Original: Annealed Importance Sampling with q-Paths
Revised: Annealed Importance Sampling with q-Paths

Original: Abstract
Revised: Abstract

Original: Annealed Importance Sampling ( AIS ) [23, 16] is the gold standard for estimatingpartition functions or marginal likelihoods, corresponding to importance samplingover a path of distributions between a tractable base and an unnormalized target.
Revised: Annealed Importance Sampling ( AIS ) [27, 18] is the gold standard for estimating partition functions or marginal likelihoods, corresponding to importance sampling over a path of distributions between a tractable base and an unnormalized target.

Original: While AIS yields an unbiased estimator for any path, existing literature has beenlimited to the geometric mixture or moment-averaged paths associated with theexponential family and KL divergence [12].
Revised: While AIS yields an unbiased estimator for any path, existing literature has been limited to the geometric mixture or moment-averaged paths associated with the exponential family and KL divergence [13].

Original: We explore AIS using q -paths, whichinclude the geometric path as a special case and are related to the homogeneouspower mean, deformed exponential family, and α -divergence [3]
Revised: We explore AIS using q -paths, which include the geometric path as a special case and are related to the homogeneous power mean, deformed exponential family, and α -divergence [3]

Original:  Introduction
Revised:  Introduction

Original: AIS [23, 16] is a method for estimating intractable normalization constants, which considers apath of intermediate distributions π t ( z ) between a tractable base distribution π 0 ( z ) and unnormalized target ˜ π T ( z ) .
Revised: AIS [27, 18] is a method for estimating intractable normalization constants, which considers a path of intermediate distributions π t ( z ) between a tractable base distribution π 0 ( z ) and unnormalized target ˜ π T ( z ) .

Original: In particular, AIS samples from a sequence of MCMC transition operators T t ( z t | z t − 1 ) which leave each π β t ( z ) =
Revised: In particular, AIS samples from a sequence of MCMC transition operators T t ( z t | z t − 1 ) which leave each π β t ( z ) =

Original: ˜
Revised: ˜

Original: π β t ( z ) /Z
Revised: π β t ( z ) /Z

Original: t invariant to estimate the ratio Z T /Z 0 .
Revised: t invariant to estimate the ratio Z T /Z 0 .

Original: Algorithm for i = 1 to N do z 0 ∼ π 0 ( z ) w ( i ) ← Zfor t = 1 to T do w ( i ) t ← w ( i ) t˜ π t
Revised: for i = 1 to N do z 0 ∼ π 0 ( z ) w ( i ) ← Zfor t = 1 to T do w ( i ) t ← w ( i ) t

Original:  ( z ( i ) t − 1 ) ˜
Revised: ˜ π t ( z ( i ) t − 1 ) ˜

Original: π t − 1 ( z ( i ) t − 1 )
Revised: π t − 1 ( z ( i ) t − 1 )

Original: z ( i ) t ∼ T t ( z t | z ( i ) t − 1 ) end end return Z T /Z[-
-]
Revised: z ( i ) t ∼ T t ( z t | z ( i ) t − 1 ) end end return Z T /Z[-
-]

Original: ≈ 1 N (cid:80) N w ( i ) T
Revised: [Equation]

Original: As shown in Algorithm 1, we can accumulate the importanceweights w ( i ) T
Revised: As shown in Algorithm 1, we can accumulate the importance weights w ( i ) T

Original: = (cid:81)
Revised: [Equation]

Original: Tt =1 ˜ π t ( z t − 1 ) / ˜ π t − 1 ( z t − 1 ) along the path.
Revised: ) along the path.

Original: Taking the expectation of w ( i ) T over sampling chains yieldsan unbiased estimate of Z T /Z 

Revised: Taking the expectation of w ( i ) T over sampling chains yields an unbiased estimate of Z T /Z

Original: Similarly, Bidirectional
Revised: Similarly, Bidirectional

Original: Monte Carlo ( BDMC ) [13, 14] provides lower and upper boundson the log partition function ratio log Z T /Z 

Revised: Monte Carlo ( BDMC ) [14, 15] provides lower and upper bounds on the log partition function ratio log Z T /Z

Original: using AIS initialized with the base or target distribution, respectively.
Revised: using AIS initialized with the base or target distribution, respectively.

Original: AIS often uses a geometric mixture path with schedule { β t } Tt =0to anneal between π 0 and˜ π T 
Revised: AIS often uses a geometric mixture path with schedule { β t } Tt = to anneal between π 0 and π T , [Equation]

Original: [Equation]
Revised: [Equation]

Original: where π β ( z ) =
Revised: where π β ( z ) =

Original: ˜
Revised: ˜

Original: π β ( z ) /Z
Revised: π β ( z ) /Z

Original: β and Z β = (cid:82) ˜ π
Revised: β and Z β = (cid:82) ˜ π

Original: 0 ( z ) 1 − β ˜ π T ( z )
Revised: 0 ( z ) 1 − β ˜ π T ( z )

Original: β dz .
Revised: β dz .

Original: Alternative paths have been discussed in [12, 11, 9], but maynot have closed form expressions for intermediate distributions.
Revised: Alternative paths have been discussed in [13, 12, 10], but may not have closed form expressions for intermediate distributions.

Original: In this work, we propose to generalizethe geometric mixture path (1) using the power mean [17, 15, 10], or q -path,
Revised: In this work, we propose to generalize the geometric mixture path (1) using the power mean [19, 17, 11], or q -path,

Original: [Equation]
Revised: [Equation]

Original: As q → 1 , we recover the geometric mixture path as a special case.
Revised: As q → 1 , we recover the geometric mixture path as a special case.

Original: The power mean also contains asa special case the q-logarithm used in non-extensive thermodynamics [27, 22, 28], which allows usto frame Eq.
Revised: The power mean also contains as a special case the q-logarithm used in non-extensive thermodynamics [31, 26, 32], which allows us to frame Eq.

