Intuitive Beliefs Jawwad Noor 1 1 Department of Economics Boston - - PowerPoint PPT Presentation

Intuitive Beliefs Belief Formation Conclusion

Intuitive Beliefs

Jawwad Noor1

1Department of Economics

Boston University

December 24, 2019

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion

Introduction

How do limited agents make decisions in a complex world? In the field, besides using tools for analysis, investors also talk about having an “instinct” or a “feel for the market” (Salas et al 2010, Hensman and Sadler-Smith 2011, Huang and Pearce 2015, Huang 2018). Keynes’ animal spirits is a “spontaneous urge to action” (Keynes, 1936) This paper: formal theory of intuition

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion

Introduction

Identify intuition with reliance on associative memory The activation of a thought or mental image activates/inhibits another

Word associations, triggered memory, image of stereotype

Associative/Intuitive/System 1 process:

Involuntary and costless mental processing Associations learnable, strength determined by frequency, salience, similarities, etc Can be strengthened (reinforcement) Can be weakened (counter-conditioning or decay)

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion

Introduction

Identify intuition with reliance on associative memory Model associative memory as an associative network

Inspiration from energy-based neural networks Hopfield net, Boltzmann machine Used for classification, object/speech recognition, etc.

Key questions studied in the paper:

Testable implications for likelihood judgements Formation of networks on the basis of data

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion

Outline

1

Intuitive Beliefs Model Characterization Results Bayesian Intuitive Beliefs

2

Belief Formation Model Illustration

3

Conclusion Related Literature Concluding Comments

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Outline

1

Intuitive Beliefs Model Characterization Results Bayesian Intuitive Beliefs

2

Belief Formation Model Illustration

3

Conclusion Related Literature Concluding Comments

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Primitives

Elements of uncertainty Γ = {1, ..i, j, ..., N}

Index set, cardinality N < ∞ E.g. assets with uncertain return: Γ = {1, 2}

Elementary state space Ωi = {xi, yi, zi...}

Abstract set for each i ∈ Γ E.g. Asset i returns are high, medium or low: Ωi = {hi, mi, li}

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Primitives

(Full) state space Ω =

• i∈Γ

Ωi

Generic element x = (x1, .., xN) E.g. vectors of asset returns (h1, l2) ∈ Ω = {h1, m1, l1} × {h2, m2, l2}

Complexity of the state space

Tension in rational decision models Intuition is a non-conscious, cognitively costless process Associations form on arbitrarily complicated state space

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Primitives

Elementary event space: algebra of events Σi = {xi, yi, ..} ⊂ 2Ωi

E.g. asset i gives a good return: xi lies in xi = {hi, mi}

(Full) event space: Σ =

• i∈Γ

Σi

generic element x = (x1, .., xN) Good (resp. bad) return for asset 1 (resp. 2), x = ({h1, m1}, {m2, l2})

Vectors of elementary events vs subsets of the state space

Each node of associative network is an elementary state Can be extended

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Primitives

Information: true state lies in z = (z1, ..., zN) ∈ Σ A belief is a normalized set-function p(·) over (Ω, Σ) conditional on z ∈ Σ: Assigns p(x|z) ∈ [0, 1] to each event x ∈ Σ and satisfies: (i) p(Ω|z) = 1 (ii) p(x|z) = 0 if xi = φ for some i (iii) p(x ∩ z|z) = p(x|z) Non-additive probability/capacity if satisfies monotonicity: x ⊂ y = ⇒ p(x|z) ≤ p(y|z),

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Primitives

Primitive: a family p of beliefs p(|z) over (Ω, Σ) for each z ∈ Σ+ where Σ+ = {z ∈ Σ : p(z|Ω) > 0}. Avoid conditioning on non-credible information Behavioral meaning of p

Scoring rules in experiments Representation of bets on x given z Connect to choice through signed Choquet integration (Waegenaere and Wakker, 2001)

