Original: Estimating Social Influence from Observational Data
Revised: Estimating Social Influence from Observational Data

Original:  Bernhard Sch¨olkopf Caroline Uhler  Kun Zhang
Revised: : Bernhard Sch¨olkopf, Caroline Uhler and Kun Zhang

Original: Abstract
Revised: Abstract

Original: We consider the problem of estimating social influence, the effect that a person’s behavior has on the future behavior of their peers.
Revised: We consider the problem of estimating social influence, the effect that a person’s behavior has on the future behavior of their peers.

Original: The key challenge is that shared behavior between friends could be equally explained by influence or by two other confounding factors: 1) latent traits that caused people to both become friends and engage in the behavior, and 2) latent preferences for the behavior.
Revised: The key challenge is that shared behavior between friends could be equally explained by influence or by two other confounding factors: 1) latent traits that caused people to both become friends and engage in the behavior, and 2) latent preferences for the behavior.

Original: This paper addresses the challenges of estimating social influence with three contributions.
Revised: This paper addresses the challenges of estimating social influence with three contributions.

Original: First, we formalize social influence as a causal effect, one which requires inferences about hypothetical interventions.
Revised: First, we formalize social influence as a causal effect, one which requires inferences about hypothetical interventions.

Original: Second, we develop Poisson Influence Factorization (PIF), a method for estimating social influence from observational data.
Revised: Second, we develop Poisson Influence Factorization (PIF), a method for estimating social influence from observational data.

Original: PIF fits probabilistic factor models to networks and behavior data to infer variables that serve as substitutes for the confounding latent traits.
Revised: PIF fits probabilistic factor models to networks and behavior data to infer variables that serve as substitutes for the confounding latent traits.

Original: Third, we develop assumptions under which PIF recovers estimates of social influence.
Revised: Third, we develop assumptions under which PIF recovers estimates of social influence.

Original: We empirically study PIF with semi-synthetic and real data from Last.fm, and conduct a sensitivity analysis.
Revised: We empirically study PIF with semi-synthetic and real data from Last.fm, and conduct a sensitivity analysis.

Original: We find that PIF estimates social influence most accurately compared to related methods and remains robust under some violations of its assumptions.
Revised: We find that PIF estimates social influence most accurately compared to related methods and remains robust under some violations of its assumptions.

Original: 1. Introduction
Revised: 1. Introduction

Original: This paper is about analyzing social network data to estimate social influence, the effect that its members have on each other.
Revised: This paper is about analyzing social network data to estimate social influence, the effect that its members have on each other.

Original: How does the past behavior of a person in a network influence the current behavior of their peers?
Revised: How does the past behavior of a person in a network influence the current behavior of their peers?

Original: Researchers across many fields have studied questions that involve social influence.
Revised: Researchers across many fields have studied questions that involve social influence.

Original: For example, Bond et al.
Revised: For example, Bond et al.

Original: (2012) study whether Facebook users influence their friends to vote in elections; Christakis and Fowler (2007) ask whether a person’s obesity status affects those of their family and friends; Bakshy et al.
Revised: (2012) study whether Facebook users influence their friends to vote in elections; Christakis and Fowler (2007) ask whether a person’s obesity status affects those of their family and friends; Bakshy et al.

Original: (2012) study whether social influence increases the effectiveness of an ad campaign.
Revised: (2012) study whether social influence increases the effectiveness of an ad campaign.

Original: We develop Poisson Influence Factorization (PIF), a new method for estimating social influence.
Revised: We develop Poisson Influence Factorization (PIF), a new method for estimating social influence.

Original: PIF uses observational data from a social network, the past behavior of its members, and the current behavior of its members.
Revised: PIF uses observational data from a social network, the past behavior of its members, and the current behavior of its members.

Original: PIF can be applied to study social influence in many settings including
Revised: PIF can be applied to study social influence in many settings including

Original: Facebook users sharing articles, Twitter users using hashtags, or e-commerce site users purchasing products.
Revised: Facebook users sharing articles, Twitter users using hashtags, or e-commerce site users purchasing products.

Original: To explain ideas in the paper, we will use the example of people purchasing items.
Revised: To explain ideas in the paper, we will use the example of people purchasing items.

Original: This paper makes three main contributions.
Revised: This paper makes three main contributions.

Original: The first is to frame the problem of estimating social influence as a causal inference, a question about a hypothetical intervention upon variables in a system.
Revised: The first is to frame the problem of estimating social influence as a causal inference, a question about a hypothetical intervention upon variables in a system.

Original: Informally, we define the social influence of a person on their peer by asking: if we could have “made” the person buy an item yesterday, would their peer purchase the item today?
Revised: Informally, we define the social influence of a person on their peer by asking: if we could have “made” the person buy an item yesterday, would their peer purchase the item today?

Original: Intuitively, for a person who has social influence, if we hypothetically intervene to make them buy an item yesterday, we expect that their peer is more likely to buy the item today.
Revised: Intuitively, for a person who has social influence, if we hypothetically intervene to make them buy an item yesterday, we expect that their peer is more likely to buy the item today.

Original: The second contribution is to introduce the assumptions from which we can estimate social influence with observational data.
Revised: The second contribution is to introduce the assumptions from which we can estimate social influence with observational data.

Original: The challenge with observational data is that we cannot intervene on people’s past purchases and observe the effect on their peers’ current purchases.
Revised: The challenge with observational data is that we cannot intervene on people’s past purchases and observe the effect on their peers’ current purchases.

