oresme attended the College of Navarre around 1340-1354 and more likely around 1355-1360. Nicole Oresme possibly did not attend the College of Navarre around 1353-1370. Sometimes, a person who lived in Middle Ages and studied at College of Navarre came from a peasant family. Usually, a philosopher born in the Middle Ages did not come from a peasant family.

  For instance, the set of interpretations of the formula P (a) ∨ P (b) is equal to {{P (a)}, {P (b)}, {P (a), P (b)}} and for P (a) ∧ P (b) is {{P (a), P (b)}}. With these notions of structures and interpretations on TMLNs, we may define the consequence relations and logical consequences over MS-FOL formulae. Definition 4 (Consequence Relation). Let φ and ψ be two MS-FOL formulae. We say that ψ is a consequence of φ, denoted by φ ψ, if for any structure St, and any interpretation I St over St, I St |= φ implies I St |= ψ. Definition 5 (Logical Consequences -Cn). Let φ ∈ MS-FOL. The function Cn(φ) is the set of all logical consequences of φ, i.e. Cn(φ) = {ψ ∈ MS-FOL | φ ψ}. The function Cn returns an infinite set of formulae, but for clarity we consider only one formula per equivalent class and only the predicates and constants appearing in the original formulae. Such as, Cn(P (a) ∨ P (b)) = {P (a) ∨ P (b)} and Cn(P (a) ∧ P (b)) = {P (a), P (b), P (a) ∧ P (b), P (a) ∨ P (b)}}.

F1

  (P erson(N O, 1320, 1382) , 1) F2 (P hilosopher(N O, 1320, 1382) , 1) F3 (LiveP eriod(N O, M A, 1320, 1382) , 1) F4 (Studied(N O, CoN, 1340, 1354) , 0.4) F5 (Studied(N O, CoN, 1355, 1360) , 0.7) F6 (¬Studied(N O, CoN, 1353, 1370) , 0.5) R1 (∀x s 2 , t

GR11

  = ((P erson(N O, 1320, 1382) ∧ LiveP eriod(N O, M A, 1320, 1382)∧ Studied(N O, CoN, 1340, 1354)) → P easantF amily(N O, tmin, tmax) , 0.4) GR12 = ((P erson(N O, 1320, 1382) ∧ LiveP eriod(N O, M A, 1320, 1382)∧ Studied(N O, CoN, 1355, 1360)) → P easantF amily(N O, tmin, tmax) , 0.5) GR2 = ((P hilosopher(N O, 1320, 1382) ∧ LiveP eriod(N O, M A, 1320, 1382)) → ¬P easantF amily(N O, tmin, tmax) , 0.8)

Principle 1 (

 1 Temporal Neutrality) Let M ∈ TMLN, M = M∪{(φ, w)} where (φ, w) is a weighted temporal formula (φ ∈ TS-FOL and w ∈ [0, 1]), such that: -τ (TF(M)) τ ({φ}), and w = 0. A semantics S ∈ Sem satisfies temporal neutrality iff, ∀I ∈ map(M, S) and ∀I ∈ map(M , S), S(I) = S(I ). The next principle ensures that one cannot decrease the strength of the MAPs of a TMLN by adding new and consistent information. The temporal consistency Con : P(TS-FOL) → { , ⊥} denotes a relation of consistence for a set of temporal formulae. Principle 2 (Consistency Monotony) Let a relation of consistence Con, M ∈ TMLN and M = M ∪ {(φ, w)} where (φ, w) is a weighted temporal formula such that: -τ (TF(M)) τ ({φ}), and -∀I ∈ map(M, S), Con({φ} ∪ TF(I)) is true and I ⊂ MI({(φ, w)} ∪ I). A semantics S ∈ Sem satisfies consistency monotony iff: ∀I ∈ map(M, S) and ∀I ∈ map(M , S), S(I) ≤ S(I ).

Principle 3 (

 3 Invariant Consistent Facts) Let a relation of consistence Con, M ∈ TMLN and M = M ∪ {(φ, w)} where (φ, w) is a TMLN fact such that: -τ (TF(M)) τ ({φ}), and -∀I ∈ map(M, S), Con({φ} ∪ TF(I)) is true. A semantics S ∈ Sem satisfies invariant consistent facts iff, ∀I ∈ map(M, S), I ∪ {(φ, w)} ∈ map(M , S).

Proposition 1 .

 1 (Complementarity: temporal consistencies) For any Φ ⊆ TS-FOL: ¬tCon(Φ) ↔ pInc(Φ) and tCon(Φ) ↔ ¬pInc(Φ). Proposition 2. (Subsumption: temporal consisencies) For any Φ ⊆ TS-FOL: pCon(Φ) → ¬tInc(Φ), tInc(Φ) → ¬pCon(Φ), ¬pCon(Φ) → pInc(Φ), and ¬pInc(Φ) → pCon(Φ).

Proposition 3 .Fig. 1 :

 31 Fig. 1: Links between Consistency and Inconsistency

Proposition 4 .

 4 Let M ∈ TMLN and ∆ x a temporal consistency validation function such that x ∈ {pCon, tCon, pInc, tInc}. For any instantiation I ⊆ MI(M):∆ tCon (I) = ∆ pInc (I) ≤ ∆ pCon (I) ≤ ∆ tInc (I)Theorem 2 shows that the strength of the temporal MAP inferences with σ and Θ on any TMLN is ranked according to the temporal consistency validation functions ∆ x .

  Example 3. An example of structure associated with the MS-FOL from Example 2 is St hist = ({T ime, Concept}, {P erson, P hilosopher, LiveP eriod, P easantF amily, Studied}, {t min , 1300, 1301, 1302, . . . , 1400, t max , N O, M A, CoN }), in which: -T ime is the set of time points, corresponding to the sort s 1 and Concept is the set of all non-temporal objects, corresponding to the sort s 2 , -P erson, P hilosopher, LiveP eriod, etc. are the predicate symbols' relations (e.g., P erson ⊆ Concept × T ime × T ime indicates which elements are a person). t min , 1300, 1301, . . ., 1400, t max are elements of the domain T ime associated with the sort s 1 , while N O (Nicolas Oresme), M A (Middle Ages) and CoN (College of Navarre) are elements of the domain Concept associated with the sort s 2 . An interpretation I St over a structure St assigns to elements of the MS-FOL vocabulary some values in the structure St. Formally, -I St (s i ) = D i , for i ∈ {1, . . . , n} (each sort symbol is assigned to a domain), -I St (P i ) = R i , for i ∈ {1, . . . , m} (each predicate symbol is assigned to a relation), -I St (a i ) = c i , for i ∈ {1, . . . , l} (each constant symbol is assigned to a value).

	Now, we show how MS-FOL formulae are interpreted.
	Definition 3 (Interpretation).