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Primitives

Notation

For x = (x1, .., xN) ∈ Σ and I ⊂ Γ, define the projection xI : = xIΩ−I. In particular

xi = (Ωj, .., xi, .., Ωj) xixk = (Ωj, .., xi, Ωi+1..., Ωk−1, xk., Ωj)

x ⊂ z is notation for xi ⊂ zi for all i x ∩ z is notation for xi ∩ zi for all i

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Model: Network

Inspired by energy-based neural networks (reviewed later) Nodes: elementary events zi ∈ Σ for each i ∈ Γ Nodes contribute to “associative energy” in the network Λ(x|z) := −[

• i<j

a(xixj) +

• i∈Γ

b(xi|z)]. Associative bias b(xi|z) is node xi association with information Symmetric associative weight a(xixj) between pairs of nodes Energy: (negative of) sum of associative weights and biases

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Model: Network

Definition An associative network is a tuple (a, b) that consists of (i) an association function a that maps each xi, xj ∈ ∪k∈ΓΣk to a symmetric associative weight a(xixj) ∈ R ∪ {−∞}, (ii) a bias function b that maps each xi ∈ ∪k∈ΓΣk and z ∈ Σ+ to some bias b(xi|z) ∈ R ∪ {−∞}. Symmetry: a(xixj) = a(xjxi) a(xixj) = −∞: occurrence of one associated with non-occurrence of other

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Model: Network

Definition An associative network (a, b) is regular if for all xi ∈ ∪k∈ΓΣk and z ∈ Σ+, the it satisfies b(xi|zi) = a(xizi) and b(xi|z) > −∞ = ⇒ a(xizj) > −∞ for all j ∈ Γ and b(xi|Ω) > −∞.

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Model: General Updating

Definition Beliefs p are Intuitive Beliefs with General Updating (IBGU) if there exists a regular associative network (a, b) and a function Z : Σ+ → R++ such that for any (x, z) ∈ Σ × Σ+ s.t. x ⊂ z, p(x|z) = 1 Z(z) × exp

 

i<j

a(xixj) +

• i∈Γ

b(xi|z)

  .

The associative network (a, b) is said to represent p. That is, p(x|z) =

1 Z(z) × exp [−Λ(x|z)]

Note Z(z) = exp [−Λ(z|z)] is determined by (a, b) Apply p(x ∩ z|z) = p(x|z) to non-nested x, z

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Model: Intuitive Updating

Definition Beliefs p are Intuitive Beliefs with Intuitive Updating (IBIU) if there exists a regular associative network (a, b) satisfying b(xi|z) =

• j∈Γ

a(xizj) for all xi ∈ ∪k∈ΓΣk and z ∈ Σ+, and moreover, for any (x, z) ∈ Σ × Σ+ s.t. x ⊂ z, p(x|z) = 1 Z(z) × exp

 

i<j

a(xixj) +

• i∈Γ
• j∈Γ

a(xizj)

  .

The association function a is said to represent p.

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Model: Intuitive Updating

IBGU silent about how information z changes energy IBIU produces all beliefs from a Regularity of (a, b) becomes: for all xi ∈ Σ and z ∈ Σ+,

• j∈Γ

a(xizj) > −∞ = ⇒

• j∈Γ

a(xiΩj) > −∞, IBGU model: only matters whether a(xizi) > −∞ IBIU model: exact value of a(xizi) matters

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Model: Boltzmann Machine

Each pixel i ∈ Γ on a screen is a node A node i can be off/on, value xi ∈ Ωi = {0, 1} A state x = (x0, ..., xN) is a configuration of activations Define energy at state x by Λ(x) = −[

• i<j

a(xixj) +

• i∈Γ

b(xi)] where xi = 0 = ⇒ a(xixj) = 0 and b(xi) = 0 Update rule for activation: prob xi = 1 is a logistic function of energy ψ(xi = 1|x) = f (Λ(x)) Update nodes in random order − → dynamic evolution of state