Original: Instead, several factors, called confounders, affect what people purchase yesterday and what their peers purchase today.
Revised: Instead, several factors, called confounders, affect what people purchase yesterday and what their peers purchase today.

Original: Confounders create a correlation between people’s behavior that is not driven by social influence.
Revised: Confounders create a correlation between people’s behavior that is not driven by social influence.

Original: We articulate the assumptions needed to distinguish the causal effect of social influence from the effects of confounders.
Revised: We articulate the assumptions needed to distinguish the causal effect of social influence from the effects of confounders.

Original: The third contribution is the PIF algorithm for estimating social influence from observed networks.
Revised: The third contribution is the PIF algorithm for estimating social influence from observed networks.

Original: The challenge with estimating social influence is that the confounders are usually unobserved.
Revised: The challenge with estimating social influence is that the confounders are usually unobserved.

Original: However, the confounders drive the observed purchases and connections in the network, which provide indirect evidence for them.
Revised: However, the confounders drive the observed purchases and connections in the network, which provide indirect evidence for them.

Original: We operationalize this insight by fitting well-studied statistical models of networks (Ball et al., 2011; Gopalan and Blei, 2013; Contisciani et al., 2020) and models such as matrix factorization Lee and Seung (2001); Gopalan et al. (2015).
Revised: We operationalize this insight by fitting well-studied statistical models of networks (Ball et al., 2011; Gopalan and Blei, 2013; Contisciani et al., 2020) and models such as matrix factorization Lee and Seung (2001); Gopalan et al. (2015).

Original: We will show how these models produce variables that contain some of the same information as the confounders.
Revised: We will show how these models produce variables that contain some of the same information as the confounders.

Original: PIF uses these variables to reduce the bias due to confounders when estimating social influence.
Revised: PIF uses these variables to reduce the bias due to confounders when estimating social influence.

Original: To understand the challenges with estimating social influence, consider the following example:
Revised: To understand the challenges with estimating social influence, consider the following example:

Original: Example 1 Yesterday, Isabela bought a sports drink and today, her friend Judy bought the same sports drink.
Revised: Example 1 Yesterday, Isabela bought a sports drink and today, her friend Judy bought the same sports drink.

Original: Did Isabela influence Judy to buy the drink or would Judy have bought it anyway?
Revised: Did Isabela influence Judy to buy the drink or would Judy have bought it anyway?

Original: Isabela might have caused Judy to buy the drink because of her social influence.
Revised: Isabela might have caused Judy to buy the drink because of her social influence.

Original: However, the shared purchasing behavior can be explained in different ways.
Revised: However, the shared purchasing behavior can be explained in different ways.

Original: Isabela and Judy became friends because of their shared interest in sports and this interest caused them each to buy the drink.
Revised: Isabela and Judy became friends because of their shared interest in sports and this interest caused them each to buy the drink.

Original: (The idea that people with shared traits are more likely to connect is known as homophily.)
Revised: (The idea that people with shared traits are more likely to connect is known as homophily.)

Original: Alternatively,
Revised: Alternatively,

Original: Isabela and Judy might both happen to like sports drinks, even though they became friends because they live in the same building.
Revised: Isabela and Judy might both happen to like sports drinks, even though they became friends because they live in the same building.

Original: It is their preference for sports drinks that drives their purchases, irrespective of the reasons they became friends.
Revised: It is their preference for sports drinks that drives their purchases, irrespective of the reasons they became friends.

Original: Thus, the shared purchasing behavior might be evidence for social influence, but it might also arise for other reasons.
Revised: Thus, the shared purchasing behavior might be evidence for social influence, but it might also arise for other reasons.

Original: The factors that cause people to form connections and purchase items are confounders of social influence.
Revised: The factors that cause people to form connections and purchase items are confounders of social influence.

Original: When we observe the confounders, we can use causal adjustment techniques (Pearl, 2009) to estimate causal effects.
Revised: When we observe the confounders, we can use causal adjustment techniques (Pearl, 2009) to estimate causal effects.

Original: However, PIF estimates social influence when the confounders are not directly observed.
Revised: However, PIF estimates social influence when the confounders are not directly observed.

Original: The main idea behind PIF is that the unobserved confounders, though not explicitly coded in the data, drive the observed connections between people and also drive their purchases.
Revised: The main idea behind PIF is that the unobserved confounders, though not explicitly coded in the data, drive the observed connections between people and also drive their purchases.

Original: PIF makes assumptions about the structure of these relationships, and then uses probabilistic models of networks and purchases to estimate variables that contain some of the same information as the unobserved confounders.
Revised: PIF makes assumptions about the structure of these relationships, and then uses probabilistic models of networks and purchases to estimate variables that contain some of the same information as the unobserved confounders.

Original: PIF uses these estimated variables in causal adjustment to estimate social influence.
Revised: PIF uses these estimated variables in causal adjustment to estimate social influence.

Original: We study PIF empirically with semi-synthetic simulations, using a novel procedure that uses real social networks to synthesize different purchasing scenarios under varying amounts of confounding.
Revised: We study PIF empirically with semi-synthetic simulations, using a novel procedure that uses real social networks to synthesize different purchasing scenarios under varying amounts of confounding.

Original: We find that PIF is more accurate at recovering influence than related methods.
Revised: We find that PIF is more accurate at recovering influence than related methods.