Original: (2) in terms of the the q -exponential family [6].
Revised: (2) in terms of the the q -exponential family [6].

Original: Further, we draw connections with theα -geometry of Amari [2, 3, 4] by showing that Eq.
Revised: Further, we draw connections with the α -integration of Amari [3, 4] by showing that Eq.

Original: (2) minimizes a mixture of α -divergences as in[3].
Revised: (2) minimizes a mixture of α -divergences as in [3].

Original: We describe properties of the geometric and q -paths in Section 2 and Section 3, respectively.
Revised: We describe properties of the geometric and q -paths in Section 2 and Section 3, respectively.

Original:  Interpretations of the Geometric Path
Revised:  Interpretations of the Geometric Path

Original: We give three complementary interpretations of the geometric path deﬁned in Eq.
Revised: We give three complementary interpretations of the geometric path deﬁned in Eq.

Original: (1), which willhave generalized analogues in Section 3.
Revised: (1), which will have generalized analogues in Section 3.

Original: Log Mixture Simply taking the logarithm of both sides of the geometric mixture (1) shows that ˜ π βcan be obtained by taking the log-mixture of ˜ π
Revised: Log Mixture Simply taking the logarithm of both sides of the geometric mixture (1) shows that ˜ π β can be obtained by taking the log-mixture of ˜ π

Original: 0 and ˜ π T with mixing parameter β ,
Revised: 0 and ˜ π T with mixing parameter β ,

Original: [Equation]
Revised: [Equation]

Original: where we may also choose to subtract a constant log Z β to enforce normalization.
Revised: where we may also choose to subtract a constant log Z β to enforce normalization.

Original: Exponential Family Distributions along the geometric path may also be viewed as coming froman exponential family [8].
Revised: Exponential Family Distributions along the geometric path may also be viewed as coming from an exponential family [9, 16].

Original: In particular, we use a base measure of ˜ π 0 ( z ) and sufﬁcient statisticsφ ( z ) = log ˜ π T / ˜ π 0 to rewrite Eq.
Revised: In particular, we use a base measure of ˜ π 0 ( z ) and sufﬁcient statistics φ ( z ) = log ˜ π T / ˜ π 0 to rewrite Eq.

Original: (1) as
Revised: (1) as

Original: [Equation]
Revised: [Equation]

Original: where the mixing parameter β appears as the natural parameter of the exponential family andψ ( β ) :
Revised: where the mixing parameter β appears as the natural parameter of the exponential family and ψ ( β ) :

Original: = log Z β .
Revised: = log Z β .

Original: The log-partition function or free energy ψ ( β ) is convex in β and induces [4, 25, 8] a
Revised: The log-partition function or free energy ψ ( β ) is convex in β and induces [4, 29, 9] a

Original: Bregman divergence over the natural parameter space equivalent to the KL divergence D KL [ π β (cid:48)
Revised: Bregman divergence over the natural parameter space equivalent to the KL divergence D KL [ π β (cid:48)

Original: ||
Revised: ||

Original: π β ] .
Revised: π β ] .

Original: Variational Representation Grosse et al.
Revised: Variational Representation Grosse et al.

Original: [12] also observe that each π β ( z ) can be viewed asminimizing a weighted sum of KL divergences to the (normalized) base and target distributions
Revised: [13] also observe that each π β ( z ) can be viewed as minimizing a weighted sum of KL divergences to the (normalized) base and target distributions

Original: [Equation]
Revised: [Equation]

Original: While the optimization in Eq.
Revised: While the optimization in Eq.

Original: (5) is over arbitrary r ( z ) , the optimal solution is the geometric mixturewith mixing parameter β , which is a member of the exponential family in Eq.
Revised: (5) is over arbitrary r ( z ) , the optimal solution is the geometric mixture with mixing parameter β , which is a member of the exponential family in Eq.

Original:  Interpretations of the q -Path
Revised:  Interpretations of the q -Path

Original: To anneal between ˜ π 0 and ˜ π T  we consider the power mean, with order parameter q, in place of thegeometric average in Eq.
Revised: To anneal between ˜ π 0 and ˜ π T , we consider the power mean with order parameter q in place of the geometric average in Eq.

Original: Analogously to Sec.
Revised: Analogously to Sec.

Original: 2 above, our generalization is associated with thedeformed log mixture, q -exponential family, and a variational representation using the α -divergence.
Revised: 2 above, our generalization is associated with the deformed log mixture, q -exponential family, and a variational representation using the α -divergence.

Original: Power Means Kolmogorov [17] proposed a generalized notion of the mean using any monotonicfunction h ( u ) , with h ( u ) =
Revised: Power Means Kolmogorov [19] proposed a generalized notion of the mean using any monotonic function h ( u ) , with h ( u ) =

Original: u corresponding to the arithmetic mean and
Revised: u corresponding to the arithmetic mean and

Original: [Equation]
Revised: [Equation]

Original: [Equation] where µ h outputs a scalar given a normalized measure { w i } over a set of elements { u i } [10].
Revised: where µ h outputs a scalar given a normalized measure { w i } over a set of elements { u i } [11].

Original: Thegeometric and arithmetic means are homogeneous , meaning they have the linear scale-free propertyµ h ( { w i , c · u i } ) =
Revised: The geometric and arithmetic means are homogeneous , meaning they have the linear scale-free property µ h ( { w i , c · u i } ) =

Original: c · µ h ( { w i , u i } ) .
Revised: c · µ h ( { w i , u i } ) .

Original: In order for the power mean to be homogenous, Hardy et al.[15] (pg. 68 or [3]) show that h ( u ) must be of the form
Revised: In order for the power mean to be homogenous, Hardy et al. [17] (pg. 68 or [3]) show that h ( u ) must be of the form

Original: [Equation]
Revised: [Equation]

Original: which we refer to as the q -power mean.
Revised: which we refer to as the q -power mean.