  φ and I St |= ψ, -I St |= φ ∨ ψ iff I St |= φ or I St |= ψ, -I St |= ¬φ iff I St |= φ. where I St,x s i ←v is a modified version of I St s.t. the variable x si is replaced by a value v in the domain D i corresponding to the sort symbol s i . Finally, if Φ is a set of formulae, then I St |= Φ iff I St |= φ for each φ ∈ Φ.

Table 1 :

 1 Example of a TMLN for Nicole Oresme.

  Otherwise ¬pCon(Φ) is true. -Φ has a total temporal consistency denoted by tCon(Φ) iff: ∀φ, ψ ∈ Cn(Φ) s.t. φ = P (x 1 ,. . . , x k , t 1 , t 1 ) and ψ = ¬P (x 1 ,. . . , x k , t 2 , t 2 ), (TI(t 1 , t 1 ) ∩ TI(t 2 , t 2 )=∅).

	Otherwise ¬tCon(Φ) is true.
	Temporal inconsistency:

t. φ = P (x 1 ,. . . , x k , t 1 , t 1 ) and ψ = ¬P (x 1 ,. . . , x k , t 2 , t 2 ), (TI(t 1 , t 1 ) \ TI(t 2 , t 2 ) = ∅) ∧ (TI(t 2 , t 2 ) \ TI(t 1 , t 1 ) = ∅).

Table 3 :

 3 TPS example: in each instantiation we omit F1, F2, F3 and formulae with a 0 weight.

Proofs are available on https://tinyurl.com/PRICAI22.

Supplementary material for Parameterisation of Reasoning on Temporal Markov Logic Networks

A Proofs

Proof (Proposition 1). Let Φ ⊆ TS-FOL such that tCon(Φ), i.e. from Definition 9: ∀φ, ψ ∈ Cn(Φ) s.t. φ = P (x 1 , • • • , x k , t 1 , t 1 ) and ψ = ¬P (x 1 , • • • , x k , t 2 , t 2 ), (TI(t 1 , t 1 ) ∩ TI(t 2 , t 2 ) = ∅) then its negation ¬tCon(Φ) is equivalent to: ¬(¬∃φ, ψ ∈ Cn(Φ) s.t. φ = P (x 1 , • • • , x k , t 1 , t 1 ) and ψ = ¬P (x 1 , • • • , x k , t 2 , t 2 ) and (TI(t 1 , t 1 ) ∩ TI(t 2 , t 2 ) = ∅))

Hence ¬tCon(Φ) is equivalent to pInc(Φ): ∃φ, ψ ∈ Cn(Φ) s.t. φ = P (x 1 , • • • , x k , t 1 , t 1 ) and ψ = ¬P (x 1 , • • • , x k , t 2 , t 2 ) and (TI(t 1 , t 1 ) ∩ TI(t 2 , t 2 ) = ∅) Moreover its negation, ¬pInc(Φ) is equivalent to: ¬(¬∀φ, ψ ∈ Cn(Φ) s.t. φ = P (x 1 , • • • , x k , t 1 , t 1 ) and ψ = ¬P (x 1 , • • • , x k , t 2 , t 2 ), (TI(t 1 , t 1 ) ∩ TI(t 2 , t 2 ) = ∅))

Therefore ¬pInc(Φ) is equivalent to tCon(Φ).

Proof (Proposition 2). Let Φ ⊆ TS-FOL, from Definition 9:

1. pCon(Φ) → ¬tInc(Φ):

-¬tInc(Φ) iff, ¬∃φ, ψ ∈ Cn(Φ) s.t. φ = P (x 1 , • • • , x k , t 1 , t 1 ),

which is equivalent to: )), therefore pCon(Φ) → ¬tInc(Φ).

tInc(Φ) → ¬pCon(Φ):

-¬pCon(Φ) iff

and

and

¬pCon(Φ) → pInc(Φ):

-

¬pInc(Φ) → pCon(Φ):

-¬pInc(Φ) is equivalent to:

-Therefore, for any TI(t Proof (Proposition 3). From Proposition 1 and 2, we know that for any Φ ⊆ TS-FOL:

Therefore from Proposition 11:

• w = 0. Given that (φ, w = 0), any new ground formula in the instantiation of M has also a zero weight: ∀(ψ, w) ∈ MI(M ) \ MI(M), w = 0 (because the instantiation take the minimum weight of the formulae building the ground formula).

If a temporal parametric semantics TPS = ∆, σ, Θ is well-behaved according to a temporal consistency relation Con, then from Definition 13:

• condition 2.(c):

Therefore, any new ground formula will have no impact in any instantiation, and so in any MAP inference: ∀I ∈ map(M, TPS) and ∀I ∈ map(M , TPS), TPS(I) = TPS(I ). -Consistency Monotony.

Let a well-behaved temporal semantics TPS = ∆, σ, Θ according to a temporal consistency relation Con, M ∈ TMLN, M = M ∪ {(φ, w)} where (φ, w) is a weighted temporal formula such that:

1. τ (TF(M)) τ ({φ}), and 2. ∀I ∈ map(M, TPS), Con({φ} ∪ TF(I)) is true and I ⊂ MI({(φ, w)} ∪ I). From Definition 13:

• condition 1.(a): ∆(I) = 1 if Con(TF(I)) is true.

Therefore, from item 2 of (φ, w) we know that any new ground formulae are consistent with each MAP, and with condition 1.(a), we know that the strength will not down to 0 from ∆. Proof (Proposition 4). . From Definition 11: Let two relations of temporal consistency r 1 , r 2 ∈ {pCon, tCon, pInc, tInc, ¬pCon, ¬tCon, ¬pInc, ¬tInc}, r 1 is considered included in r 2 if:

From Proposition 3:

Hence, for any set of formulae Φ ⊆ TS-FOL:

From Definition 14: Let M ∈ TMLN, an instantiation I ⊆ MI(M) and x ∈ {pCon, tCon, pInc, tInc}. Therefore using the case equal to 1 and given that (tCon(Φ) ↔ ¬pInc(Φ)) → ¬pInc(Φ) → pCon(Φ) → ¬tInc(Φ), for any instantiation I ⊆ MI(M):

Proof (Theorem 2). Let x ∈ {pCon, tCon, pInc, tInc}, M ∈ TMLN and ∀I ⊆ MI(M), we know from Proposition 4 that:

From Definition 12, a temporal parametric semantics is defined as follows:

Consequently, ∆(I) is a coefficient of the combination of σ and Θ. Thus, for any I, σ and Θ, we may order the results according to the ∆(I) coefficient.

Finally, given that the MAP Inference returns the instantiation with the maximum strength, all strengths of the MAP inferences are equal.

Therefore, if we denote by:

-

Hence: ∀I tCon ∈ map(M, TPS tCon ), ∀I pInc ∈ map(M, TPS pInc ), ∀I pCon ∈ map(M, TPS pCon ), ∀I tInc ∈ map(M, TPS tInc ), TPS tCon (I tCon ) = TPS pInc (I pInc ) ≤ TPS pCon (I pCon ) ≤ TPS tInc (I tInc ).