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Model: Boltzmann Machine

Distribution over states converges to an equilibrium

Called thermal equilibrium (physics reference)

The equilibrium distribution is of the Boltzmann-Gibbs form p(x|Ω) = 1 Z exp[

• i<j

a(xixj) +

• j∈Γ

b(xj)] Improves on the Hopfield network

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Model: Boltzmann Machine

Probability of an event obtained additively: p(x|Ω) = 1 Z

• x∈x

exp[

• i<j

a(xixj) +

• j∈Γ

b(xj)], Posteriors by Bayesian conditioning: for any x ⊂ z, p(x|z) = p(x|Ω) p(z|Ω) =

• x∈x exp[

i<j a(xixj) + j∈Γ b(xj)]

• z∈x exp[

i<j a(zizj) + j∈Γ b(zj)] .

The key differences

bias is information-dependent Non-additive, Non-Bayesian Interpret belief over states vs probability of state

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Outline

1

Intuitive Beliefs Model Characterization Results Bayesian Intuitive Beliefs

2

Belief Formation Model Illustration

3

Conclusion Related Literature Concluding Comments

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Properties of IBGU

Characterization of IBGU model Statistical independence: p(x|z) =

j∈Γ p(xj|z), for all x, z

Measure of statistical dependence SD(x|z): SD(x|z) := p(x|z)

• j∈Γ p(xj|z).

p(x|z)

• j∈Γ p(xj|z) ≥ 1 −

→ (x1, .., xN) are positively related

p(x|z)

• j∈Γ p(xj|z) ≤ 1 −

→ (x1, .., xN) are negatively related

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Properties of IBGU

Definition Beliefs p satisfy Aggregated Statistical Dependence if for any x, z ∈ Σ+, p(x|z)

• j∈Γ p(xj|z) =
• i<j

p(xixj|z) p(xi|z)p(xj|z). SD(x|z) built from SD(xixj|z), but no structure on SD(xixj|z) Information theory: pairwise mutual information

p(XY ) p(X)p(Y )

between pairs of RV’s

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Properties of IBGU

Next: deviation from Bayesian update Bayesian update: pBU(x|z) := p(x ∩ z|Ω) p(z|Ω) .

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Properties of IBGU

Definition Beliefs p satisfy Aggregated Updating Bias if for any x, z ∈ Σ+ s.t. xiz−i ∈ Σ+ for all i, p(x|z) pBU(x|z) =

• i∈Γ

p(xi|z) pBU(xi|z). Bias

p(x|z) pBU(x|z) built from bias of marginals p(xi|z) pBU(xi|z)

No structure on

p(xi|z) pBU(xi|z)

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Properties of IBGU

Definition Beliefs p satisfy Regularity if for any x ∈ Σ and z ∈ Σ+ s.t. x ⊂ z, p(x|z) > 0 ⇐ ⇒ p(x|Ω) > 0 and p(xi|z) > 0 for all i ∈ Γ. Minimal relationship between posteriors and prior Satisfied by Bayesian beliefs

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Properties of IBGU

Definition Beliefs p satisfy Information Regularity if for any i ∈ Γ, xi ∈ Σ and z ∈ Σ+ s.t. xi ⊂ zi, p(xi|z) > 0 = ⇒ p(xi|zi) > 0. If xi is possible given z then also possible under less information Satisfied by Bayesian beliefs

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Characterization

Theorem Beliefs p are Intuitive Beliefs with General Updating if and only if they satisfy Aggregated Statistical Dependence, Aggregated Updating Bias, Regularity and Information Regularity.