Original: Finally, we apply
Revised: Finally, we apply

Original: PIF to real data from the song-sharing platform Last.fm to perform an exploratory study.
Revised: PIF to real data from the song-sharing platform Last.fm to perform an exploratory study.

Original: We find evidence of correlated song-listening behavior between friends, but the results from PIF suggest that most of it is attributed to shared preferences and influence plays a small role.
Revised: We find evidence of correlated song-listening behavior between friends, but the results from PIF suggest that most of it is attributed to shared preferences and influence plays a small role.

Original: The code and data to reproduce the empirical results are available at https://github.com/blei-lab/ poisson-influence-factorization .
Revised: The code and data to reproduce the empirical results are available at https://github.com/blei-lab/ poisson-influence-factorization .

Original: Related work.
Revised: Related work.

Original: This paper relates to other work on estimating social influence from observed data.
Revised: This paper relates to other work on estimating social influence from observed data.

Original: There is a line of work that develops theory and methods for estimating social influence when all the confounders are observed (Ogburn et al., 2017; Sharma and Cosley, 2016; Eckles and Bakshy, 2020; Aral et al., 2009).
Revised: There is a line of work that develops theory and methods for estimating social influence when all the confounders are observed (Ogburn et al., 2017; Sharma and Cosley, 2016; Eckles and Bakshy, 2020; Aral et al., 2009).

Original: However, Shalizi and Thomas (2011) caution that is usually difficult to observe all confounders; unobserved factors that drive connections and purchases can still bias our estimates of social influence.
Revised: However, Shalizi and Thomas (2011) caution that is usually difficult to observe all confounders; unobserved factors that drive connections and purchases can still bias our estimates of social influence.

Original: PIF works in the context of this problem.
Revised: PIF works in the context of this problem.

Original: In using latent variable models to estimate variables that help causal inference, PIF adapts and extends ideas from Wang and Blei (2019a); Shalizi and McFowland III (2016).
Revised: In using latent variable models to estimate variables that help causal inference, PIF adapts and extends ideas from Wang and Blei (2019a); Shalizi and McFowland III (2016).

Original: 1 Building on their work, we show that models of both networks and purchases are needed to adjust for the bias due to confounders.
Revised: 1 Building on their work, we show that models of both networks and purchases are needed to adjust for the bias due to confounders.

Original: Similar to PIF, Chaney et al.
Revised: Similar to PIF, Chaney et al.

Original: (2015) extend latent variable models to model purchases based on both social networks and unobserved preferences, but they do not consider causal inference of social influence.
Revised: (2015) extend latent variable models to model purchases based on both social networks and unobserved preferences, but they do not consider causal inference of social influence.

Original: Guo et al.
Revised: Guo et al.

Original: (2020); Veitch et al.
Revised: (2020); Veitch et al.

Original: (2019) have also used inferred properties of social networks to estimate different types of causal effects but they do not consider social influence.
Revised: (2019) have also used inferred properties of social networks to estimate different types of causal effects but they do not consider social influence.

Original: At the intersection of causality and networks, there are two other lines of work that are distinct to this paper.
Revised: At the intersection of causality and networks, there are two other lines of work that are distinct to this paper.

Original: First, there is work on the bias introduced by social networks when estimating causal effects (Ogburn et al., 2017; Sherman and Shpitser, 2018; Sherman et al., 2020).
Revised: First, there is work on the bias introduced by social networks when estimating causal effects (Ogburn et al., 2017; Sherman and Shpitser, 2018; Sherman et al., 2020).

Original: In this line of work, social influence undermines valid inference, inducing dependence across the networked samples.
Revised: In this line of work, social influence undermines valid inference, inducing dependence across the networked samples.

Original: In this paper, however, social influence is itself the target of causal inference.
Revised: In this paper, however, social influence is itself the target of causal inference.

Original: Second, there is work on estimating social influence from time series data (Soni et al., 2019;
Revised: Second, there is work on estimating social influence from time series data (Soni et al., 2019;

Original: Anagnostopoulos et al., 2008;
Revised: Anagnostopoulos et al., 2008;

Original: La and Neville, 2010) or randomized experiments (Aral and Walker, 2012; Aral et al., 2013; Taylor et al., 2013; Toulis and Kao, 2013).
Revised: La and Neville, 2010) or randomized experiments (Aral and Walker, 2012; Aral et al., 2013; Taylor et al., 2013; Toulis and Kao, 2013).

Original: In contrast, we focus on observational settings.
Revised: In contrast, we focus on observational settings.

Original: 2. Social Influence
Revised: 2. Social Influence

Original: In this section
Revised: In this section

Original: [Figure]
Revised: [Figure]

Original: we introduce a causal model of social influence.
Revised: we introduce a causal model of social influence.

Original: We formalize social influence as a causal quantity.
Revised: We formalize social influence as a causal quantity.

Original: Then, we use the assumed causal model show how social influence can be identified in an ideal setting where the confounders are observed.
Revised: Then, we use the assumed causal model show how social influence can be identified in an ideal setting where the confounders are observed.

Original: [Figure]
Revised: [Figure]

Original: The estimation of social influence involves n people connected in a social network, their purchases across m items “yesterday” and “today”.
Revised: The estimation of social influence involves n people connected in a social network, their purchases across m items “yesterday” and “today”.

Original: The social network is represented by an adjacency matrix a where each entry a ij ∈ { 1 , 0 } indicates whether person i and person j are connected or not.
Revised: The social network is represented by an adjacency matrix a where each entry a ij ∈ { 1 , 0 } indicates whether person i and person j are connected or not.