0): 0 cannot increase with any power and it is the identity element of the addition.

it is a sum of positive element and the addition is monotonic. (e) Θ sum,α is symmetric: the addition is monotonic.

where

is greater than the subtraction of the product w 1 • w 2 , i.e. w 1 + w 2 ≥ w 1 • w 2 . (e) Θ psum is symmetric: the addition and the product are symmetric.

The three functions Θ sum , Θ sum,α and Θ psum are well-behaved.

) τ ({φ}): the function return the initial weight, then any weight doesn't change and the 0 is added.

Con(TF({(φ 1 , w 1 ), • • • , (φ k , w k ), (φ k+1 , w k+1 )})): same reason as in item (c).

(e) for any well behaved Θ: Θ σ id ({(φ 1 , w 1 ),

τ ({φ k+1 }) and Con(TF({(φ 1 , w 1 ),

same reason as in item (c) and (d).

-σ thresh,α ({(φ 1 , w 1 ),

every 0 still 0 and doesn't change the rest of the other weights.

and Con(TF({(φ 1 , w 1 ), • • • , (φ k , w k ), (φ k+1 , w k+1 )})).

Any weight is added.

(e) for any well behaved Θ: Θ σ thresh,α ({(φ 1 , w 1 ),

Any weight is added and the previous weights are independent to new weight then the aggregation cannot decrease.

Any new weight is either added with its weight or with 0 but it is added then ∀k ≥ 1, n = k. Add a weighted temporal formula with a weight of 0 will still be 0 in any case.

(d) σ rule ({(φ 1 , w 1 ), • • • , (φ k , w k )}) ⊂ σ rule ({(φ 1 , w 1 ), • • • , (φ k , w k ), (φ k+1 , w k+1 )}) if φ k+1 is a ground temporal formula, τ (TF({φ 1 , • • • , φ k })) τ ({φ k+1 }) and Con(TF({(φ 1 , w 1 ), • • • , (φ k , w k ), (φ k+1 , w k+1 )})). Add any new consistent information doesn't change the other information then we have a strict inclusion.

(e) for any well behaved Θ: Θ σ rule ({(φ 1 , w 1 ), • • • , (φ k , w k )}) ≤ Θ σ rule ({(φ 1 , w 1 ), • • • , (φ k , w k ), (φ k+1 , w k+1 )}) if τ (TF({φ 1 , • • • , φ k }))

τ ({φ k+1 }) and Con(TF({(φ 1 , w 1 ), • • • , (φ k , w k ), (φ k+1 , w k+1 )})).

Add a new information has a weight belonging to 0 and 1, it cannot decrease the aggregation.

The three functions σ id , σ thresh,α and σ rule are well-behaved.


Jean-Baptiste Coulomb
email: jb.coulomb@orange.fr

Fabrice Larceneux
email: fabrice.larceneux@dauphine.psl.eu

Arnaud Simon
email: arnaud.simon@dauphine.psl.eu

Choosing annuities for a home reversion: the case of the French viager market

Keywords: Viager, Home reversion, Annuitization puzzle, Lifetime annuity, Ageing

In this article, we analysed the preference for annuitization when retired people extract cash from their home. Based on 2,608 viager (home reversion) transactions closed in France between 2010 and 2018, we used a three-stage least squares (3SLS) model and a moderation model to study the relations among annuitization, negotiation, extraction, age, gender, and matrimonial status. We found that difficulties in selling the property result, generally, in increases in the annuitization. However, for the group of older sellers, if negotiations are more difficult, they nevertheless tend to sign contracts with lower annuities. The single men's group endures a gender inequality, suffering a limitation on their possibility to extract wealth and to annuitize, as well as an additional price discount while negotiating. Young single men, compared to young single women and young couples, must consent to a substantial price reduction when they prefer high down-payment, and limited price reduction when they prefer annuities. Elderly single men, compared to young single men, have less capacity to negotiate, a concern that is reinforced when they prefer annuities.

Introduction

Over the last two decades, the increase in life expectancy, along with the massive cohort of babyboomers entering retirement, has caused an ageing shock in numerous developed countries. To live long is a chance, but financing pensions has become increasingly challenging, whether for the public system, with the relative evolution of the retired and the active populations, or for the privately funded system, given the very low level on the bond markets, for instance. At the same time, the illiquid housing stock value represents an important wealth, of which the retired population possesses a significant part (1,060 over 5,660 billion euros for France [START_REF] Lefebvre | Richesse immobilière et encours des prêts immobiliers: Quelles divergences territoriales aujourd'hui?[END_REF]).

In this situation, the home equity extraction products (e.g., reverse mortgage, equity withdrawal, home reversion, etc.) offer to the elderly homeowner a means to complete their retirement. Home reversion, in particular, enables the extraction of cash while staying at home until death, or entry into a care home. A viager is a specific kind of home reversion, in which the extracted amount is a direct function of the seller's age. Extracted wealth is paid through annuities and a down-payment, in a proportion negotiated and freely decided between buyers and sellers.

Favouring annuities is a way to hedged against the risk of insufficient revenue in case of longevity, so it appears suitable or rational to elderly individuals as a risk-reducing strategy. However, annuitizing is neither as obvious nor as rational as it seems, and there may be reluctance; in fact, [START_REF] Benartzi | Annuitization puzzles[END_REF] talked about an annuitization puzzle. In this article, by studying 2,608 viager transactions closed in France between 2010 and 2018, we can document the factors that influence the preference for annuities over lump-sum transactions. In particular, we explore the effects of age, gender, and matrimonial status on this preference. We highlight the significant effect of interactions between annuitization, expected length, and gender in the context of explaining the negotiation.

The rest of this article is organized as follows. Section 2 is devoted to the literature review. Section 3 presents the principle of a viager, the database, and descriptive statistics. Section 4 presents the models and the estimation strategy: a three-stage least squares (3SLS) approach and an interaction model. Section 5 presents the results and the relevant interpretation. Section 6 concludes.

Literature Review

During the 1990s, an important debate emerged, discussing whether the baby-boom, and later, the elder-boom, would affect housing prices [START_REF] Mankiw | The baby boom, the baby bust, and the housing market[END_REF][START_REF] Holland | The baby boom and the housing market: Another look at the evidence[END_REF][START_REF] Peek | The baby boom, "pent-up" demand, and future house prices[END_REF][START_REF] Ohtake | The effect of demographics on the Japanese housing market[END_REF]. Based on the life-cycle theory, the hypotheses under consideration explained that the large cohorts of baby-boomers would first raise prices by a demand effect, and upon arriving at retirement age, would contribute to decrease prices, by a supply effect. This conversation has gained reinforced attention today, in a context of high real estate prices and ageing societies (for Canada, [START_REF] Fortin | Déterminants du prix réel des logements au Canada[END_REF] for the Netherlands, [START_REF] Neuteboom | Demography and housing demand-Dutch cohort evidence[END_REF]for Japan, Nakamura & Saita, 2007;for Scotland and England, Levin et al., 2009;for France, Simon & Essafi, 2017). Published in the Journal of Housing Economics, the article of [START_REF] Takáts | Aging and house prices[END_REF], a panel approach to 21 countries over the period 1970-2009, deserves special attention.