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Characterization

Theorem Beliefs p are Intuitive Beliefs with General Updating if and only if they satisfy Regularity, Aggregated Statistical Dependence and Aggregated Updating Bias. Content: beliefs over “simple” events are building blocks of beliefs Observable implication of reliance on associative network Can weaken Aggregated Statistical Dependence Next consider characterization of IBIU

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Properties of IBIU

Definition Beliefs p satisfy Intuitive Updating if for any distinct i, j ∈ Γ and xizj ∈ Σ+, p(xi|zj) pBU(xi|zj) = p(xizj|Ω) p(xi|Ω)p(zj|Ω). Key property: Updating strength depends on statistical dependence

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Properties of IBIU

Definition Beliefs p satisfy Information Separability if for any distinct i, j ∈ Γ, xi, z ∈ Σ+ s.t. xi ⊂ z and p(xi|Ωjz−j) > 0, p(xi|zjz−j) p(xi|Ωjz−j) = p(xi|zj) p(xi|Ω). Dependence on information is separable Expresses additive separability of associative bias

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Properties of IBIU

Definition Beliefs p satisfy Strong Information Regularity if for any i, j ∈ Γ, xi ∈ Σ and z ∈ Σ+ s.t. xi ⊂ zi, p(xi|z) > 0 = ⇒ p(xi|Ωjz−j) > 0. If xi is possible given z then it is possible given less information

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Properties of IBIU

Theorem Suppose beliefs p are Intuitive Beliefs with General Updating. Then p are Intuitive Beliefs with Intuitive Updating if and only if they satisfy Intuitive Updating, Information Separability and Strong Information Regularity.

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Uniqueness

To sidestep some (uninteresting) non-uniqueness that arise when p(xi|Ω) = 0, assume: for any xi ∈ ∪l∈ΓΣl and i = j ∈ Γ, a(xiΩj) = −∞ or b(xi|Ω) = −∞ = ⇒ a(xizk) = −∞ for all zk ∈ ∪l∈ΓΣl. (1) Issue: p(xi|Ω) = 0 ⇐ ⇒

j a(xiΩj) = −∞ or b(xi|Ω) = −∞

By Regularity, p(xi|Ω) = 0 implies p(xi|z) = 0 Value of any a(xiΩj), a(xizj) > −∞ is immaterial

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Uniqueness

Theorem Consider two regular IBGU representations (a, b) and (α, β) that satisfy (1). Then (a, b) and (α, β) represent the same p if and only if there exist real-valued functions (x, i, j) → γ(xiΩj) and (z, i) → ψ(zi|z) such that for any distinct i, j ∈ Γ, x ∈ Σ and z ∈ Σ+ for which x ⊂ z, a(xizj) = α(xizj) + [γ(xiΩj) + γ(zjΩi) − γ(ΩiΩj)], and b(xi|z) = β(xi|z) + ψ(zi|z) −

• i=j∈Γ

[γ(xiΩj) − γ(ziΩj)] .

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Uniqueness

Theorem Consider two regular IBIU representations a and α that satisfy (1). Then a and α represent the same p if and only if (i) there exists a real-valued function (x, i, j) → γ(xiΩj) satisfying, for each i ∈ Γ and xi ∈ Σ+, γ(xiΩi) = γ(ΩiΩi) − 2

• i=j∈Γ

[γ(xiΩj) − γ(ΩiΩj)] , (ii) there exists a function (i, j, zi) → λ(zi, j) ∈ R satisfying λ(zi, j) = 0 if j = i, and (iii) for any i, j ∈ Γ, x ∈ Σ and z ∈ Σ+ s.t. x ⊂ z, a(xizj) = α(xizj) + [γ(xiΩj) + γ(zjΩi) − γ(ΩiΩj)] + λ(zi, j).