Original: Notable examples of the power mean include the arithmeticmean at q = 0 , geometric mean as q → 1 , and the min or max operation as q → ±∞ .
Revised: Notable examples of the power mean include the arithmetic mean at q = 0 , geometric mean as q → 1 , and the min or max operation as q → ±∞ .

Original: For q = 1+ α 2 ,h q ( u ) matches the α -embedding of Amari [4].
Revised: For q = 1+ α 2 , h q ( u ) matches the α -representation of Amari [4][5, 7].

Original: In App. A, we show that h q ( u ) yields the same power mean µ h q (6), for any choice of a and b . Usingthe power mean to generalize geometric mean, we propose the q -path of intermediate unnormalizeddensities in AIS as [Equation]
Revised: = 0.

Original: where { w i } = { 1 − β, β } and { u i } = { ˜ π 0 , ˜ π T } .

Original: = 0.= 0.
Revised: = 0.

Original: = 0.11 =
Revised: = 0.11 =

Original: 0.22 = 0.
Revised: 0.22 = 0.

Original: = 0.44 = 0.56 = 0.
Revised: = 0.44 = 0.56 = 0.

Original: = 0.78 = 0.
Revised: = 0.78 = 0.

Original: 10 x = 1.
Revised: 10 x = 1.

Original: (a) q == 0.
Revised: (a) q == 0.

Original: = 0.
Revised: = 0.

Original: = 0.11 =
Revised: = 0.11 =

Original: 0.22 = 0.
Revised: 0.22 = 0.

Original: = 0.44 = 0.56 = 0.
Revised: = 0.44 = 0.56 = 0.

Original: = 0.78 = 0.
Revised: = 0.78 = 0.

Original: 10 x = 1.
Revised: 10 x = 1.

Original: (b) q = 0 .= 0.
Revised: (b) q = 0 .= 0.

Original: =0.
Revised: =0.

Original: = 0.11 =
Revised: = 0.11 =

Original: 0.22 = 0.
Revised: 0.22 = 0.

Original: = 0.44 = 0.56 = 0.
Revised: = 0.44 = 0.56 = 0.

Original: = 0.78 = 0.
Revised: = 0.78 = 0.

Original: 10 x = 1.
Revised: 10 x = 1.

Original: (c) q = 0 .= 0.
Revised: (c) q = 0 .= 0.

Original: =1.
Revised: =1.

Original: = 0.11 =
Revised: = 0.11 =

Original: 0.22 = 0.
Revised: 0.22 = 0.

Original: = 0.44 = 0.56 = 0.
Revised: = 0.44 = 0.56 = 0.

Original: = 0.78 = 0.
Revised: = 0.78 = 0.

Original: 10 x = 1.
Revised: 10 x = 1.

Original: (d) q =
Revised: (d) q =

Original: Deformed Log Mixture
Revised: Using the power mean to generalize geometric mean, we propose the q -path of intermediate unnormalized densities ˜ π ( q ) β ( z ) for AIS . In App.

Revised: A, we show that for any choice of a and b , h q ( u ) yields the same power mean

Revised: [Equation]

Revised: [Equation]

Revised: where we have chosen { w i }

Original:  The deformed, or q -logarithm [22], which plays a crucial role in nonextensive thermodynamics [27, 28], is a particular special case of h q ( u ) in Eq.
Revised: = { 1 − β, β } and { u i } = { ˜ π 0 , ˜ π T } in (6). Deformed Log Mixture The deformed, or q -logarithm [26], which plays a crucial role in nonextensive thermodynamics [31, 32], is a particular special case of h q ( u ) in Eq.

Original: (16), with
Revised: (16), with

Original: [Equation]
Revised: [Equation]

Original:  where we have also deﬁned the q -exponential with exp q ( u ) =
Revised: q [Equation] where we have also deﬁned the q -exponential with exp q ( u ) =

Original: ln 
Revised: ln − 1 q ( u ) and [ x ] + = max { 0 , x } ensuring g ( u ) is non negative. Note that lim q → 1 ln q ( u ) = log u and lim q → 1 exp q ( u )

Original: [Equation]
Revised: = exp u .

Original: Applying h q ( u )
Revised: Applying h q ( u )

Original: =
Revised: =

Original: ln q ( u ) to both sides of Eq.
Revised: ln q ( u ) to both sides of Eq.

Original: (6) or (8), we can write ˜ π ( q ) β as a deformed log-mixture
Revised: (6) or (8), we can write ˜ π ( q ) β as a deformed log-mixture

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original:  with mixing weight β .
Revised: (10) with mixing weight β .

Original: We also provide detailed derivations for Eq.
Revised: We also provide detailed derivations for Eq.

Original: (10) in App.
Revised: (10) in App.

Original: B.1.
Revised: B.1.

Original: q -Exponential Family
Revised: q -Exponential Family

Original: The q -exponential in Eq.
Revised: The q -exponential in Eq.

Original: (9) may be used to deﬁne a q -exponential familyof distributions [6, 22].
Revised: (9) may be used to deﬁne a q -exponential family of distributions [6, 26].

Original: Using θ as the natural parameter,
Revised: Using θ as the natural parameter,

Original: [Equation]
Revised: [Equation]

Original:  which recovers the standard exponential family at q → 1 .
Revised: (11) which recovers the standard exponential family at q → 1 .

Original: In App.
Revised: In App.

Original: B.2 we show that the q -mixture˜ π ( α ) β in Eq.
Revised: B.2 we show that the q -mixture ˜ π ( α ) β in Eq.

Original: (8) can be rewritten in terms of the q -exponential family
Revised: (8) can be rewritten in terms of the q -exponential family

Original: [Equation]
Revised: [Equation]

Original: q

Original: [Equation]
Revised: [Equation]

Original: β [Equation] [Equation] with  φ q ( z ) =
Revised: with sufﬁcient statistic φ q ( z ) =

Original: ln q ˜ π T / ˜ π 0 and natural parameter β .
Revised: ln q ˜ π T / ˜ π 0 and natural parameter β .

