Open Access

Erratum to: Generalized Thompson sampling for sequential decision-making and causal inference

Complex Adaptive Systems Modeling20142:4

DOI: 10.1186/s40294-014-0004-x

Received: 24 June 2014

Accepted: 7 August 2014

Published: 1 October 2014

The original article was published in Complex Adaptive Systems Modeling 2014 2:2

Abstract

No abstract.

Decisions in the presence of latent variables

We correct errors in equations (14), (15) and (19) of the main text.

Equations (14) and (15)

Nature’s probability of flipping either coin does not actually depend on the agent’s prediction, so we can replace the conditional probabilities p0(θ|x) by p0(θ). We have then an inner variational problem:
arg max p ~ ( θ | x ) Σ θ p ~ ( θ | x ) - 1 β log p ~ ( θ | x ) p 0 ( θ ) + U ( x , θ )
(14)
with the solution
p ( θ | x ) = 1 Z β ( x ) p 0 ( θ ) exp β U ( x , θ )
(15)

and the normalization constant Z β ( x ) = Σ θ p 0 ( θ ) exp β U ( x , θ ) and an outer variational problem as described by equation (16) in the main text. Note that deliberation renders the two variables x and θ dependent.

Equation (19)

In the case of α = β and uniform prior p 0 ( x ) = U ( x ) , equation (17) reduces to
p ( x ) = Σ θ p 0 ( θ ) e α U ( x , θ ) Z α ,
(19)
where Z α = Σ x Σ θ p 0 ( θ ) e α U ( x , θ ) . Note that eα U(x,θ)/Z α is in general not a conditional distribution. However, equation (19) can be equivalently rewritten as
p ( x ) = Σ θ p 0 ( θ ) Σ x ' e α U ( x ' , θ ) Z α e α U ( x , θ ) Σ x , e α U ( x ' , θ ) = Σ θ p ( θ ) p ( x | θ ) ,

where we have expanded the fraction by Σ x ' e α U ( x ' , θ ) .

This last equality can also be obtained by stating the same variational problem in reverse causal order of x and θ, which is the natural statement of the Thompson sampling problem. The nested variational problem then becomes
arg max p ~ ( x , θ ) Σ θ p ~ ( θ ) - 1 β log p ~ ( θ ) p 0 ( θ ) + Σ x p ~ ( x | θ ) U ( x , θ ) - 1 α log p ~ ( x | θ ) p 0 ( x )
with the solutions
p ( x | θ ) = p 0 ( x ) e α U ( x , θ ) Σ x ' p 0 ( x θ ) e α U ( x ' , θ )
(i)
and
p ( θ ) = 1 Z β α p 0 ( θ ) exp β α log Σ x p 0 ( x ) e α U ( x , θ )
(ii)

with normalization constant Z β α = Σ θ p 0 ( θ ) exp β / α log Σ x p 0 ( x ) e α U ( x , θ ) . In the limit α and β → 0, the Thompson sampling agent is determined by the solutions p ( x | θ ) = δ ( x - arg max x ' U ( x ' , θ ) ) and p(θ)=p0(θ). Sampling an action from p ( x ) = Σ θ p ( θ ) p ( x | θ ) is much cheaper than sampling an action from equation (18) because of the reversed causal order in θ and x, which implies that β/α→ 0 in equation (ii) instead of β/α as in equation (17).

In the case of α=β the solutions for the two different causal orders of x and θ are equivalent. Assuming again a uniform prior p 0 ( x ) = U ( x ) , we can compute the Thompson sampling agent from equation (i) and equation (ii) for α=β to be
p ( x ) = Σ θ p ( θ ) p ( x | θ ) = Σ θ p 0 ( θ ) Σ x ' e α U ( x ' , θ ) Σ x ' Σ θ ' p 0 ( θ ' ) e α U ( x ' , θ ' ) e α U ( x , θ ) Σ x θ e α U ( x ' , θ ) ,

which is exactly equivalent to p(x) in equation (19). To sample from equation (19), we draw θ~p0(θ) and accept x ~ p 0 ( x ) = U ( x ) if ueα U(x,θ)/e α T , where u ~ U [ 0 ; 1 ] .

Notes

Authors’ Affiliations

(1)
GRASP Laboratory, Electrical and Systems Engineering Department, University of Pennsylvania
(2)
Max Planck Institute for Biological Cybernetics and Max Planck Institute for Intelligent Systems

References

  1. Ortega, PA, Braun DA: Generalized Thompson sampling for sequential decision-making and causal inference. Complex Adaptive Systems Modeling 2014, 2: 2.View ArticleGoogle Scholar

Copyright

© Ortega and Braun; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.