Predicting Preference Reversals via Gaussian Process Uncertainty Aversion

References
Predicting Preference Reversals
via Gaussian Process Uncertainty Aversion
Rikiya Takahashi1
Tetsuro Morimura2
1SmartNews, Inc.
rikiya.takahashi@smartnews.com
2IBM Research - Tokyo
tetsuro@jp.ibm.com
May 10, 2015
AISTATS 2015 Predicting Preference Reversals via Gaussian Process Uncertainty

References
Discrete Choice Modelling
Goal: predict prob. of choosing an option from a choice set.
Why solving this problem?
For business: brand positioning among competitors
For business: sales promotion (yet involving some abuse)
To deeply understand how human makes decisions

References
Random Utility Theory
Each human is a maximizer of random utility.
i’s choice from Si = arg max
j∈Si
fi (vj )
mean utility
+ εij
random noise
Si : choice set for i, vj : vector of j’s attributes, fi : i’s
mean utility function
Assuming independence among every option’s attractiveness
For both mean and noise: (e.g., logit (McFadden, 1980))
For only mean: (e.g., nested logit (Williams, 1977))

References
Why Random Utility Theory has been Used?
Voices from friends (machine learners & econometricians)
1 Rationality of independence assumption
Attributes of unchosen options are irrelevant to the
chosen option’s benefit.
I bought diamond. This is the best. It’s ridiculous to
think that other dirty stones affected my final choice.
2 Computational practicality
Unless scoring each option, how to decide the best one?
Formalizing data likelihood is straight and easy.

References
Complexity of Real Human’s Choice
An example of choosing PC (Kivetz et al., 2004)
Each subject chooses 1 option from a choice set
A B C D E
CPU [MHz] 250 300 350 400 450
Mem. [MB] 192 160 128 96 64
Choice Set #subjects
{A, B, C} 36:176:144
{B, C, D} 56:177:115
{C, D, E} 94:181:109
Can random utility theory still explain the preference reversals?
B C or C B?

References
Agenda
1 Introduction of the Goal and Issues
2 Irrational Context Effects
Similarity Effect
Attraction Effect
Compromise Effect
Prior Work
3 Proposing a Bayesian Model of Mental Conflict
4 Numerical Studies
5 Conclusion

References
Similarity Eﬀect (Tversky, 1972)
Top-share choice can change due to correlated utilities.
E.g., one color from {Blue, Red} or {Violet, Blue, Red}?

References
Attraction Eﬀect (Huber et al., 1982)
Introduction of an absolutely-inferior option A−
(=decoy)
causes irregular increase of option A’s attractiveness.
Despite the natural guess that decoy never aﬀects the choice.
If D A, then D A A−
.
If A D, then A is superior to both A−
and D.

References
Compromise Eﬀect (Simonson, 1989)
Moderate options within each chosen set are preferred.
Diﬀerent from non-linear utility function involving
diminishing returns (e.g.,
√
inexpensiveness+
√
quality).

References
Positioning of Our Work in Literature
Sim.: similarity, Attr.: attraction, Com.: compromise
Sim. Attr. Com. Mechanism Predict. for Likelihood
Test Set Maximization
SPM OK NG NG correlation OK MCMC
MDFT OK OK OK dominance & indifference OK MCMC
PD OK OK OK nonlinear pairwise comparison OK MCMC
MMLM OK NG OK none OK Non-convex
NLM OK NG NG hierarchy NG Non-convex
BSY OK OK OK Bayesian OK MCMC
LCA OK OK OK loss aversion OK MCMC
MLBA OK OK OK nonlinear accumulation OK Non-convex
Proposed OK NG OK Bayesian OK Convex
MDFT: Multialternative Decision Field Theory (Roe et al., 2001)
PD: Proportional Difference Model (González-Vallejo, 2002)
MMLM: Mixed Multinomial Logit Model (McFadden and Train, 2000)
SPM: Structured Probit Model (Yai, 1997; Dotson et al., 2009)
NLM: Nested Logit Models (Williams, 1977; Wen and Koppelman, 2001)
BSY: Bayesian Model of (Shenoy and Yu, 2013)
LCA: Leaky Competing Accumulator Model (Usher and McClelland, 2004)
MLBA: Multiattribute Linear Ballistic Accumulator Model (Trueblood, 2014)

References
Agenda
Utility Estimation as Dual Personality
Irrationality by Bayesian Shrinkage
Convex Optimization when using Posterior Mean
4 Numerical Studies
5 Conclusion

References
Utility Estimation as Dual Personality
How about regarding utilities as samples in statistics?
Assumption 1: Utility function is partially disclosed to DMS.
1 UC computes the sample value of every option’s utility,
and sends only these samples to DMS.
2 DMS statistically estimates the utility function.