Original: Yesterday’s purchases are represented by a binary matrix x , where an entry x jk ∈ { 1 , 0 } indicates whether person j bought item k yesterday or not.
Revised: Yesterday’s purchases are represented by a binary matrix x , where an entry x jk ∈ { 1 , 0 } indicates whether person j bought item k yesterday or not.

Original: Today’s purchases are represented by a matrix y , where each entry y
Revised: Today’s purchases are represented by a matrix y , where each entry y

Original: ik is a count of the units of item k that person i bought today.
Revised: ik is a count of the units of item k that person i bought today.

Original: 2.1. A Causal Model of Social Influence
Revised: 2.1. A Causal Model of Social Influence

Original: The causal graphical model in Figure 1 captures the assumptions about how the variables are drawn.
Revised: The causal graphical model in Figure 1 captures the assumptions about how the variables are drawn.

Original: A connection a ij between person i and person j is driven by the per-person latent variables { z i , ρ i } and { z j , ρ j } .
Revised: A connection a ij between person i and person j is driven by the per-person latent variables { z i , ρ i } and { z j , ρ j } .

Original: Each per-person variable (e.g., ρ i ) is a vector of traits that capture the reasons why person i forms connections in the network.
Revised: Each per-person variable (e.g., ρ i ) is a vector of traits that capture the reasons why person i forms connections in the network.

Original: Yesterday’s purchase x jk is driven by per-person latent variables { θ j , ρ j } and a per-item latent variable τ k .
Revised: Yesterday’s purchase x jk is driven by per-person latent variables { θ j , ρ j } and a per-item latent variable τ k .

Original: The variable τ k is a vector of attributes that capture people’s reasons for buying item k .
Revised: The variable τ k is a vector of attributes that capture people’s reasons for buying item k .

Original: The variables θ j and ρ j capture person j ’s preferences for those attributes.
Revised: The variables θ j and ρ j capture person j ’s preferences for those attributes.

Original: We distinguish between the per-person variables ρ j , θ j and z j .
Revised: We distinguish between the per-person variables ρ j , θ j and z j .

Original: The variable ρ i captures traits that affect both purchases and connections while the variables θ i and z i only affect purchases or connections (but not both).
Revised: The variable ρ i captures traits that affect both purchases and connections while the variables θ i and z i only affect purchases or connections (but not both).

Original: Today’s purchase y ik is driven by the same per-person and per-item variables that drove yesterday’s purchases but it also depends on the social influence from peers.
Revised: Today’s purchase y ik is driven by the same per-person and per-item variables that drove yesterday’s purchases but it also depends on the social influence from peers.

Original: This dependence is captured by the edges x jk → y ik and a ij → y ik in the causal model (Figure 1).
Revised: This dependence is captured by the edges x jk → y ik and a ij → y ik in the causal model (Figure 1).

Original: More precisely, the purchase y ik depends on all of person i ’s connections, a i = { a i 1 , . .
Revised: More precisely, the purchase y ik depends on all of person i ’s connections, a i = { a i 1 , . .

Original: , a in } , and all of the purchases of item k , x k = { x 1 k , . .
Revised: , a in } , and all of the purchases of item k , x k = { x 1 k , . .

Original: , x nk } .
Revised: , x nk } .

Original: 2.2. Defining Social Influence
Revised: 2.2. Defining Social Influence

Original: We now define the social influence of each person j as a causal effect, formalized with the language of interventions.
Revised: We now define the social influence of each person j as a causal effect, formalized with the language of interventions.

Original: Specifically, the distribution p ( y ik ; do( x jk = x )) is an interventional distribution over the purchase
Revised: Specifically, the distribution p ( y ik ; do( x jk = x )) is an interventional distribution over the purchase

Original: y
Revised: y

Original: ik where the do( · ) operator indicates that the variable x jk was set to the value x by intervening (Pearl, 2009).
Revised: ik where the do( · ) operator indicates that the variable x jk was set to the value x by intervening (Pearl, 2009).

Original: The ideal intervention may be hypothetical.
Revised: The ideal intervention may be hypothetical.

Original: Nonetheless, interventional distributions allow us to define causal effects.
Revised: Nonetheless, interventional distributions allow us to define causal effects.

Original: First, we define the social influence of person j with respect to a single item k and a single person i that is connected to person j .
Revised: First, we define the social influence of person j with respect to a single item k and a single person i that is connected to person j .

Original: It is the causal effect,
Revised: It is the causal effect,

Original: [Equation]
Revised: [Equation]

Original: These expectations are over interventional distributions, the first where person j is “made” to purchase item k yesterday and the second where person j is “prevented” from purchasing it yesterday.
Revised: These expectations are over interventional distributions, the first where person j is “made” to purchase item k yesterday and the second where person j is “prevented” from purchasing it yesterday.

Original: They are distributions over the purchase y
Revised: They are distributions over the purchase y

Original: ik under these two interventions (and conditioned on person i and j being connected).
Revised: ik under these two interventions (and conditioned on person i and j being connected).

Original: Using the causal quantity ψ ijk , the main causal effect of interest in this paper is the average social influence of a person j .
Revised: Using the causal quantity ψ ijk , the main causal effect of interest in this paper is the average social influence of a person j .

Original: It is defined with respect to the n j peers of person j and the m items,
Revised: It is defined with respect to the n j peers of person j and the m items,

Original: [Equation]
Revised: [Equation]

Original: This quantity captures person j ’s average influence across their peers and all items.
Revised: This quantity captures person j ’s average influence across their peers and all items.