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Uniqueness

Corollary For any representation (α, β) of an IBGU p there exists another representation a that is normalized in that it satisfies (1) and the following properties: (a) a(xiΩj) = 0 for any x ∈ Σ+ and distinct i, j ∈ Γ, (b) a(xixi) = 0 for any x ∈ Σ+ and i ∈ Γ, (c) b(zi|z) = 0 for any z ∈ Σ+ and i ∈ Γ. For any representation α of an IBIU p there exists another representation a that is normalized in that it satisfies (1) and properties (a) and (b). Uniqueness result yields this useful normalization

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Identification

Proposition For IBGU p, the following statements hold: (i) For any x ∈ Σ and z ∈ Σ+ s.t. x ⊂ z and any distinct i, j ∈ Γ s.t. p(xi|z)p(xj|z) > 0, p(xixj|z) p(xi|z)p(xj|z) = exp [a(xixj) − a(xizj) − a(zixj) + a(zizj)] (ii) For any i ∈ Γ and xiz−i, z ∈ Σ+ s.t. xi ⊂ zi, p(xi|z) BU(xi|z) = exp[b(xi|z) − b(zi|z) − b(xi|Ω) + b(zi|Ω)]. Content:

Marginal updating bias reflects b Binary Statistical Dependence reflects a

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Identification

Proposition For IBGU p with normalized (a, b), p(xixj|Ω) p(xi|Ω)p(xj|Ω) = exp [a(xixj)] for any distinct i, j ∈ Γ and xixj ∈ Σ s.t. p(xi|Ω)p(xj|Ω) > 0, and p(xi|zi) = exp[a(xizi)] for any i ∈ Γ and xi ∈ Σ and zi ∈ Σ+ s.t. xi ⊂ zi. Finally, p(xi|z)pBU(xi|zi) pBU(xi|z) = exp[b(xi|z)] for any i ∈ Γ and xi, z ∈ Σ+ s.t. xiz−i ∈ Σ+.

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Outline

1

Intuitive Beliefs Model Characterization Results Bayesian Intuitive Beliefs

2

Belief Formation Model Illustration

3

Conclusion Related Literature Concluding Comments

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Monotonicity

Primitive: normalized set function A capacity if it satisfies Monotonicity By ASD, beliefs are built from beliefs about 2-d events Characterization requires more than Monotonicity wrt 2-d events

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Monotonicity

Proposition The following are equivalent for IBGU p: for any z ∈ Σ+, (i) p(|z) satisfies monotonicity. (ii) For any x, y ∈ Σ s.t. p(x|z) > 0, if xi ⊂ yi and y =yix−i, then p(xi|z)×

• i=j∈Γ s.t. xj=zj

p(xixj|z) p(xi|z) ≤ p(yi|z)×

• i=j∈Γ s.t. xj=zj

p(yixj|z) p(yi|z) . (iii) The associative network (a, b) satisfies: for any x, yi ∈ Σ s.t. p(x|z) > 0, xi ⊂ yi = ⇒

• i=j

a(xixj) + b(xi|z) ≤

• i=j

a(yixj) + b(yi|z).

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Bayesian Updating

Can intuitive beliefs be Bayesian? p is Bayesian if it satisfies: for all (x, z) ∈ Σ × Σ+, p(x|z) = pBU(x|z) := p(x ∩ z|Ω) p(z|Ω) Aggregated Updating Bias provides a relationship: p(x|z) = BU(x|z)

• i∈Γ

p(xi|z) BU(xi|z)

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Bayesian Updating

Proposition The following are equivalent for IBGU p: (i) p is Bayesian. (ii) For all xi, z ∈ Σ+ s.t. xi ⊂ zi, p(xi|z) = pBU(xi|z) > 0. (iii) In any representation (a, b), for all xi, z ∈ Σ+ s.t. xi ⊂ zi and p(xi|z) > 0, b(xi|z) − b(zi|z) = b(xi|Ω) − b(zi|Ω).