Whether renters or homeowners, retired people, in general, wish for "ageing in place" [START_REF] Costa-Font | Ageing in place'? Exploring elderly people's housing preferences in Spain[END_REF]. However, issues about the suitability or unsuitability of places must be considered [START_REF] Severinsen | Ageing in unsuitable places[END_REF], along with home maintenance difficulties [START_REF] Coleman | Older adults' experiences of home maintenance issues and opportunities to maintain ageing in place[END_REF]. Child proximity [START_REF] Lee | Housing tenure transitions of older households: What is the role of child proximity?[END_REF] and widowhood [START_REF] Bonnet | The effect of widowhood on housing and location choices[END_REF] are also discussed in the literature to better understand housing choices made by the elders.

For retired homeowners, one way to finance ageing in place consists of using home equity conversion products, such as reverse mortgage, home reversion, or sale of the bare ownership, especially for those with the profile of "house-rich, cash-poor" [START_REF] Shan | Reversing the trend: The recent expansion of the reverse mortgage market[END_REF]. Such interest in those products was identified very early (Higgins & Folt, 1992), and sometimes presented as a way to reduce poverty among elders [START_REF] Kutty | The scope for poverty alleviation among elderly home-owners in the United States through reverse mortgages[END_REF]. Obstacles to the developments of these products were discussed by [START_REF] Leviton | Reverse mortgage decision-making[END_REF], by [START_REF] Weber | The reverse mortgage and its ethical concerns[END_REF] in the context of ethical concerns, and by [START_REF] Tsay | An approximation approach for valuing reverse mortgages[END_REF] in the context of pricing issues.

The viager is a member of the family of home equity conversion product and is a kind of home reversion. Rather, it is oriented toward lifetime annuities; in that sense, it meets the annuitization puzzle issue documented by [START_REF] Benartzi | Annuitization puzzles[END_REF]. Annuities should indeed be popular for many reasons, but people are often reluctant to annuitize a substantial share of their wealth and tend to prefer lump sums. This preference for the present is a well-known subject in the economic literature, as surveyed by [START_REF] Frederick | Time discounting and time preference: A critical review[END_REF] and explored by [START_REF] Warner | The personal discount rate: Evidence from military downsizing programs[END_REF]. Personal discount rate differences between rich and poor households have been studied, for instance, by [START_REF] Lawrance | Poverty and the rate of time preference: Evidence from panel data[END_REF] and, more recently, by [START_REF] Klawitter | Savings and personal discount rates in a matched savings program for low-income families[END_REF]. If [START_REF] Lahav | Subjective time discount rates among teenagers and adults: Evidence from Israel[END_REF] found differences between teenagers and adults regarding their subjective time discount rates, we may consider that elders also express specificities in this regard. This is all the more relevant in the case of the viager; the choice between a lifetime annuity and a lump sum is made in a context of mortality salience [START_REF] Kasser | Of wealth and death: Materialism, mortality salience, and consumption behavior[END_REF][START_REF] Fransen | Rest in peace? Brand-induced mortality salience and consumer behavior[END_REF][START_REF] Partouche-Sebban | L'impact de la saillance de mortalité sur les stratégies régulatrices et sur les préférences produits promotion et prévention[END_REF]. It is also interesting to keep in mind that this kind of transaction involves something that may unconsciously be associated with an illusion of immortality: a home [START_REF] Erbs | La médiation de la pierre ou l'illusion de l'immortalité[END_REF].

The viager: Principle, database, and descriptive statistics a. Principle of the viager

A viager is a home reversion transaction, in which the extracted amount is fixed. It consists of selling a property in return for a down-payment and a lifetime annuity that the buyer must pay until the seller's death [START_REF] Février | Testing for asymmetric information in the viager market[END_REF]. Older than 200 years of age, the viager product is used and standardized in many countries, such as France, Belgium, Germany ('leibrente'), Italy ('vitalizio'), the UK ('home reversion') and Spain ('vitalicio'). It belongs to the family of structured products that allows retired homeowners to increase their pensions by extracting cash from their home equity. It differs from a reverse mortgage in that it consists of a true sale of a property, without bank involvement or issues related to interest rates.

France is an interesting case study for this kind of product. On one hand, about 75 % of the French retired population are homeowners (compared to a national average of 58%), but a significant proportion receives low pension. On the other hand, the total housing wealth can be estimated at 5,660 billion euros [START_REF] Lefebvre | Richesse immobilière et encours des prêts immobiliers: Quelles divergences territoriales aujourd'hui?[END_REF], of which around 1,060 billion euros are owned by the retirees. As in numerous OECD countries, retired individuals are often in the "house rich, cash poor" situation.

A viager sale functions as follows. First, an occupied discount is applied to the estimated market value of the property. This discount reflects the fact that the property is occupied and that the seller retains the usufruct (fructus and usus), or more simply, the right of use and habitation (usus), which permits them to continue living in their home until death. The remainder, the bare ownership (abusus), is transferred to the buyer at the contract signature. The economic value of the bare ownership, purchased by the buyer, corresponds to the wealth extracted by the seller. The payment is split into two parts: a down-payment at the signature (bouquet in French), and a commitment to a lifetime annuity (rente in French), which is often paid monthly. The occupied discount, or equivalently, the extracted amount, derives from an actuarial table. It is supposed to depend solely on the seller's age and should rely neither on the preferences of the buyer nor the preferences of the seller. However, as we shall see, practice differs from theory. In terms of the split between downpayment and annuities, it is freely determined by the parties. When the seller dies, the buyer becomes full owner of the property and ceases to owe further annuities.

b. Database

Our database comprises 2,608 transactions, 55% apartments and 45% houses, from January 2010 to April 2018, provided by Renée Costes Viager, the main French brokerage chain for this product (market share around 25%). We included in the study apartments with a surface area between 8 and 200 m 2 , houses with a surface area between 20 and 300 m 2 and associated land of less than 10,000 m 2 . Properties were all built after 1700, have a minimum value of 10,000 euros and fewer than 10 rooms. We excluded properties for which the asking price was revised more than five times and that had a time-on-market longer than three years (71 properties removed). We also filtered out observations when the ratio of the final total estimated value of the property and the initial total estimated value was lower than 50% or higher than 115% (70 properties removed).