Original: If person j has social influence, on average across their peers and items, we expected the difference in purchasing rate across interventions to be large.
Revised: If person j has social influence, on average across their peers and items, we expected the difference in purchasing rate across interventions to be large.

Original: 2.3. Challenges to Estimating Social Influence
Revised: 2.3. Challenges to Estimating Social Influence

Original: The challenge with estimating average social influence (Eq.
Revised: The challenge with estimating average social influence (Eq.

Original: (2)) is that the terms ψ ijk (Eq.
Revised: (2)) is that the terms ψ ijk (Eq.

Original: (1)) involve hypothetical interventions.
Revised: (1)) involve hypothetical interventions.

Original: With the observed data, we could approximate the expected conditional difference,
Revised: With the observed data, we could approximate the expected conditional difference,

Original: [Equation]
Revised: [Equation]

Original: Informally, this captures whether person j ’s purchases yesterday are correlated with the purchases of their peers today.
Revised: Informally, this captures whether person j ’s purchases yesterday are correlated with the purchases of their peers today.

Original: However, this difference is not necessarily the causal quantity ψ ijk Eq.
Revised: However, this difference is not necessarily the causal quantity ψ ijk Eq.

Original: This is because in the observed data, the purchases x jk and y ik are correlated for reasons other than the social influence of person j on their peer i , as we discussed in Section 1.
Revised: This is because in the observed data, the purchases x jk and y ik are correlated for reasons other than the social influence of person j on their peer i , as we discussed in Section 1.

Original: For example, in the assumed model in Figure 1, the purchases x jk and y ik share causes such as τ k , item k ’s attributes.
Revised: For example, in the assumed model in Figure 1, the purchases x jk and y ik share causes such as τ k , item k ’s attributes.

Original: More generally, the observed variables are affected by confounders , variables that induce a noncausal dependence between purchases yesterday and those of today.
Revised: More generally, the observed variables are affected by confounders , variables that induce a noncausal dependence between purchases yesterday and those of today.

Original: The presence of these variables means that when we observe a correlation between the purchasing habits of a person yesterday and those of their peers today, we cannot attribute it to the person’s social influence alone.
Revised: The presence of these variables means that when we observe a correlation between the purchasing habits of a person yesterday and those of their peers today, we cannot attribute it to the person’s social influence alone.

Original: Causal graphical models clarify which variables create confounding bias, based on backdoor paths in the graph that induce non-causal associations; see Pearl (2009) for full details.
Revised: Causal graphical models clarify which variables create confounding bias, based on backdoor paths in the graph that induce non-causal associations; see Pearl (2009) for full details.

Original: The graph in
Revised: The graph in

Original: Figure 1 shows two backdoor paths between the intervened variable x ik and the variable y jk : (1) via item attributes, x ik ← τ k → y jk ; (2) via traits involved in homophily, x
Revised: Figure 1 shows two backdoor paths between the intervened variable x ik and the variable y jk : (1) via item attributes, x ik ← τ k → y jk ; (2) via traits involved in homophily, x

Original: ik ← ρ i → a ij ← ρ j → y jk .
Revised: ik ← ρ i → a ij ← ρ j → y jk .

Original: (Note that it is because we condition on the social network in Eq.
Revised: (Note that it is because we condition on the social network in Eq.

Original: (1) that the second backdoor path is opened.)
Revised: (1) that the second backdoor path is opened.)

Original: The variables τ k and ρ i , which appear along backdoor paths, are confounders of social influence, the causal effect represented by the x jk → y ik (Figure 1).
Revised: The variables τ k and ρ i , which appear along backdoor paths, are confounders of social influence, the causal effect represented by the x jk → y ik (Figure 1).

Original: 2.4. Estimating Social Influence with Observed Confounders
Revised: 2.4. Estimating Social Influence with Observed Confounders

Original: When the confounders are observed, we can use causal adjustment (Pearl, 2009) to estimate causal effects.
Revised: When the confounders are observed, we can use causal adjustment (Pearl, 2009) to estimate causal effects.

Original: As a step towards estimating social influence in the presence of unobserved confounders, we consider the easier setting where we observe the per-person variables ρ 1: n and per-item variables τ 1: m .
Revised: As a step towards estimating social influence in the presence of unobserved confounders, we consider the easier setting where we observe the per-person variables ρ 1: n and per-item variables τ 1: m .

Original: In this setting, we rewrite each interventional quantity ψ ijk in terms of the observed data distribution.
Revised: In this setting, we rewrite each interventional quantity ψ ijk in terms of the observed data distribution.

Original: This derivation involves two ideas.
Revised: This derivation involves two ideas.

Original: The first idea is that social influence ψ ijk is the marginal effect of yesterday’s purchase x jk on today’s purchase y ik , marginalizing out all other causes of the purchase y ik .
Revised: The first idea is that social influence ψ ijk is the marginal effect of yesterday’s purchase x jk on today’s purchase y ik , marginalizing out all other causes of the purchase y ik .

Original: The causal model (Figure 1) shows that calculating the social influence of person j requires marginalizing out the connections a − j i
Revised: The causal model (Figure 1) shows that calculating the social influence of person j requires marginalizing out the connections a − j i

Original: = a i \ { a ij } of person
Revised: = a i \ { a ij } of person

Original: i and the purchases x − j k = x
Revised: i and the purchases x − j k = x

Original: k \ { x jk } of item k that do not involve person j .
Revised: k \ { x jk } of item k that do not involve person j .