References
Mental Conflict as Bayesian Shrinkage
Assumption 2: DMS does Bayesian shrinkage estimation.
i ∈{1, . . . , n}: context, yi ∈{1, . . . , m[i]}: final choice
Xi (xi1 ∈RdX , . . . , xim[i]) : features of m[i] options
Objective Data: values of random utilities
vi (vi1, . . . , vim[i]) ∼N µi , σ2
Im[i] , vij = b+wφ φ (xij )
µi : Rm[i]: vec. of the true mean utility, σ2: noise level
b: bias term, φ : RdX →Rdφ : mapping function. wφ: vec. of coefficients
Subjective Prior: choice-set-dependent Gaussian process
µi ∼ N 0m[i], σ2
K(Xi ) s.t. K(Xi ) = (K(xij , xij ))∈Rm[i]×m[i]
µi ∈Rm[i]: vec. of random utilities, K(·, ·): similarity between options
Final choice: based on (Posterior mean u∗
i + i.i.d. noise) as
u∗
i = K(Xi ) Im[i]+K(Xi )
−1
b1m[i]+Φi wφ ,
yi = arg max
j
(u∗
ij + εij ) where ∀j εij ∼ Gumbel.

References
Irrationality by Bayesian Shrinkage
Implication of (1): similarity-dependent discounting
u∗
i = K(Xi ) Im[i]+K(Xi )
−1
shrinkage factor
b1m[i]+Φi wφ
vec. of utility samples
. (1)
Under RBF kernel K(x, x ) = exp(−γ x − x 2
),
an option dissimilar to others involves high uncertainty.
Strongly shrunk into prior mean 0.
Context eﬀects as Bayesian uncertainty aversion
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 2 3 4
FinalEvaluation
X1=(5-X2)
DA
- A
{A,D}
{A,A
-
,D}
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 2 3 4
FinalEvaluation
X1=(5-X2)
DCBA
{A,B,C}
{B,C,D}

References
Convex Optimization when using Posterior Mean
Global ﬁtting of the parameters using data (Xi , yi )n
i=1
Fix the mapping and similarity functions during updates.
Shrinkage factor Hi K(Xi )(Im[i] + K(Xi ))−1
is constant!
Obtaining a MAP estimate is convex w.r.t. (b, wφ).
max
b,wφ
n
i=1
( bHi 1m[i]+Hi Φi wφ
Context−speciﬁc Hi is multiplied.
, yi ) −
c
2
wφ
2
Exploiting the log-concavity of multinomial logit
(u∗
i , yi ) log
exp(u∗
iyi
)
m[i]
j =1 exp(u∗
ij )

References
Agenda
4 Numerical Studies
5 Conclusion

References
Experimental Settings
Evaluates accuracy & log-likelihood for real choice data.
Dataset #1: PC (n=1, 088, dX =2)
Dataset #2: SP (n=972, dX =2)
Subjects are asked of choosing a speaker.
A B C D E
Power [Watt] 50 75 100 125 150
Price [USD] 100 130 160 190 220
Choice Set #subjects
{A, B, C} 45:135:145
{B, C, D} 58:137:111
{C, D, E} 95:155: 91
Dataset #3: SM (n=10, 719, dX =23)
SwissMetro dataset (Antonini et al., 2007)
Subjects are asked of choosing one transportation, either
from {train, car, SwissMetro} or {train, SwissMetro}.
Attribute of option: cost, travel time, headway, seat
type, and type of transportation.

References
Cross-Validation Performances
High predictability in addition to the interpretable mechanism.
For SP, successfully detected combination of compromise
eﬀect & prioritization of power.
1st best for PC & SP.
2nd best for higher-dimensional SM: slightly worse than
highly expressive nonparametric version of mixed
multinomial logit (McFadden and Train, 2000).
-1.1
-1
-0.9
-0.8
AverageLog-Likelihood
Dataset
PC SP SM
LinLogit
NpLogit
LinMix
NpMix
GPUA
0.3
0.4
0.5
0.6
0.7
ClassificationAccuracy
Dataset
PC SP SM
LinLogit
NpLogit
LinMix
NpMix
GPUA
2
3
4
100 150 200
Evaluation Price [USD]
EDCBA
Obj. Eval.
{A,B,C}
{B,C,D}
{C,D,E}

References
Conclusion
Introduced a simple & interpretable Bayesian choice model.
Bayesian shrinkage involving mental conflict
Irrational choice-set-dependent Gaussian process prior
Uncertain aversion as a cause of context effects
Accurate prediction when absolute preference and
compromise effect are mixed.