First results showed that the expected length of the viager is, on average, approximately 11-12 years, higher for houses than for apartments (Table 1). If the seller is single, the expected length is equal to the residual life expectancy. If the sellers are a couple, the length corresponds to their joint residual life expectancy (cf. Appendix 1). Computation of these lengths is based on the mortality tables TH/TF 00-021 . The time-on-market is quite important, compared to basic housing sales, and should be around 8-10 months. The price/m 2 is twice as high for apartments, compared to houses. Most sellers of apartments are single women, whereas distribution is more homogenous for houses. Globally, two-thirds of the sellers were single.

c. Intermediate price measures

The observable financial elements in the database are the initial estimated value of the full property (IEV), the agreed down-payment and the agreed monthly amount of the annuity. The initial estimated value is generally proposed by the broker and indicated in the first version of the listing. Down-payment and annuities are the results of a negotiation. To decompose the selling process and analyse its steps, we reconstruct the three implicit measures that correspond to the viager's logic:

-FEV (final estimated value): the estimation of the value of the full property at the contract signature -PVA (present value of annuities): the present value of annuities at the contract signature -EXTRACT: the amount of wealth extracted at the contract signature

The PVA is calculated for each transaction from the agreed annuity by discounting the value of annuities over the average residual lifetime of the individual (or the joint residual lifetime for couples). This average residual lifetime is derived from the mortality tables TH/TF 00-02, depending on the age and on the gender of the individual (available in the database). An annual inflation of the annuity amount of 1% per year is applied -it is indeed a common practice in the viager market to adjust the annual amount according to inflation. The discount rates for each month derive from the zero-coupon curves provided by the French Institute of Actuaries. For each viager, we use the zerocoupon curve at the contract signature, and we actualize the monthly amount paid on a given date, by the corresponding rate for the same date (cf. Appendix 2).

The amount of wealth extracted, EXTRACT, is obtained by adding the down-payment to the present value of annuities:

EXTRACT = PVA + Down-payment
At last, we reconstruct the final estimated value (FEV), reversing the Daubry scale. This scale serves as the reference for the viager agencies in France. It gives, for each viager, the occupied discount2 , depending on the age and gender of the seller(s). More precisely, we calculate:

d. Financial horizon for the payments

To better qualify the temporal structure of the cash-flows paid by the buyer, for each viager, we also calculate the Macaulay duration, assuming the viager ends at its expected horizon (cf. Appendix 3). Figure 1 presents the distribution of the ratio between duration and expected length. The temporal cash flow structure is strongly compressed, with a mean around 35% and a discontinuity at zero that corresponds to the viager with a 100% down-payment (2% of the whole set). If the expected length of the viager is around 11 years, the financial horizon for the payments is actually much smaller. It is also interesting to compare, for each viager, the Macaulay duration of the signed transaction with its equivalent, assuming that all payments would be due in lifetime annuities (Figure 2). It stems from that down-payments, on average, reduce by 30% the duration, compared to the full-annuities choice. Discontinuities at 0% and 100% again suggest that the extreme choices (full down-payment (2% of the whole set) or full annuities (4%)) constitute two specific sub-segments. We decompose the transaction process in three fictitious stages (Figure 3). First, we consider negotiation by dividing the final estimated value (FEV) by the initial estimated value (IEV). On average, this ratio is around 85%, which means that the price reduction due to negotiation is high, around 15%. The closer to 100% this ratio is, the greater the seller's capacity for negotiation.

We then consider extraction, dividing the extracted wealth (EXTRACT) by the final estimated value (FEV). The extracted amount is important, and it is, on average, around 60%, and ranges between 30% and 80%.

The last element, annuitization, consists of an arbitrage between the down-payment and the lifetime annuities. We consider this stage by dividing the present value of annuities (PVA) by the extracted wealth (EXTRACT). If annuities are generally preferred and represent, on average, 70%, the ratio, however, varies between 0% (full down-payment) and 100% (full annuities); the greater the ratio, the stronger the preference for annuitization. As the choice between down-payment and lifetime annuities is free, the viager provides an interesting opportunity to better understand the determinants of this choice in a context of mortality salience. The extraction ratio reflects the consumption-investment duality of housing. Those two sides interact dynamically over the life-cycle and it is well-known that household wealth has a significant effect on consumption. This balance is particularly interesting to study among retired people. Based on a survey, [START_REF] Dillingh | Who wants to have their home and eat it too? Interest in reverse mortgages in the Netherlands[END_REF] documented that reverse mortgage in the Netherlands is more consumption-oriented for people under 65 years, whereas the bequest motivation increased after that age. In theory, for the viager, the extracted wealth percentage is not freely decided; it is directly and uniquely determined by the seller's age, using actuarial tables. However, as the three phases (negotiation, extraction, annuitization) are realized simultaneously, interaction effects may exist. As for the negotiation ratio, it is centrally associated with the commercialization process, but without necessarily representing the all commercialization aspects, as discussed hereafter. The literature has documented a strong relation between price reduction and time-on-market. For instance, [START_REF] Martel | Quality, price, and time-on-market[END_REF]

b. A 3SLS model explaining negotiation, wealth extraction and annuitization

Our objective is to explore the factors that influence those three dimensions. Here, an important point should be discussed. Figure 3 may suggest that negotiation, extraction, and annuitization are decided sequentially, one after the other. However, this is fictitious and, in practice, they are decided simultaneously. This may produce interaction effects; for instance, a full down-payment may be easier to commercialize, resulting in a smaller discount and a greater negotiation ratio. To document those potential crossed effects, we introduce, in the annuitization equation, the negotiation ratio, and in the negotiation equation, we introduce the annuitization ratio. The annuitization and negotiation ratios are also introduced in the extraction equation. Regarding the extraction ratio, it is not included in the two other equations because they already include the expected length, which is strongly correlated with the extraction ratio (-0.98).

Given the interaction between equations and variables, we jointly estimate those three equations with a three-stage least squares procedure (3SLS). Using a 3SLS further deals with the potential correlation's issues between the exogenous variables 3 . At last, as mentioned in Section 3.d, the full down-payment and the full lifetime annuities viagers may constitute specific sub-segments. Therefore, we also test the robustness of our results by removing each of these segments separately.

The explanatory variables are as follow. The time-on-market is introduced using dummy variables per quarter (the removed reference variable corresponds to the less-than-three-months commercialization). We also control for market conditions with dummies for each year (removed reference 2018). Dwellings characteristics are the year of construction (removed reference 'before 1945'), the number of rooms, the apartment/house distinction, and the price per square meters.

Local conditions are rendered by the percentage of owners, and the median income in the municipality. Information on sellers is captured with the expected length of the viager based on sellers' age, and the single/couple distinction, crossed with gender (removed reference 'couple').

c. Moderation analysis for the capacity of negotiation

In a second step, we conduct a more in-depth investigation of our results regarding negotiation, by implementing a moderation model. A moderator variable W is a variable that alters the strength of the causal relationship between X and Y. It consists of including an interaction effect between W and X in the regression model. Studying the effect of annuitization on negotiation, we propose a moderation model that allows for the impact of two moderators: length and status. This model has been undertaken with Process modelling and a bootstrap technique (n = 5000) to identify simple and interaction effects [START_REF] Preacher | Addressing moderated mediation hypotheses: Theory, methods, and prescriptions[END_REF].