Original: That is,
Revised: That is,

Original: [Equation]
Revised: [Equation]

Original: The inner distribution is with respect to the purchase y ik .
Revised: The inner distribution is with respect to the purchase y ik .

Original: The outer distribution over connections and purchases a − j i and x − j k is specified by the assumed causal model in Figure 1.
Revised: The outer distribution over connections and purchases a − j i and x − j k is specified by the assumed causal model in Figure 1.

Original: The second idea is that when the confounders for a causal effect of interest are observed, we can rewrite an unobserved interventional distribution in terms of an observed conditional distribution using a formula called backdoor adjustment (Pearl, 2009).
Revised: The second idea is that when the confounders for a causal effect of interest are observed, we can rewrite an unobserved interventional distribution in terms of an observed conditional distribution using a formula called backdoor adjustment (Pearl, 2009).

Original: First, define,
Revised: First, define,

Original: [Equation]
Revised: [Equation]

Original: This function of conditional expected outcomes involves the per-item and per-person confounders.
Revised: This function of conditional expected outcomes involves the per-item and per-person confounders.

Original: The backdoor adjustment allows us to rewrite the interventional distributions in Eq.
Revised: The backdoor adjustment allows us to rewrite the interventional distributions in Eq.

Original: (4) in terms of the functions µ ik ( · ) ,
Revised: (4) in terms of the functions µ ik ( · ) ,

Original: [Equation]
Revised: [Equation]

Original: Intuitively, this formula “adjusts” for the effects that a person’s traits
Revised: Intuitively, this formula “adjusts” for the effects that a person’s traits

Original: ρ i and an item’s attributes τ k have on their purchasing.
Revised: ρ i and an item’s attributes τ k have on their purchasing.

Original: The average social influence of person j is the remaining difference in purchases among their peers when person j buys items versus when they do not.
Revised: The average social influence of person j is the remaining difference in purchases among their peers when person j buys items versus when they do not.

Original: In the next section, we develop one estimator for the quantity in Eq.
Revised: In the next section, we develop one estimator for the quantity in Eq.

Original: (6), using regression to fit the function µ ik ( · ).
Revised: (6), using regression to fit the function µ ik ( · ).

Original: Roadmap.
Revised: Roadmap.

Original: In this paper, the confounders – per-person variables ρ i and per-item variables τ k – are latent and we cannot directly use the strategy given in Eq.
Revised: In this paper, the confounders – per-person variables ρ i and per-item variables τ k – are latent and we cannot directly use the strategy given in Eq.

Original: (6) to estimate each person’s average social influence ψ j .
Revised: (6) to estimate each person’s average social influence ψ j .

Original: In the next section, we develop Poisson Influence Factorization (PIF), a method for estimating social influence that addresses the challenge presented by latent confounders.
Revised: In the next section, we develop Poisson Influence Factorization (PIF), a method for estimating social influence that addresses the challenge presented by latent confounders.

Original: To develop PIF, we exploit the fact that the confounders, though latent, are common causes of multiple social network connections and purchases.
Revised: To develop PIF, we exploit the fact that the confounders, though latent, are common causes of multiple social network connections and purchases.

Original: By fitting latent variable models of networks and multivariate data, we can construct variables that contain some of the same information as the confounders.
Revised: By fitting latent variable models of networks and multivariate data, we can construct variables that contain some of the same information as the confounders.

Original: These variables can then serve as substitutes when estimating social influence.
Revised: These variables can then serve as substitutes when estimating social influence.

Original: 3. Poisson Influence Factorization
Revised: 3. Poisson Influence Factorization

Original: Poisson Influence Factorization (PIF) involves two main ideas.
Revised: Poisson Influence Factorization (PIF) involves two main ideas.

Original: First, we show how to construct variables, called substitutes, that contain some of the same information contained in the latent confounders.
Revised: First, we show how to construct variables, called substitutes, that contain some of the same information contained in the latent confounders.

Original: Second, we develop an estimator of social influence that uses the substitutes to adjust for some of the bias due to confounders.
Revised: Second, we develop an estimator of social influence that uses the substitutes to adjust for some of the bias due to confounders.

Original: The result is a practical algorithm for estimating social influence from observational data.
Revised: The result is a practical algorithm for estimating social influence from observational data.

Original: 3.1. Substitutes for Confounders
Revised: 3.1. Substitutes for Confounders

Original: The first step of PIF is to construct substitutes for the per-person latent variables
Revised: The first step of PIF is to construct substitutes for the per-person latent variables

Original: ρ i and per-item latent variables τ k by fitting latent variable models to the social network and yesterday’s purchases.
Revised: ρ i and per-item latent variables τ k by fitting latent variable models to the social network and yesterday’s purchases.

Original: The idea is that the per-person variables ρ i and z i drive each observed connection a ij in the social network.
Revised: The idea is that the per-person variables ρ i and z i drive each observed connection a ij in the social network.

Original: If we fit a latent variable model to the observed social network that has this same structure, the inferred latent variables will contain some of the same information contained by the confounder
Revised: If we fit a latent variable model to the observed social network that has this same structure, the inferred latent variables will contain some of the same information contained by the confounder

Original: ρ i .
Revised: ρ i .

Original: The same idea applies to the observed data of purchases from yesterday and the confounder τ k .
Revised: The same idea applies to the observed data of purchases from yesterday and the confounder τ k .