Original: The expression in (12) may be used to directlyestimate the normalization constant Z ( q ) β .
Revised: The expression in (12) might be used to directly estimate the normalization constant Z ( q ) β via Monte Carlo approximation.

Original:  However, Matsuzoe et al.
Revised: As for the standard exponential family, the q -free energy ψ q ( θ ) in Eq. (11) is convex in θ and can be used to construct a Bregman divergence over normalized q -exponential family distributions [6]. However, to normalize (12) using the q -free energy, a non-linear mapping θ ( β ) between parameterizations is required.

Original: [19] [26] highlight the delicateissue of normalization in the q -exponential family, which we discuss in detail in App. B.3. 
Revised: This delicate issue of normalization in the q -exponential family has been noted in [22, 30, 26], and we provide more detailed discussion in App. B.3. [Figure]

Original: Inparticular, log Z β = ψ ( β ) does not hold for ln q Z ( q ) β (cid:54) = ψ q ( β ) with q (cid:54) = 1 . As in the standard exponential family, the q -free energy ψ q ( θ ) , which normalizes Eq.
Revised: [Figure]

Original: (11), is convexin θ and can be used to construct a Bregman divergence for the q -exponential family [6].
Revised: [Table]

Original:  Variational Representation using the α -Divergence We ﬁrst recall the deﬁnition of the αdivergence over unnormalized measures ˜ q ( z ) and ˜ p
Revised: [Figure] Variational Representation using the α -Divergence Since we do not have access to normalization constants in the AIS setting we focus on the α -divergence [2, 4] over unnormalized measures ˜ q ( z ) and ˜ p

Original: ( z ) [2, 4]
Revised: ( z ) .

Revised: We ﬁrst recall the deﬁnition,

Original: [Equation]  [Equation]
Revised: [Equation] [Table] [Equation]

Original:  which is an f -divergence [1] for the generator f ( u )
Revised: [Table] [Equation] which is an f -divergence [1] for the generator f ( u )

Original: =
Revised: =

Original: [Equation]
Revised: [Figure]

Revised: − α 2 (cid:0) 1 − α 2 + 1+ α 2 u − u 1+ α 2 (cid:1) [5, 4].

Original: Note thatlim
Revised: Note that lim α

Original: [Equation]
Revised: → 1 D α [˜ q ( z ) :

Original: [Equation] ˜ 
Revised: ˜ p ( z )]

Original: [Equation] ˜ [Equation]
Revised: = D KL [˜ p ( z ) :

Original:  z )] and lim [Equation] D [Equation]
Revised: ˜ q ( z )] and lim α →− 1 D α

Original: [˜ [Equation] ( z [Equation]
Revised: [˜ q ( z ) :

Original: ˜ p [Equation]
Revised: ˜ p ( z )]

Original: z [Equation] [˜ [Equation]
Revised: = D KL [˜ q ( z ) :

Original: ˜ p [Equation] z 
Revised: ˜ p ( z )] .

Original: [Equation]
Revised: In App.

Original: For q = 1+ α 2 as in [4] (Ch. 4), we show in App. C that ˜ π ( q ) β minimizes the expected α -divergence [Equation]
Revised: C, we follow similar derivations as Amari [3] to show that, for q = 1+ α 2 ([4] Ch. 4), the q -path density ˜ π ( q ) β minimizes the expected α -divergence to the endpoints

Original: [Equation]
Revised: [Equation]

Original: where the optimization is over arbitrary ˜ r ( z ) .
Revised: where the optimization is over arbitrary ˜ r ( z ) .

Original: While Amari [3] derive a similar result for normalizedπ 0 and π T , we focus on the unnormalized case since the goal of AIS is to estimate intractable Z T /Z [Figure] [Figure]
Revised: This variational representation generalizes Eq. (5), since the KL divergence is recovered (with the order of the arguments reversed) as α → 1 or q → 1 .

Original: Moment-Matching Procedures
Revised: Moment-Matching Procedures

Original: At q = 0 , the solution to the optimization (13) correponds to the arithmetic mean or mixture distribution ˜ π ) t ( z ) =
Revised: At q = 0 , the solution to the optimization (13) correponds to the arithmetic mean, or mixture distribution ˜ π (0) t ( z ) =

Original: (1 − β ) ˜ π
Revised: (1 − β ) ˜ π

Original:  + β ˜ π 1 .
Revised: 0 + β ˜ π 1 .

Original: While the ‘moment-averaged’
Revised: While the ‘moment-averaged’

Original: AIS path [12] appears related to the q = 0 case, we clarify in App.
Revised: AIS path [13] appears related to the q = 0 case, we clarify in App.

Original: C.1 that Grosse et al.
Revised: C.1 that Grosse et al.

Original: [12] restrictto optimization within an exponential family of distributions.
Revised: [13] restrict to optimization within an exponential family of distributions.

Original: Generalizing this approach to theα -divergence,  Minka [21] (Sec. 3.1-2) and Bui [9] derive the moment-matching condition
Revised: Generalizing this approach to the α -divergence, Bui [10] follows Minka [24] (Sec. 3.1-2) to derive the moment-matching condition

Original: [Equation]
Revised: [Equation]

Original: where ˜ r ( z ) comes from an exponential family with sufﬁcient statistics φ ( z ) .
Revised: where ˜ r ( z ) comes from an exponential family with sufﬁcient statistics φ ( z ) .

Original: However, we note that our q -path is more general than these approaches, since the optimization in
Revised: However, we note that our q -path is more general than these approaches, since the optimization in

Original: Eq.
Revised: Eq.

Original: (13) is over all unnormalized distributions.
Revised: (13) is over all unnormalized distributions.

Original: Unlike the moment matching conditions above, ourclosed form expression for ˜ π ( q ) β can be directly used as an energy function for MCMC sampling.
Revised: Unlike the moment matching conditions above, our closed form expression for ˜ π ( q ) β can be directly used as an energy function for MCMC sampling.