We designed the following equation to explain negotiation:

With:

3 The correlations between those variables are low: with a minimum of -0.53 and a maximum of 0.61.

-NEGOTIATION representing the capacity to succeed in a negotiation, i.e. the negotiation discount (FEV / IEV) -ANNUITIZATION representing the preference for annuitization (PVA / EXTRACT) -LENGTH representing the expected length (referring to the age(s) of the decision-maker(s)) -STATUS as a control variable to avoid potential confounding effects. It corresponds to the household status (single man vs. not-single man (= single woman, or couple))

And:

-a 1 referring to the simple effect of ANNUITIZATION on NEGOTIATION -a 4 referring to a moderation effect, the effect of ANNUITIZATION on NEGOTIATION, moderated by LENGTH -a 7 referring to a moderation effect, the effect of ANNUITIZATION on NEGOTIATION, moderated by both LENGTH and STATUS

Results and interpretations a. 3SLS estimation results

Table 2 presents the results for the full dataset, the sample without the full down-payment viager, and the sample without the full annuities viager. In theory, according to the viager's practice, the extracted amount should be defined as a direct and unique function of the sellers' age (alternatively referring to the expected length). However, it is surprising to observe that several other variables also have an effect, although this is not supposed to occur. This means, in particular, that the seller's capacity for extraction exists and may fluctuate. Expected length (thus, the sellers' age) also affects annuitization significantly. When the seller is one year older, the portion of annuities in the extracted amount decreases by 1.33%. As for negotiation, when the seller is one year older, the price decreases by 0.87%, relative to the initial estimated value (IEV).

The time-on-market logically affects negotiation: after one year, the price decreases by 9.78%. However, at the same time, it also reduces the lifetime annuities portion by 10.4%. When a property is difficult to sell, adjustments are made by reducing the financial horizon, increasing the downpayment part.

In terms of the characteristics of the dwelling, the period of construction is non-significant or has very limited effects and the apartment/house distinction has no impact, whereas a greater number of rooms tends to increase the negotiation discount slightly. The market conditions of each year mainly and unsurprisingly affect negotiation. The magnitude of the effects of the local conditions (ownership rate and median income in the municipality) is very limited. A more expensive price per square meter tends to slightly decrease annuities, in favour of down-payment, thereby moderately reducing the negotiation discount, whereas the impact on extraction is negligible.

More interesting are the effects of matrimonial status and gender. Among singles, the portion of annuities decreases by 2.76% for women, this decrease is even greater for men, at 4.93%. Additionally, compared to couples and single women, the amount extracted by single men is 1.03% less, and they also must consent to an additional price reduction of 2.86%.

We must discuss the crossed effects. Although negotiation has no effect on extraction, an indirect relationship still obtains between them: greater the negotiation discount is, the higher the annuitization: an additional price reduction of 1% does, in fact, increase the portion of annuities by 0.50%. The impact of the annuitization ratio on extraction is almost null, but not on negotiation: The dwelling price for a full annuities viager (PVA/EXTRACT = 1) is 10.3% lower, compared to the same property sold through a full down-payment viager. This result is an estimation of the risk premium associated with the riskiest viager cash flows (full annuities), compared to the non-risky cash flows case (full down-payment).

b. Moderation analyses

To deepen our understanding of the interactions between negotiation and annuitization, this section discusses the results of the moderation model presented in section 4.c. The mean, standard error, and correlations of the variables of interest are shown in Table 3. Results of the regressions models are presented in Table 4; Model 1 is a regression without moderation effects, Model 2 includes the moderation effects. The R² difference with and without moderation effects is significant (0.017***). Preference for annuitization shows a negative direct effect: Sellers who prefer annuitization negotiate less well than those who prefer down-payment, as seen in the previous section. By contrast, the expected length has a positive effect: Younger sellers negotiate more successfully, obtaining higher prices than older sellers. This positive effect is reinforced by the preference for annuitization: Young sellers who want to annuitize more negotiate even more successfully. Regarding status, we find that single women negotiate more successfully than couples, who negotiate more successfully than single men (Model 1). Gender strongly influences the capacity to negotiate: Women show more negotiation power than single men, alone or in a couple. By contrast, being a single man is detrimental to the capacity of negotiation, directly but also as a moderation factor: If a single man prefers annuities and wants to annuitize more, the negotiation is even more difficult (Model 2): Among the young sellers' group (Figure 4), single men encounter significantly greater difficulty to get a high price if they want a higher down-payment, compared to women or couples (negotiation = 85%, versus 89%4 ). This difference is reduced when annuitizing more ( positive and significant in Model 2), but single men are still less capable of negotiating the price (negotiation = 86%, versus 89%5 ). Among the single men's group (Figure 5), the capacity for negotiation is, on average, lower for older men. The magnitude of this effect depends on the preference for annuitization: When old single men prefer annuitizing, this effect is clearly reinforced (M = 76% versus M = 81% 7 for higher downpayment). For young single men, their annuitization choice does not significantly impact their negotiation capacity. As a consequence, when male sellers get older, they have an interest in choosing greater down-payment to secure a higher price. In this case, the difference observed with young men is limited (negotiation = 81%, versus 85% 8 ), compared to the annuitization cases (negotiation = 76%, versus 86% 9 ).

Conclusion

In this article we analysed the determinants of viager transactions, considering horizon, negotiation, extraction, and annuitization. We first established that the financial horizon of the cash-flows paid by the buyer is much lower than the expected horizon: It is approximately 35% of the expected horizon. Moreover, the effect of down-payment on the financial horizon is substantial, imposing a decrease of 30%. Although the extracted amount of wealth in a viager is intended to depend solely on the sellers' age, we found evidence indicating that several other variables bear influence and that bargaining, in this respect, is a reality.

However, the main point in a viager's transactions is the relation between the annuitization choice and negotiation. When a property is difficult to sell, and has substantial time-on-market, a price discount logically emerges, but the negotiation also involves an increase of the annuity part. Therefore, to a seller aiming at a significant portion of down-payment, this result suggests that it may be advantageous to avoid deploying overly aggressive commercialization strategies with a high initial asked price. Negotiations are generally more difficult for older sellers, and they tend to sign the viager's contract with lower annuities.

The single men's group appears as very specific and subject to a gender inequality. On average, they receive lower annuities (-4.93%), they extract less wealth (-1.03%)), and they must suffer an additional price discount in the negotiation (-2.86%). Young single men, compared to young single women and young couples, must consent to a substantial price reduction when they prefer high down-payment, and to a limited price reduction when they prefer annuities. Old single men, compared to young single men, have less capacity to negotiate, a deficit that is reinforced when they prefer annuities. In that sense, when single men grow older, they find it advantageous to choose a higher down-payment if they want to limit their losses.