Original: To implement this strategy, we fit models of networks (Holland et al., 1983; De Bacco et al., 2017;
Revised: To implement this strategy, we fit models of networks (Holland et al., 1983; De Bacco et al., 2017;

Original: Ball et al., 2011; Contisciani et al., 2020; Peixoto, 2019; Gopalan and Blei, 2013; Hoff, 2008) and matrix factorization methods (Lee and Seung, 2001; Gopalan et al., 2013, 2015).
Revised: Ball et al., 2011; Contisciani et al., 2020; Peixoto, 2019; Gopalan and Blei, 2013; Hoff, 2008) and matrix factorization methods (Lee and Seung, 2001; Gopalan et al., 2013, 2015).

Original: The idea behind substitutes was developed in Wang and Blei (2019a), in the deconfounder algorithm for multivariate treatments.
Revised: The idea behind substitutes was developed in Wang and Blei (2019a), in the deconfounder algorithm for multivariate treatments.

Original: Here, we extend these ideas to the social influence setting, where the social network and yesterday’s purchases (“treatments” in our setting) are both needed to capture the information in the confounders.
Revised: Here, we extend these ideas to the social influence setting, where the social network and yesterday’s purchases (“treatments” in our setting) are both needed to capture the information in the confounders.

Original: We illustrate the technical ideas behind substitutes by defining them for the per-person variable ρ i .
Revised: We illustrate the technical ideas behind substitutes by defining them for the per-person variable ρ i .

Original: We fit a latent variable model to the observed network a .
Revised: We fit a latent variable model to the observed network a .

Original: Standard generative models for networks model the likelihood of observing a connection between person i and person j using Kdimensional latent variables
Revised: Standard generative models for networks model the likelihood of observing a connection between person i and person j using Kdimensional latent variables

Original: c i and c j on nodes, often called community membership.
Revised: c i and c j on nodes, often called community membership.

Original: For concreteness, we will focus on a generative model with a Poisson likelihood, which has been well-studied for modeling social networks (Gopalan and Blei, 2013).
Revised: For concreteness, we will focus on a generative model with a Poisson likelihood, which has been well-studied for modeling social networks (Gopalan and Blei, 2013).

Original: The model assumes that each connection in the network is conditionally independent given the latent variables,
Revised: The model assumes that each connection in the network is conditionally independent given the latent variables,

Original: [Equation]
Revised: [Equation]

Original: We perform inference in this model over the latent variables c 1: n .
Revised: We perform inference in this model over the latent variables c 1: n .

Original: Each inferred variable ˆ c
Revised: Each inferred variable ˆ c

Original: i is a substitute for some of the per-person confounders ρ i .
Revised: i is a substitute for some of the per-person confounders ρ i .

Original: We call the variables ˆ c 1: n per-person substitutes.
Revised: We call the variables ˆ c 1: n per-person substitutes.

Original: [Equation]
Revised: [Equation]

Original: ia in … c i
Revised: ia in … c i

Original: Per-person substitute
Revised: Per-person substitute

Original: [Equation]
Revised: [Equation]

Original: ia in …
Revised: ia in …

Original: [Equation]
Revised: [Equation]

Original: i
Revised: i

Original: Per-person confounder
Revised: Per-person confounder

Original: [Equation]
Revised: [Equation]

Original: Why are substitutes valid?
Revised: Why are substitutes valid?

Original: Figure 2 illustrates the justification behind per-person substitutes.
Revised: Figure 2 illustrates the justification behind per-person substitutes.

Original: Consider the connections a i made by person i , and for ease of explanation, suppose that there are no traits z i , which only affect the connections a i .
Revised: Consider the connections a i made by person i , and for ease of explanation, suppose that there are no traits z i , which only affect the connections a i .

Original: Then, Figure 2 shows that the perperson confounder ρ i can be separated into the traits u
Revised: Then, Figure 2 shows that the perperson confounder ρ i can be separated into the traits u

Original: i (outlined in green) that affect more than one connection in the set a i and traits v i (outlined in red), which affect only one connection.
Revised: i (outlined in green) that affect more than one connection in the set a i and traits v i (outlined in red), which affect only one connection.

Original: The assumed causal model on the left in Figure 2 implies that each connection a ij ∈ a i is conditionally independent given the per-person variable
Revised: The assumed causal model on the left in Figure 2 implies that each connection a ij ∈ a i is conditionally independent given the per-person variable

Original: u i .
Revised: u i .

Original: That is,
Revised: That is,

Original: [Equation]
Revised: [Equation]

Original: Consider the generative model on the right in Figure 2.
Revised: Consider the generative model on the right in Figure 2.

Original: We see that it posits the same conditional independence of observed variables a i given the latent variables c i .
Revised: We see that it posits the same conditional independence of observed variables a i given the latent variables c i .

Original: Assume that the generative model well-approximates the empirical distribution of the network, captured by each row a i of the adjacency matrix.
Revised: Assume that the generative model well-approximates the empirical distribution of the network, captured by each row a i of the adjacency matrix.

Original: Then, the per-person variables c i that render each observation a ij conditionally independent of the others must coincide with the variables u
Revised: Then, the per-person variables c i that render each observation a ij conditionally independent of the others must coincide with the variables u

Original: i posited in the causal model in Figure 1.
Revised: i posited in the causal model in Figure 1.

Original: We make this idea concrete with a formal result and proof in the appendix.
Revised: We make this idea concrete with a formal result and proof in the appendix.

Original: When does this not work?
Revised: When does this not work?