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Characterization Results Bayesian Intuitive Beliefs

Bayesian Updating

Proposition The following are equivalent for IBIU p: (i) p is Bayesian. (ii) For any x, z ∈ Σ+ s.t. x ⊂ z and p(x|z) > 0, p(x|z) =

• i∈Γ p(xi|Ω)
• i∈Γ p(zi|Ω)

(iii) In any normalized representation, for any distinct i, j ∈ Γ and events xi, zi, xizj ∈ Σ+ s.t. xi ⊂ zi, a(xizj) = 0 and a(xizi) − a(zizi) = a(xiΩi) − a(ziΩi). Intuition can be Bayesian only if perceived uncertainty has simple structure

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Outline

1

Intuitive Beliefs Model Characterization Results Bayesian Intuitive Beliefs

2

Belief Formation Model Illustration

3

Conclusion Related Literature Concluding Comments

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Model of Belief Formation

Agent experiences events according to a Bayesian system of

• bjective probabilities

q∗ = {q∗(·|z) over (Ω, Σ) : z ∈ Σ s.t. q∗(z|Ω) > 0}. How are Intuitive Beliefs p shaped by q∗? Model after machine learning Beliefs minimize distance wrt some loss function

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Model of Belief Formation

Fix some class P of beliefs Definition Prior beliefs p(|Ω) are trained by q∗(|Ω) if they solve: minp∈P, p(|Ω)∈p L((x, q∗(x|Ω), p(x|Ω))x∈Σ), for some real-valued function L. Content: the prior is trained Permits training only on simple events Presumption: agent experiences q∗(|Ω) for a long period

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Model of Belief Formation

Common priors if unique solution + posteriors pinned down Non-uniquess → heterogeneous beliefs Even if unique prior, the posteriors may not be determined Bayesian: unique posterior for every prior IBGU: general updating rule IBIU: a(xizi) for xi ⊂ zi not determined by training Some strengthening desirable

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Model of Belief Formation

Definition Beliefs are semi-deliberative if they are Bayesian with respect to any single source: for any xi ∈ Σ and zi ∈ Σ+, p(xi|zi) = p(xi|Ω) p(zi|Ω). Rational updating along any given dimension

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Model of Belief Formation

Definition Beliefs p are strongly trained by q∗ if they solve minp∈Pmaxz∈Σ+ L((x, q∗(x|z), p(x|z))x∈Σ). Beliefs of an expert Extensively experiences the objective posterior conditional on information Network is shaped by entire family of objective distributions

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Outline

1

Intuitive Beliefs Model Characterization Results Bayesian Intuitive Beliefs

2

Belief Formation Model Illustration

3

Conclusion Related Literature Concluding Comments

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Illustration

Suppose only two dimensions, Γ = {1, 2} Loss function L is the supnorm: L(q∗(|Ω), p(|Ω)) = supx∈Σ|q∗(x|Ω) − p(x|Ω)|. Restrict attention to IBIU, semi-deliberative Results in terms of objective prior statistical dependence: SD∗(xixj|Ω) := q∗(xixj|Ω) q∗(xi|Ω)q∗(xj|Ω).

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Illustration

Semi-deliberative IBIU do not always exist Under some condition on q∗, existence and uniqueness: Proposition Suppose that for all x, z ∈ Σ+ s.t. x ⊂ z, q∗(zi|Ω) q∗(xi|Ω) ≤

• q∗(zizj|Ω)

q∗(xizj|Ω)

1.5

Then there exists a unique family of semi-deliberative Intuitive Beliefs with Intuitive Updating that is trained by q∗.

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Illustration

Proposition Moreover: (i) The prior is perfectly trained in the sense that p(|Ω) = q∗(|Ω). (ii) The posterior is given by p(xixj|z) = [q∗(xi|zi)q∗(xj|zj)]

• SD∗(xixj|Ω)

SD∗(zizj|Ω) SD∗(xizj|Ω) SD∗(zizj|Ω) SD∗(zixj|Ω) SD∗(zizj|Ω)

• .