At last, the global negative effect of annuitization on negotiation may open a path to further research. We may hypothesize that, when the seller is reluctant to consider the idea of their death, to refuse the salience of mortality, they tend to ask for greater annuities. However, as a consequence, they must consent to additional price reductions. We observe that point indirectly in our article, with young sellers and couples preferring annuities, compared to more elderly sellers and singles. Indeed, for young retirees, the idea of death is probably still distant, whereas older sellers may have a clearer awareness of their finitude. Likewise, singles, who are often widows and widowers, have already endured the death of their partner and also may have a clearer awareness of their own mortality.

Figure 1 :

 1 Figure 1: Distribution of the ratio Macaulay duration/Expected length

Figure 2 :

 2 Figure 2: Distribution of the ratio Macaulay duration/Macaulay duration all annuities

Figure 3 :

 3 Figure 3: Decomposition of the viager transaction

  proposed a theoretical model, incorporating quality, price, and time-on-market, in which sale prices fall along with time-on-market. It is also possible to refer to the state of the art, realized by[START_REF] Benefield | A review of literature utilizing simultaneous modeling techniques for property price and time-on-market[END_REF] about the (property price -marketing time) tradeoff.

Figure 4 :

 4 Figure 4: Capacity of negotiation for young sellers, according to the household status and the preference for annuitization 6

Figure 5 :

 5 Figure 5: Capacity of negotiation for single men, according to the expected length and the preference for annuitization

Table 1 : Descriptive statistics APARTMENTS HOUSES Mean St. Dev. Min Max Mean St. Dev. Min Max Expected length (TF/TH 00-02)

 1

			11.21	4.42	2.49	28.40	12.18	4.83	3.05	31.09
	Time-on-market (days)	251	161	22	1 056	299	193	23	1 086
	Year of construction	1 956	41	1 700 2 015	1 954	46	1 700 2 015
	% owners in the municipality	47	15	22	92	67	16	19	96
	Median income in the municipality (€)	23 977 5 657	NA	43 350 20 951 4 189	NA	41 236
	Price/m	2	4 132	2 658	415 17 785 1 857	1 047	363 12 979
	% single males	23	42	-	-	23	42	-	-
	% single females	57	50	-	-	38	48	-	-
	% couples	21	40	-	-	40	49	-	-
	Final estimated value/Initial estimated value FEV/IEV	0.86	0.11	0.50	1.15	0.84	0.12	0.50	1.13
	Extracted wealth /Final estimated value EXTRACT/FEV	0.59	0.09	0.30	0.78	0.57	0.10	0.28	0.80
	Present value of annuities/Extracted wealth PVA/EXTRACT	0.64	0.21	0.00	1.00	0.69	0.20	0.00	1.00

Table 2 : 3SLS estimations for annuitization, extraction and negotiation

 2

			Full dataset		Without full down-payment viager	Without full lifetime annuities viager
		Annuitization	Wealth extraction	Negotiation	Annuitization	Wealth extraction	Negotiation	Annuitization	Wealth extraction	Negotiation
		PVA / EXTRACT	EXTRACT / FEV	FEV / IEV	PVA / EXTRACT	EXTRACT / FEV	FEV / IEV	PVA / EXTRACT	EXTRACT / FEV	FEV / IEV
	(Intercept)	1.13 ***	8.16e-01 ***	9.39e-01 ***	1.06 ***	8.14e-01 ***	9.44e-01 ***	1.14 ***	8.17e-01 ***	9.47e-01 ***
	Expected length	1.33e-02 ***	-2.05e-02 ***	8.67e-03 *** 1.41e-02 ***	-2.06e-02 ***	9.26e-03 *** 1.27e-02 ***	-2.06e-02 ***	8.69e-03 ***
	Time-on-market 3-6 months	-1.81e-02	2.54e-03	-1.44e-02	-2.50e-03	2.65e-03	-1.37e-02	-1.85e-02	2.57e-03	-1.51e-02
	Time-on-market 6-9 months	-3.19e-02	3.08e-03	-3.04e-02 *	-1.04e-02	3.10e-03	-2.97e-02 *	-3.17e-02	3.31e-03	-2.99e-02 *
	Time-on-market 9-12 months	-6.64e-02 *	3.64e-03 .	-5.54e-02 ***	-4.58e-02	3.53e-03 .	-5.50e-02 ***	-6.62e-02 *	3.80e-03 .	-5.70e-02 ***
	Time-on-market >12 months	-1.04e-01 **	3.52e-03 .	-9.78e-02 *** -7.86e-02 **	3.59e-03 .	-9.86e-02 *** -9.96e-02 **	3.72e-03 .	-9.82e-02 ***
	Year of construction 1945-1980	-7.39e-03	-3.61e-04	-9.44e-03 .	-1.13e-02	-9.13e-05	-1.01e-02 *	-2.70e-03	-1.75e-04	-8.88e-03 .
	Year of construction 1980-2000	-1.34e-02	-7.14e-04	-1.28e-02 *	-1.63e-02	-7.74e-04	-1.37e-02 *	-1.35e-02	-5.57e-04	-1.25e-02 *
	Year of construction 2000-2018	1.95e-02	-7.73e-04	-2.39e-03	6.78e-03	-4.51e-04	-3.69e-03	1.92e-02	-1.28e-03	2.09e-04
	Year of construction not indicated	-2.13e-03	-4.94e-04	5.59e-03	-1.34e-02	-3.28e-04	3.96e-03	2.21e-03	-2.52e-04	5.77e-03
	% owners in the municipality	2.89e-02	-2.87e-03	3.21e-02 *	2.13e-02	-2.89e-03	3.10e-02 *	2.49e-02	-2.75e-03	2.94e-02 *
	Median income in the municipality	-1.24e-06	6.51e-08	-2.69e-06 ***	-9.23e-07	4.55e-08	-2.51e-06 ***	-1.35e-06	4.99e-08	-2.65e-06 ***
	Price/m2	-1.63e-05 ***	-3.32e-07 .	1.22e-05 *** -1.65e-05 ***	-2.51e-07	1.13e-05 *** -1.41e-05 ***	-2.67e-07	1.17e-05 ***
	Is single man	-4.93e-02 ***	-1.03e-02 ***	-2.86e-02 *** -3.09e-02 **	-1.05e-02 ***	-2.70e-02 *** -4.63e-02 ***	-1.02e-02 ***	-2.61e-02 ***
	Is single woman	-2.76e-02 **	4.28e-04	4.86e-04	-2.40e-02 **	4.59e-04	4.68e-04	-2.24e-02 *	4.17e-04	2.00e-03
	Is year of

Table 3 -Mean, standard error, and correlations

 3

		Mean	SD	Expected length	Single man
	Preference for annuitization	66.3%	0.208	0.214**	-0.060**
	Expected length	11.7	4.63	1	-0.160**
	Single man	23.0%	0.419		1

Table 4 -Effects of annuitization, expected length, and gender on negotiation

 4

	Capacity of negotiation	Model 1	Model 2
	R R 2 Adjusted R 2 Difference of R 2	0.395 0.156 0.155	0.416 0.173 0.170 0.017***
	Constant	78.4***	84.9***
	Annuitization	-0.063***	-0.160***
	Expected length	0.920***	0.338*
	Single man (1)	-2.01***	ns
	Single woman (1)	1.34**	1.31**
	Annuitization x Expected length		0.009***
	Annuitization x Single man		-0.161**
	Expected length x Single man		-0.617*
	Annuitization x Expected length x Single man		0.014**
	***<0.001, **<0.01, *<0.05; ns: not significant; (1) ref variable: couple

These tables are life tables, commonly used in France, built from INSEE

[2000][2001][2002] tables, for males and females, respectively.