Original:  Experiments
Revised:  Experiments

Original: We consider q -paths between π 0 = N ( − 4 , 3) and π T = N (4 , 1) to estimate Z T /Z 

Revised: We consider q -paths between π 0 = N ( − 4 , 3) and π T = N (4 , 1) to estimate Z T /Z

Original: = 1 , and useparallel runs of Hamiltonian Monte Carlo ( HMC ) [24] to obtain accurate, independent samples from˜ π ( q ) t ( z ) linearly spaced between β 0 = 0 and β T = 1 .
Revised: = 1 , and use parallel runs of Hamiltonian Monte Carlo ( HMC ) [28] to obtain accurate, independent samples from ˜ π ( q ) t ( z ) linearly spaced between β 0 = 0 and β T = 1 .

Original: For all experiments, we use 10k samples fromeach intermediate distribution and average results across 20 seeds.
Revised: For all experiments, we use 10k samples from each intermediate distribution and average results across 20 seeds.

Original: In Fig.
Revised: In Fig.

Original: 2, we report BDMC upper and lower bound estimates of log Z T /Z 1 for various q and T .
Revised: 2, we report BDMC upper and lower bound estimates of log Z T /Z 1 for various q and T .

Original: Weobserve that the choice of q can impact performance, with q = 0 .
Revised: We observe that the choice of q can impact performance, with q = 0 .

Original: 9 obtaining tighter estimates at small
Revised: 9 obtaining tighter estimates at small

Original: T and q = 0 .
Revised: T and q = 0 .

Original: 5 converging more quickly as T increases.
Revised: 5 converging more quickly as T increases.

Original: Both outperform the baseline geometric pathat q = 1 .
Revised: Both outperform the baseline geometric path at q = 1 .

Original: In Table 1, we estimate Z T /Z 

Revised: In Table 1, we estimate Z T /Z

Original: using AIS for T = 100 , and observe that our the q = 0 .
Revised: using AIS for T = 100 , and observe that our the q = 0 .path can achieve a lower error than the geometric path. Finally, in App.

Revised: E, we provide additional analysis for annealing between two Student- t distributions.

Original: 9path can achieve a lower error than the geometric path.
Revised: The Student- t family can be shown to correspond to a q -exponential family [21], with the same sufﬁcient statistics as a Gaussian, and a degrees of freedom parameter ν that induces heavier tails and sets the value of q . As q → 1 or ν → ∞ , the standard Gaussian is recovered. In Fig.

Revised: 3-4, we compare annealing between two Student- t distributions in the q = 2 family to the Gaussian case of q = 1 , and observe that the same q -path can induce different qualitative behavior based on properties of the endpoint distributions.

Original:  Conclusion
Revised:  Conclusion

Original: In this work, we propose q -paths to generalize the geometric mixture path commonly used in AIS ,and show that modifying the path can improve AIS and BDMC for a ﬁxed mixing schedule on a toy
Revised: In this work, we propose q -paths to generalize the geometric mixture path commonly used in AIS , and show that modifying the path can improve AIS and BDMC for a ﬁxed mixing schedule on a toy

Original: Gaussian example.
Revised: Gaussian example.

Original: We interpreted our q -paths using the deformed logarithm, q -exponential family,and α -divergences, which may suggest further connections with non-extensive thermodynamics andinformation geometry.
Revised: We interpreted our q -paths using the deformed logarithm, q -exponential family, and α -divergences, which may suggest further connections in non-extensive thermodynamics and information geometry.

Original: Choosing a schedule for a given q -path, understanding how the choice of qdepends on properties of the initial and target distributions, and exploring the use of q -paths in relatedmethods such as the thermodynamic variational objective ( TVO ) [18, 8] remain interesting questionsfor future work.
Revised: Choosing a schedule for a given q -path, understanding how the choice of q depends on properties of the initial and target distributions, and exploring the use of q -paths in related methods such as the thermodynamic variational objective ( TVO ) [20, 9] remain interesting directions for future work.

Original: A Abstract Mean is Invariant to Afﬁne Transformations
Revised: A Abstract Mean is Invariant to Afﬁne Transformations

Original: In this section, we show that h q ( u ) is invariant to afﬁne transformations.
Revised: In this section, we show that h q ( u ) is invariant to afﬁne transformations.

Original: That is, for any choice of aand b ,
Revised: That is, for any choice of a and b ,

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: yields the same expression for the abstract mean µ h α .
Revised: yields the same expression for the abstract mean µ h α .

Original: First, we note the expression for the inverseh − 1 q ( u ) at q (cid:54) =[-
-]
Revised: First, we note the expression for the inverse h − 1 q ( u ) at q (cid:54) =[-
-]

Original: [Equation]
Revised: [Equation]

Original: Recalling that (cid:80) i w i = 1 , the abstract mean then becomes
Revised: Recalling that (cid:80) i w i = 1 , the abstract mean then becomes

Original: [Equation]
Revised: [Equation]

Original: which is independent of both a and b .
Revised: which is independent of both a and b .

Original: B Derivations of the q -PathB.1 Deformed Log Mixture
Revised: B Derivations of the q -Path B.1 Deformed Log Mixture

Original: In this section, we show that the unnormalized ln q mixture
Revised: In this section, we show that the unnormalized ln q mixture

Original: [Equation]
Revised: [Equation]

Original: (21) reduces to the form of the q -path intermediate distribution in (2) and (8).
Revised: (21) reduces to the form of the q -path intermediate distribution in (2) and (8).