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Illustration: Correlation Perception

Proposition For any x, z ∈ Σ+, p(xixj|z) p(xi|z)p(xj|z) = q∗(xixj|z) q∗(xi|z)q∗(xj|z). Posteriors deviate from objective posterior, but cancel out Feature of 2-dimensions More generally, IB unable to perceive high dimensional dependence Reason: simple binary nature of associations

• Eg. 2-d independence implies belief in independence

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Illustration: Base-Rate Neglect/Conservatism

Proposition For any x, z ∈ Σ+, p(x|z) ≥ pBU(x|z) ⇐ ⇒ SD∗(xizj|Ω)×SD∗(zixj|Ω) ≥ SD∗(zizj|Ω)2. Over/undershooting Bayes (Tversky and Kahneman, 1974) Referred to as base rate neglect/conservatism Over-updating iff x and z are associated strongly enough SD∗(zizj|Ω)2 is the threshold a(xizi) drops because semi-deliberative

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Illustration: Conjunction Fallacy

Non-monotonicity (Tversky and Kahneman, 1974) Subjects given information1 about “Linda” then Asked which is more likely:

Linda is a bank teller Linda is a bank teller that is active in the feminist movement

Subjects deem the conjunction of events as more likely

1“Linda is 31 years old, single, outspoken, and very bright. She majored in

• philosophy. As a student, she was deeply concerned with issues of

discrimination and social justice, and also participated in anti-nuclear demonstrations”.

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Illustration: Conjunction Fallacy

Proposition For any yixj, xixj, z ∈ Σ+ s.t. xi ⊂ yi, p(yixj|z) ≥ p(xixj|z) ⇐ ⇒ q∗(yixj|Ω) ≥ q∗(xixj|Ω)×SD∗(xizj|Ω) SD∗(yizj|Ω). Objective q∗(yixj|Ω) ≥ q∗(xixj|Ω) If xi is strongly associated with zj then SD∗(xizj|Ω)

SD∗(yizj|Ω) > 1

Possible to get p(xixj|z) > p(yixj|z)

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Model Illustration

Illustration: Other

Gambler’s Fallacy/Hot-Hand Effect as in literature: Updating of underlying parameter with information

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Outline

1

Intuitive Beliefs Model Characterization Results Bayesian Intuitive Beliefs

2

Belief Formation Model Illustration

3

Conclusion Related Literature Concluding Comments

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Related Literature

Associations and memory modelled as networks in cognitive psychology using spreading activation networks (Collins and Loftus 1975, Anderson 1983). More advanced modelling of associative memory taken up in computer science Hopfield (1982) shows how associative memory can be modelled using an energy-based network Boltzmann machine is stochastic version of Hopfield network (Hinton and Sejnowski 1983, Ackley, Hinton and Sejnowski 1985) Formal structure inspired Intuitive Beliefs structure But this literature is not about beliefs

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Related Literature

(Non-formal) Paradigm in Psychology: Heuristics and Biases program of Kahneman-Tversky

Likelihood judgements based on heuristics that create systematic biases Representativeness, Availability, and Anchoring and Adjustment Heuristics

Morewedge and Kahneman (2010) posit that intuitive judgements are made through automatic, non-deliberative “System 1” processing, of which associative memory is a part. Intuitive beliefs model accommodates Representativeness and Availability Associations are a coarse category (subsuming similarity, salience/recallability) But provides empirical mileage - relevant for economics

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Related Literature

Economics literature: Non-Bayesian Updating Model particular biases: Rabin 2002, Gennaioli and Shleifer 2010, Rabin and Vayanos (2010), Benjamin et al (2019)... Other: Epstein et al 2008, Ortoleva 2012.... Intuitive Beliefs are a framework

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Related Literature

Economics literature: Memory Gilboa and Schmeidler (1995) (uncertainty) and Bordalo et al (2019) (certainty) Memory: sets of past cases Similarity determines what is considered/recalled Intuitive Beliefs: memory takes form of weights in a network