The occupied discount corresponds to the right of use and habitation or the usufruct.

t(535) = 3.9, p <0.000

t(728) = 2.6, p = 0.011

To facilitate the understanding of these interaction effects, we dichotomized expected length and preference for annuitization, relative to their median point (Figure4and Figure5).

Appendix 1: Joint life expectancy

The joint life expectancy is the life expectancy of two sellers until the last survivor. It has been calculated using a mortality

Appendix 2: Present value of annuities (PVA)

For each viager, we have calculated the following: the residual life expectancy (single or joint); the term annuity coefficient on a monthly basis, paid until the last survivor with the zero-coupon curve and an inflation of 1% per year; as well as, the present value of annuities and the economic value.

The extracted wealth is calculated at the contract signature with the following formula:

With:

-: annuity coefficient until the last survivor ( , : individuals), on a monthly basis -: lifetime annuity at the contract signature, on a monthly basis -: number of age individuals in the mortality table (we assume that at time , all individuals are dead in the mortality table .  and -: real discount rate -i: inflation (1%) -r: discount rate obtained from the zero-coupon curve observed at the contract signature until the maturity of the investment (maturity = contract signature's date + residual life expectancy ×365.25)

Note: On a yearly basis and for a single individual, we would have

Appendix 3: Durations

The Macaulay duration for a given viager is calculated with the following formula:

-: cash-flow at time o lifetime annuity for o down-payment for -: real discount rate

The Macaulay duration for its equivalent with full annuities is calculated with the following formula:

--: equivalent in monthly lifetime annuities of down-payment at the contract signature.


Faten Jaouahdou

Florian Enez

Christophe Eklouh-Molinier

Pierre Boudry

Vincent Segura

Virginie Nazabal

Florence Phocas

Marc Vandeputte

François Allal

Pierrick Haffray

Sophie Brard-Fudulea

Phénomix : Estimation de l'héritabilité de spectres Raman et premiers résultats d'évaluation phénomique chez la truite

?

Objectifs du programme Phénomix Première application à la truite

• Valorisation des données du programme Omega-truite (FEAMP)

• Biométrie réalisée en 2018 sur une génération de truites issue de 183 parents

• 1372 individus :

• Phénotypés sur des caractères de production-découpe (poids, longueur, FAT, rendements…)

• Analysés par spectrométrie Raman (tissu adipeux abdominal) pour estimation de la teneur en différents acides gras, sur 2 fenêtres spectrales :

•

Conclusions & Perspectives

• Les prétraitements permettent une amélioration de l'héritabilité le long des spectres Raman

  Principe de l'évaluation phénomique 5èmes Journées Techniques Interfilières du SYSAAF, 4 et 5 octobre 2022,

•

  Fenêtre 1 : [550, 1800] → 1845 nombres d'onde • Fenêtre 2 : [2610, 3100] → 1005 nombres d'onde ➢ Génotypages exploitables sur 1661 individus (génération phénotypée + parents), 29K SNP 5èmes Journées Techniques Interfilières du SYSAAF, 4 et 5 octobre 2022, Rennes 4 Démarche expérimentale • Prétraitements des spectres • Estimation d'h² le long des spectres pour choix du meilleur prétraitement 5èmes Journées Techniques Interfilières du SYSAAF, 4 et 5 octobre 2022, Réalisation de validations croisées pour estimer la précision des évaluations pedigree, génomique et phénomique • Construction d'une matrice de ressemblance basée sur les spectres prétraités 𝑐𝑜𝑟𝑟(𝑉𝑎𝑙𝑒𝑢𝑟𝑠 𝑔é𝑛é𝑡𝑖𝑞𝑢𝑒𝑠 𝑒𝑠𝑡𝑖𝑚é𝑒𝑠, 𝑃ℎé𝑛𝑜𝑡𝑦𝑝𝑒𝑠 𝑐𝑜𝑟𝑟𝑖𝑔é𝑠) • Critère de validation : Une héritabilité faible à modérée le long des spectres bruts, localement améliorée par les prétraitements 5èmes Journées Techniques Interfilières du SYSAAF, 4 et 5 octobre 2022, phénomique permet de bien prédire les phénotypes dérivés des spectres 5èmes Journées Techniques Interfilières du SYSAAF, 4 et 5 octobre 2022, phénomique ne surpasse les évaluations classique et génomique que sur peu de caractères 5èmes Journées Techniques Interfilières du SYSAAF, 4 et 5 octobre 2022,

•

  Comparer la précision de la sélection phénomique à la précision de la sélection classique et de la sélection génomique.• Tester la possibilité de combiner des spectres acquis dans des générations différentes • Tester l'effet sur la précision de la technologie d'acquisition des spectres, du tissu, des

	prétraitement réalisés sur les spectres
	• Estimer la dominance
	• Identifier les régions les plus héritables des spectres et susceptibles de reproduire la
	matrice d'apparentement génomique
	5èmes Journées Techniques Interfilières du SYSAAF, 4 et 5 octobre 2022, Rennes	3

•

  Les intensités des spectres Raman acquis sur le tissu adipeux abdominal des truites sont peu à moyennement héritables Possibilité de vérifier la répétabilité des résultats et de combiner les jeux de données ➢ Acquisition de spectres NIRS, pour test de l'effet du type de spectroscopie • Deux autres jeux de donnée à analyser : NIRS et Raman sur daurade, et NIRS sur huitre creuse

• L'évaluation phénomique ne donne de meilleurs résultats que sur peu de caractères 5èmes Journées Techniques Interfilières du SYSAAF, 4 et 5 octobre 2022, Rennes 10

• Une biométrie a été réalisée en 2022 sur une seconde génération de truites génétiquement connectée à la première : ➢

• Possibilité de construire une matrice combinant pedigree et phénomique ?

Merci pour votre attention • Financeurs • Partenaires 5èmes Journées Techniques Interfilières du SYSAAF, 4 et 5 octobre 2022, Rennes 11

5èmes Journées Techniques Interfilières du SYSAAF, 4 et 5 octobre 2022, Rennes