References
Future Directions
More active Bayesianism for realistic human models
Integration with other Bayesian discrete choice models
(e.g., (Shenoy and Yu, 2013))
Explaining attraction effect
Current limitation: decoy gets high share due to
symmetric similarity to target option.
Extension to time-series decision making models
E.g., emulating how human plays multi-armed bandit
(Zhang and Yu, 2013)
Choice-set optimization avoiding irrational context effects
News channel = set of news articles
Diversified item recommendation (Ziegler et al., 2005)
Via linear submodular bandits (Yue and Guestrin, 2011)

References
References I
Antonini, G., Gioia, C., Frejinger, E., and Thémans, M. (2007).
Swissmetro: description of the data.
http://biogeme.epfl.ch/swissmetro/examples.html.
Dotson, J. P., Lenk, P., Brazell, J., Otter, T., Maceachern, S. N.,
and Allenby, G. M. (2009). A probit model with structured
covariance for similarity effects and source of volume
calculations. http://ssrn.com/abstract=1396232.
González-Vallejo, C. (2002). Making trade-offs: A probabilistic and
context-sensitive model of choice behavior. Psychological
Review, 109:137–154.
Huber, J., Payne, J. W., and Puto, C. (1982). Adding
asymmetrically dominated alternatives: Violations of regularity
and the similarity hypothesis. Journal of Consumer Research,
9:90–98.

References
References II
Kivetz, R., Netzer, O., and Srinivasan, V. S. (2004). Alternative
models for capturing the compromise effect. Journal of
Marketing Research, 41(3):237–257.
McFadden, D. and Train, K. (2000). Mixed MNL models for
discrete response. Journal of Applied Econometrics,
15:447 –470.
McFadden, D. L. (1980). Econometric models of probabilistic
choice among products. Journal of Business, 53(3):13–29.
Roe, R. M., Busemeyer, J. R., and Townsend, J. T. (2001).
Multialternative decision field theory: A dynamic connectionist
model of decision making. Psychological Review, 108:370–392.
Shenoy, P. and Yu, A. J. (2013). A rational account of contextual
effects in preference choice: What makes for a bargain? In
Proceedings of the Cognitive Science Society Conference.

References
References III
Simonson, I. (1989). Choice based on reasons: The case of
attraction and compromise eﬀects. Journal of Consumer
Research, 16:158–174.
Trueblood, J. S. (2014). The multiattribute linear ballistic
accumulator model of context eﬀects in multialternative choice.
Psychological Review, 121(2):179– 205.
Tversky, A. (1972). Elimination by aspects: A theory of choice.
Psychological Review, 79:281–299.
Usher, M. and McClelland, J. L. (2004). Loss aversion and
inhibition in dynamical models of multialternative choice.
Psychological Review, 111:757– 769.
Wen, C.-H. and Koppelman, F. (2001). The generalized nested
logit model. Transportation Research Part B, 35:627–641.

References
References IV
Williams, H. (1977). On the formulation of travel demand models
and economic evaluation measures of user beneﬁt. Environment
and Planning A, 9(3):285–344.
Yai, T. (1997). Multinomial probit with structured covariance for
route choice behavior. Transportation Research Part B:
Methodological, 31(3):195–207.
Yue, Y. and Guestrin, C. (2011). Linear submodular bandits and
their application to diversiﬁed retrieval. In Shawe-taylor, J.,
Zemel, R., Bartlett, P., Pereira, F., and Weinberger, K., editors,
Advances in Neural Information Processing Systems 24, pages
2483–2491.

References
References V
Zhang, S. and Yu, A. J. (2013). Forgetful Bayes and myopic
planning: Human learning and decision-making in a bandit
setting. In Burges, C., Bottou, L., Welling, M., Ghahramani, Z.,
and Weinberger, K., editors, Advances in Neural Information
Processing Systems 26, pages 2607–2615. Curran Associates,
Inc.
Ziegler, C.-N., McNee, S. M., Konstan, J. A., and Lausen, G.
(2005). Improving recommendation lists through topic
diversiﬁcation. In Proceedings of the 14th international
conference on World Wide Web (WWW 2005), pages 22–32.
ACM.

Predicting Preference Reversals via Gaussian Process Uncertainty Aversion

Recomendados

Recomendados

Mais conteúdo relacionado

Mais procurados

Mais procurados (20)

Semelhante a Predicting Preference Reversals via Gaussian Process Uncertainty Aversion

Semelhante a Predicting Preference Reversals via Gaussian Process Uncertainty Aversion (20)

Último

Último (20)

Predicting Preference Reversals via Gaussian Process Uncertainty Aversion