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

  • 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 eαU(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 ueαU(x,θ)/eαT, where u~U[0;1].

References

  1. 1.

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

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Correspondence to Pedro A Ortega or Daniel A Braun.

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The online version of the original article can be found at 10.1186/2194-3206-2-2

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Ortega, P.A., Braun, D.A. Erratum to: Generalized Thompson sampling for sequential decision-making and causal inference. Complex Adapt Syst Model 2, 4 (2014). https://doi.org/10.1186/s40294-014-0004-x

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