Original: Taking exp q of both sides,˜ π ( q ) β
Revised: Taking exp q of both sides, ˜ π ( q ) β

Original: ( z ) =
Revised: ( z ) =

Original: exp q { (1 − β ) ln q ˜ π 

Revised: exp q { (1 − β ) ln q ˜ π

Original: ( z ) + β ln q ˜ π 1 ( z ) } =[Equation] + [Equation]
Revised: ( z ) + β ln q ˜ π 1 ( z ) } =

Original:  − q [Equation] (ln q ˜ π 0 ( z 
Revised: [1 + (1 − q ) (ln q ˜ π 0 ( z ) +

Original: [Equation]
Revised: β

Original: + β (ln [Equation] ˜ 
Revised: (ln q ˜ π 1 ( z ) − ln q ˜ π

Original: [Equation]
Revised: [Equation]

Original: B.2 q -Exponential Family
Revised: B.2 q -Exponential Family

Original: Here, we show that the unnormalized q -path reduces to a form of the q -exponential family
Revised: Here, we show that the unnormalized q -path reduces to a form of the q -exponential family

Original: [Equation]
Revised: [Equation]

Original: Deﬁning φ ( z ) =
Revised: Deﬁning φ ( z ) =

Original: ln q
Revised: [Equation] q

Original: [Equation]
Revised: [Equation]

Original: π 1 (z )
Revised: π 1 (

Original: [Equation]
Revised: [Equation]

Original:  π 0{+
+} ( z ) and introducing a multiplicative normalization factor Z q ( β ) , we arrive at
Revised: ˜ π {+
+} ( z ) and introducing a multiplicative normalization factor Z q ( β ) , we arrive at

Original: [Equation]
Revised: [Equation]

Original: B.3 Normalization in q-Exponential Families
Revised: B.3 Normalization in q-Exponential Families

Original: The q -exponential family can also be written using the q -free energy ψ q ( θ ) for normalization [6, 22],
Revised: The q -exponential family can also be written using the q -free energy ψ q ( θ ) for normalization [6, 26],

Original: [Equation]
Revised: [Equation]

Original: However, since exp q { x + y } = exp q { y } · exp q { x 1+(1 − q ) y } (see [26] or App.
Revised: However, since exp q { x + y } = exp q { y } · exp q { x 1+(1 − q ) y } (see [30] or App.

Original: D below) instead ofexp { x + y } = exp { x } · exp { y } for the standard exponential, we can not easily move between theseways of writing the q -family
Revised: D below) instead of exp { x + y } = exp { x } · exp { y } for the standard exponential, we can not easily move between these ways of writing the q -family

Original: Mirroring the derivations of Naudts [22] pg. 108, we can rewrite (28) using the above identity forexp q { x + y } , as
Revised: Mirroring the derivations of Naudts [26] pg. 108, we can rewrite (28) using the above identity for exp q { x + y } , as

Original: [Equation]
Revised: [Equation]

Original: Our goal is to express π ( q ) θ ( z ) using a normalization constant Z ( q ) β instead of the q -free energy ψ q ( θ ) .
Revised: Our goal is to express π ( q ) θ ( z ) using a normalization constant Z ( q ) β instead of the q -free energy ψ q ( θ ) .

Original: While the exponential family allows us to freely move between ψ ( θ ) and log Z θ , we must adjust thenatural parameters (from θ to β ) in the q -exponential case.
Revised: While the exponential family allows us to freely move between ψ ( θ ) and log Z θ , we must adjust the natural parameters (from θ to β ) in the q -exponential case.

Original: Deﬁning
Revised: Deﬁning

Original: [Equation]
Revised: [Equation]

Original: 32) we can obtain a new parameterization of the q -exponential family, using parameters β and multiplicative normalization constant Z ( q ) β ,
Revised: we can obtain a new parameterization of the q -exponential family, using parameters β and multiplicative normalization constant Z ( q ) β ,

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Revised: See Matsuzoe

Original: C Minimizing α -divergences
Revised: C Minimizing α -divergences

Original: Amari [3] show that the α -mixture π α t minimizes the expected divergence to a single point fornormalized measures.
Revised: Amari [3] show that the α -mixture π α t minimizes the expected divergence to a single point for normalized measures.

Original: We repeat similar derivations but for the case of unnormalized { ˜ p
Revised: We repeat similar derivations but for the case of unnormalized { ˜ p

Original: i } and ˜ r ( z )
Revised: i } and ˜ r ( z )

Original: [Equation]
Revised: [Equation]

Original: 35) where
Revised: 35) where

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: d

Original: [Equation]
Revised: [Equation]

Original: This result is similar to a general result about Bregman divergences in Banerjee et al.
Revised: This result is similar to a general result about Bregman divergences in Banerjee et al.

Original: [7] Prop. 
Revised: [8] Prop. 1.

Original: 1,although D α is not a Bregman divergence over normalized distributions.
Revised: although D α is not a Bregman divergence over normalized distributions.

Original: C.1 Arithmetic Mean ( q = 0 )
Revised: C.1 Arithmetic Mean ( q = 0 )

Original: The moment-averaging path from Grosse et al.
Revised: The moment-averaging path from Grosse et al.

Original: [12] is not a special case of the α -mean path of
Revised: [13] is not a special case of the α -mean path of

Original: Amari [3].
Revised: Amari [3].

Original: While both minimize a convex combination of reverse KL divergences, Grosse et al.
Revised: While both minimize a convex combination of reverse KL divergences, Grosse et al.

Original: [12]minimize within the constrained space of exponential families , while Amari [3] optimizes over allnormalized distributions.
Revised: [13] minimize within the constrained space of exponential families , while Amari [3] optimizes over all normalized distributions.

Original: More formally, consider minimizing the functional
Revised: More formally, consider minimizing the functional

Original: [Equation]
Revised: [Equation]

Original: We will show how Grosse et al.
Revised: We will show how Grosse et al.

Original: [12] and Amari [3] minimize (43).
Revised: [13] and Amari [3] minimize (43).

Original: Solution within Exponential Family Grosse et al.
Revised: Solution within Exponential Family Grosse et al.

Original: [12] constrains r ( z ) 
Revised: [13] constrains r ( z ) = [Equation]

Original: = 1 Z ( θ ) h ( z ) exp( θ T g ( z ))to be a (minimal) exponential family model and minimizes 43 w.r.t r ’s natural parameters θ (cf.
Revised: to be a (minimal) exponential family model and minimizes (43) w.r.t r ’s natural parameters θ (cf.