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Related Literature

Economics literature: Belief formation Case-based inductive inference (Gilboa and Schmeidler 2003)

• utcome A more likely than B if similar past cases support A

more than B. past cases are a primitive

Bounded rationality (Spiegler 2016)

Agent’s model posits causal relations + naiveté Obtains corresponding marginals from data and then constructs belief using chain rule Bayesian networks: probabilistic graph representation of conditional independence between variables Bayesian agent with incorrect prior

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Related Literature

Similarities:

Network holds the memory of cases, frequencies shape beliefs Energy-based vs Bayesian networks: CS literature on stochastic networks

Differences:

Intuitive vs rational Associations can be formed by copies of data as well

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Outline

1

Intuitive Beliefs Model Characterization Results Bayesian Intuitive Beliefs

2

Belief Formation Model Illustration

3

Conclusion Related Literature Concluding Comments

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Summary

Motivation:

Rationality, simplification plays role in economic decisions Intuitive assessment also plays a role

Beliefs are intuitive if rely on associative memory Associative memory modelled as energy-based network

Study observable manifestation Study uniquness and identification

Belief formation: training of network, as in machine learning Intuitive updating, consistent with evidence

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Future Directions

Many directions for future research: Intuitive learning (in progress)

Intuition is an informative signal, but how informative? Under what conditions can intuition learn the true state? Updating beliefs vs updating network with information

Extend to hypergraphs so that associations are not necessarily binary

Non-Bayesian due to limit on dimensionality of associative connections

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Future Directions

Decision-theoretic analysis

betting preferences condition on signals

The formation of beliefs:

Limited attention can bias training

• Eg. if already hold an asset, then attention restricted to

relevant events Repetition can form associations (use multi-sets)

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Future Directions

Decision-theoretic analysis

betting preferences condition on signals

The formation of beliefs:

Limited attention can bias training

• Eg. if already hold an asset, then attention restricted to

relevant events Repetition can form associations (use multi-sets)

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

The End

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Model: Other Formulations

Log-Additive Intuitive Beliefs: p(x|z) = 1 Z(z) × exp

• x∈x
• z∈z

Λ(x|z)

• =

1 Z(z) × exp

 

x∈x

• z∈z
• i<j

a(xixj) +

• i∈Γ

a(xi|z)

 

Left for future research

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Implicit Restrictions

Recall reduced form p(x|z) =

 

i∈Γ

p(xi|zi)

   

i<j

SD(xixj|Ω) SD(zizj|Ω)

   

i∈Γ

• i=j∈Γ

SD(xizj|Ω) SD(zizj|Ω)

 

For any z = Ω, p(x|z) ≤ 1 requires

 

i∈Γ

p(xi|zi)

  ≤  

i<j

SD(xixj|Ω) SD(zizj|Ω)

   

i∈Γ

• i=j∈Γ

SD(xizj|Ω) SD(zizj|Ω)

 

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Implicit Restrictions

More care needed for prior: p(x|Ω) =

 

i∈Γ

p(xi|Ω)

  ×  

i<j

SD(xixj|Ω)

 

Note: the expression follows directly from ASD Sufficient conditions:

Jawwad Noor Intuitive Beliefs

Intuitive Beliefs Belief Formation Conclusion Related Literature Concluding Comments

Implicit Restrictions

Proposition For any x ∈ Σ+, we have

 

i∈Γ

p(xi|Ω)

  ×  

i<j

SD(xixj|Ω)

  ≤ 1

for each x ∈ Σ if any of the two (equivalent) statements hold: (ii) n−1

i=1 p(xixn|Ω) p(xi|Ω)p(xn|Ω) ≤ p(xn|Ω)−1 for each 2 < n ≤ N.

(iii) a(xnΩn) + n−1

i=1 a(xixn) ≤ 0 for each 2 < n ≤ N.

Jawwad Noor Intuitive Beliefs