Original: [12]appendix 2.2):
Revised: [13] appendix 2.2):

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: where the last line follows because π 0 ( z ) and π 1 ( z ) are assumed to be correctly normalized.
Revised: where the last line follows because π 0 ( z ) and π 1 ( z ) are assumed to be correctly normalized.

Original: Then toarrive at the moment averaging path, we compute the partials ∂J
Revised: Then to arrive at the moment averaging path, we compute the partials ∂J

Original: ( θ ) ∂θ i and set to zero:
Revised: ( θ ) ∂θ i and set to zero:

Original: [Equation]
Revised: [Equation]

Original: where we have used the exponential family identity ∂ log Z( θ ) ∂θ i =
Revised: where we have used the exponential family identity ∂ log Z

Original: E r θ [ g i ( z )] in the ﬁrst line.
Revised: [Equation] i [Equation] in the ﬁrst line.

Original: General Solution Instead of optimizing in the space of minimal exponential families, Amari [3]instead adds a Lagrange multiplier to (43) and optimizes r directly (cf.
Revised: General Solution Instead of optimizing in the space of minimal exponential families, Amari [3] instead adds a Lagrange multiplier to (43) and optimizes r directly (cf.

Original: [3] eq.
Revised: [3] eq.

Original: [Equation]
Revised: [Equation]

Original: 50 can be minimized using the Euler-Lagrange equations or using the identity
Revised: Eq. (50) can be minimized using the Euler-Lagrange equations or using the identity

Original: [Equation]
Revised: [Equation]

Original: (51) from [20].
Revised: (51) from [23].

Original: We compute the functional derivative of J (cid:48)
Revised: We compute the functional derivative of J (cid:48)

Original: [ r ] using 51 and solve for q :
Revised: [ r ] using (51) and solve for r :

Original: [Equation]
Revised: [Equation]

Original: Therefore
Revised: Therefore

Original: [Equation]
Revised: [Equation]

Original: [Equation] which corresponds to our q -path at q = 0 , or α = − 1 in Amari [3].
Revised: 55) which corresponds to our q -path at q = 0 , or α = − 1 in Amari [3].

Original: Thus, while both Amari [3]and Grosse et al.
Revised: Thus, while both Amari [3] and Grosse et al.

Original: [12] start with the same objective, they arrive at different optimum because theyoptimize over different spaces.
Revised: [13] start with the same objective, they arrive at different optimum because they optimize over different spaces.

Original: D Sum and Product Identities for q -Exponentials
Revised: D Sum and Product Identities for q -Exponentials

Original: In this section, we prove two lemmas which are useful for manipulation expressions involvingq -exponentials, for example in moving between Eq.
Revised: In this section, we prove two lemmas which are useful for manipulation expressions involving q -exponentials, for example in moving between Eq.

Original: (29) and Eq.
Revised: (29) and Eq.

Original: (30) in either direction.
Revised: (30) in either direction.

Original: Lemma 1.
Revised: Lemma 1.

Original: Sum identity
Revised: Sum identity

Original: [Equation]
Revised: [Equation]

Original: [Equation]

Original: Lemma 2.
Revised: Lemma 2.

Original: Product identity
Revised: Product identity

Original: N 
Revised: N (cid:89) n = exp q ( x n ) = exp q (cid:32) N (cid:88) n = x n · n − 1 (cid:89) i =

Original: [Equation] [Equation]
Revised: (1 + (1 − q ) x i ) (cid:33) (57)

Original: D.1 Proof of Lemma1
Revised: D.1 Proof of Lemma

Original: Proof.
Revised: Proof.

Original: We prove by induction.
Revised: We prove by induction.

Original: The base case ( N = 1 ) is satisﬁed using the convention (cid:80) bi = a x i= 0if
Revised: The base case ( N = 1 ) is satisﬁed using the convention (cid:80) bi = a x i

Original:  b < a so that the denominator on the RHS of Eq.
Revised: =if b < a so that the denominator on the RHS of Eq.

Original: (56) is 1 .
Revised: (56) is 1 .

Original: Assuming Eq.
Revised: Assuming Eq.

Original: (56) holds for N ,exp q
Revised: (56) holds for N , exp q

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: the inductive hypothesis
Revised: hypothesis

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: D.2 Proof of Lemma2
Revised: D.2 Proof of Lemma

Original: Proof.
Revised: Proof.

Original: We prove by induction.
Revised: We prove by induction.

Original: The base case ( N = 1 ) is satisﬁed using the convention (cid:81) bi = a x i= 1if
Revised: The base case ( N = 1 ) is satisﬁed using the convention (cid:81) bi = a x i

Original:  b < a .
Revised: =if b < a .

Original: Assuming Eq.
Revised: Assuming Eq.

Original: (57) holds for N , we will show the N + 1 case.
Revised: (57) holds for N , we will show the N + 1 case.

Original: To simplify notation wedeﬁne y N := (cid:80)
Revised: To simplify notation we deﬁne y N := (cid:80)

Original: Nn =1 x n · (cid:81) n − 1 i
Revised: Nn =1 x n · (cid:81) n − 1 i

Original: =1 (1 + ¯ qx i ) and ¯ q = (1 − q )  .
Revised: =1 (1+ = 1 − q ) x i ) .

Original: Then
Revised: Then,

Original: N +1 (cid:89)
Revised: N +1 (cid:89)

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation] [Equation]
Revised: reindex

Original: [Equation]
Revised: [Equation]

Original: =
Revised: inductive hypothesis

Original: [Equation] 
Revised: [Equation] [Equation]

Original: Next we use the deﬁnition of y N and rearrange= exp q
Revised: Next we use the deﬁnition of y N and rearrange = exp q

Original: [Equation]
Revised: [Equation]

Original:  2 (1 + ¯ q  · x 1 ) + ... + x N ·
Revised: (cid:32) x 1 + x 2 (1 + (1 − q ) · x 1 ) + ... + x N ·

Original: N − 1 (cid:89) i
Revised: N − 1 (cid:89) i

Original: =
Revised: =

Original: (1 + ¯ q  · x i ) (cid:33)(cid:33)(70) =
Revised: (1 + (1 − q ) · x i ) (cid:33)(cid:33) =

Original: exp q (cid:32) N (cid:88) n = x n · n − 1 (cid:89) i =
Revised: exp q (cid:32) N (cid:88) n = x n · n − 1 (cid:89) i =

Original: (1 + ¯ qx i ) (cid:33) .
Revised: (1 + (1 − q ) x i ) (cid:33) .

Original: Then reindexing n → n + 1 establishes
Revised: Then reindexing n → n + 1 establishes

Original: N +1 (cid:89) n = exp q ( x n ) = exp q (cid:32) N +1 (cid:88) n = x n · n − 1 (cid:89) i
Revised: N +1 (cid:89) n = exp q ( x n ) = exp q (cid:32) N +1 (cid:88) n = x n · n − 1 (cid:89) i

Original: =
Revised: =

Original: (1 + (1 − q ) x i ) (cid:33) . 
Revised: (1 + (1 − q ) x i ) (cid:33) . = 0. = 0.

Original: {+= 0.11 = 0.
+}
Revised: {+= 0.11 = 0.
+}

Revised: = 0.33 = 0.

Revised: = 0.56 = 0.67 = 0.78 = 0.

Revised: 10 x = 1.

Revised: (a) q == 0.

Revised: = 0.

Original: {+= 0.11 = 0.
+}
Revised: {+= 0.11 = 0.
+}

Revised: = 0.33 = 0.

Revised: = 0.56 = 0.67 = 0.78 = 0.

Revised: 10 x = 1.0 (b) q = 0 .= 0.

Revised: =0.

Original: {+= 0.11 = 0.
+}
Revised: {+= 0.11 = 0.
+}

Revised: = 0.33 = 0.

Revised: = 0.56 = 0.67 = 0.78 = 0.

Revised: 10 x = 1.

Revised: (c) q = 0 .= 0.

Revised: =1.

Original: {+= 0.11 = 0.
+}
Revised: {+= 0.11 = 0.
+}

Revised: = 0.33 = 0.

Revised: = 0.56 = 0.67 = 0.78 = 0.

Revised: 10 x = 1.

Revised: (d) q =(e) q =

Revised: E Annealing between Student- t Distributions E.1 Student- t Distributions and q -Exponential Family

Revised: The Student- t distribution appears in hypothesis testing with ﬁnite samples, under the assumption that the sample mean follows a Gaussian distribution.

Revised: In particular, the degrees of freedom parameter ν = n − 1 can be shown to correspond to an order of the q -exponential family with ν = (3 − q ) /

Revised: ( q − 1) (in 1-d), so that the choice of q is linked to the amount of data observed.

Revised: We can ﬁrst write the multivariate Student- t density, speciﬁed by a mean vector µ , covariance Σ , and degrees of freedom parameter ν , in d dimensions, as

Revised: [Equation]

Revised: where Z ( ν, Σ ) =

Revised: Γ( ν + d 2 ) / Γ( ν 2 ) · | Σ | − 1 / 2 ν − d 2 π − d 2 .

Revised: Note that ν > 0 , so that we only have positive values raised to the − ( ν + d ) / 2 power, and the density is deﬁned on the real line.

Revised: The power function in (70) is already reminiscent of the q -exponential, while we have ﬁrst and second moment sufﬁcient statistics as in the Gaussian case.

Revised: We can solve for the exponent, or order parameter q , that corresponds to − ( ν + d ) / 2 using − (cid:0)

Revised: ν

Revised: + d 2 (cid:1) =

Revised: 11 − q .

Revised: This results in the relations

Revised: [Equation]

Revised: We can also rewrite the ν − 1 ( x − µ ) T Σ − 1 ( x − µ ) using natural parameters corresponding to { x, x 2 } sufﬁcient statistics as in the Gaussian case (see, e.g. Matsuzoe and Wada [21] Example 4).

Revised: Note that the Student- t distribution has heavier tails than a standard Gaussian, and reduces to a multivariate Gaussian as q → 1 and exp q ( u ) → exp( u ) .

Revised: This corresponds to observing n → ∞ samples, so that the sample mean and variance approach the ground truth [25].

Revised: E.2 Annealing between 1-d Student- t Distributions

Revised: Since the Student- t family generalizes the Gaussian distribution to q (cid:54) = 1 , we can run a similar experiment annealing between two Student- t distributions.

Revised: We set q = 2 , which corresponds to ν =with ν = (3 − q ) / ( q − 1) , and use the same mean and variance as the Gaussian example in Fig. 4, with π 0 ( z ) =

Revised: t ν =1 ( − 4 , 3) and π 1 ( z ) =

Revised: t ν =1 (4 , 1) .

Revised: We visualize the results in Fig.

Revised: For this special case of both endpoint distributions within a parametric family, we can ensure that the q = 2 path stays within the q -exponential family of

Revised: Student- t distributions.

Revised: We make a similar observation for the Gaussian case and q = 1 in Fig.

Revised: Comparing the q = 0 .

Revised: 5 and q = 0 .

Revised: 9 Gaussian path with the q = 1 . 0 and q = 1 . 5 path, we observe that mixing behavior appears to depend on the relation between the q -path parameter and the order of the q -exponential family of the endpoints.

Revised: As q → ∞ , the power mean (6) approaches the min operation as 1 − q → −∞ .

Revised: In the Gaussian case, we see that, even at q = 2 , intermediate densities for all β appear to concentrate in regions of low density under both π 0 and π T .

Revised: However, for the heavier-tailed Student- t distributions, we must raise the q -path parameter signiﬁcantly to observe similar behavior.