Original: The per-person substitute ˆ c i cannot capture the confounder v i , which only affects one of the connection a i .
Revised: The per-person substitute ˆ c i cannot capture the confounder v i , which only affects one of the connection a i .

Original: This fact is because a per-person substitute ˆ c i can still render the observed connections
Revised: This fact is because a per-person substitute ˆ c i can still render the observed connections

Original: a i conditionally independent without capturing the trait that affects a single connection.
Revised: a i conditionally independent without capturing the trait that affects a single connection.

Original: Put differently, such traits leave no observable implications in the data.
Revised: Put differently, such traits leave no observable implications in the data.

Original: What information is captured?
Revised: What information is captured?

Original: The per-person substitute ˆ c i contains information about both variables ρ
Revised: The per-person substitute ˆ c i contains information about both variables ρ

Original: i and z i , since both drive the connections a i and render them conditionally independent (Figure 1).
Revised: i and z i , since both drive the connections a i and render them conditionally independent (Figure 1).

Original: However, only the per-person variables ρ i are confounders.
Revised: However, only the per-person variables ρ i are confounders.

Original: To target ρ -specific information, we can instead fit a joint factor model,
Revised: To target ρ -specific information, we can instead fit a joint factor model,

Original: [Equation]
Revised: [Equation]

Original: which uses the community membership c to jointly model the network and yesterday’s purchases.
Revised: which uses the community membership c to jointly model the network and yesterday’s purchases.

Original: We can construct per-person substitutes with inferred variables ˆ c i as before.
Revised: We can construct per-person substitutes with inferred variables ˆ c i as before.

Original: We will compare both methods for constructing per-person substitutes in the empirical studies.
Revised: We will compare both methods for constructing per-person substitutes in the empirical studies.

Original: We will use the same ideas to find substitutes for the per-person confounders τ k that capture the attributes of an item k that cause people to purchase it (or not).
Revised: We will use the same ideas to find substitutes for the per-person confounders τ k that capture the attributes of an item k that cause people to purchase it (or not).

Original: As we noted, finding substitutes relies on models that posit the conditional independence of observed data given latent variables (e.g., Eq.
Revised: As we noted, finding substitutes relies on models that posit the conditional independence of observed data given latent variables (e.g., Eq.

Original: For heterogeneous data such as people’s purchases of multiple products, one such a model is Poisson matrix factorization (Gopalan et al., 2013).
Revised: For heterogeneous data such as people’s purchases of multiple products, one such a model is Poisson matrix factorization (Gopalan et al., 2013).

Original: Thus, to construct substitutes for some of the latent confounders
Revised: Thus, to construct substitutes for some of the latent confounders

Original: τ k , we fit Poisson matrix factorization to yesterday’s purchases,
Revised: τ k , we fit Poisson matrix factorization to yesterday’s purchases,

Original: [Equation]
Revised: [Equation]

Original: where Q is the number of factors.
Revised: where Q is the number of factors.

Original: After performing inference over the latent variables d 1: n and w 1: m , each inferred variable ˆ w k is a substitute for the subset of per-item variables τ k that affect more than one purchase in x k = { x 1 k , . . .
Revised: After performing inference over the latent variables d 1: n and w 1: m , each inferred variable ˆ w k is a substitute for the subset of per-item variables τ k that affect more than one purchase in x k = { x 1 k , . . .

Original: , x nk } .
Revised: , x nk } .

Original: We call each variable ˆ w k a per-item substitute.
Revised: We call each variable ˆ w k a per-item substitute.

Original: 3.2. Estimation with Substitutes
Revised: 3.2. Estimation with Substitutes

Original: The goal is to estimate the average social influence
Revised: The goal is to estimate the average social influence

Original: ψ j (Eq.
Revised: ψ j (Eq.

Original: (2)) of a person j , which is defined based on (hypothetical) interventions to person j ’s purchases yesterday.
Revised: (2)) of a person j , which is defined based on (hypothetical) interventions to person j ’s purchases yesterday.

Original: The second stage of PIF will use the constructed per-person and per-item substitutes to construct an estimator of average social influence.
Revised: The second stage of PIF will use the constructed per-person and per-item substitutes to construct an estimator of average social influence.

Original: The idea will be to use the substitutes ˆ c
Revised: The idea will be to use the substitutes ˆ c

Original: i and ˆ w k as drop-in replacements for the unobserved confounders ρ i and τ k .
Revised: i and ˆ w k as drop-in replacements for the unobserved confounders ρ i and τ k .

Original: First, we define the substitutes more formally to be ˆ c i =
Revised: First, we define the substitutes more formally to be ˆ c i =

Original: E [ c i | a ] and ˆ w k = E
Revised: E [ c i | a ] and ˆ w k = E

Original: [ w k | x ] .
Revised: [ w k | x ] .

Original: They are expected values of the posterior distributions p ( c 1: n | a ) and p ( w 1: m | x ) .
Revised: They are expected values of the posterior distributions p ( c 1: n | a ) and p ( w 1: m | x ) .

Original: Since calculating such posteriors exactly is generally intractable, we approximate them with variational inference.
Revised: Since calculating such posteriors exactly is generally intractable, we approximate them with variational inference.

Original: Specifically, we adopt a Bayesian perspective by placing sparse Gamma priors on the latent variables, constraining them to be non-negative.
Revised: Specifically, we adopt a Bayesian perspective by placing sparse Gamma priors on the latent variables, constraining them to be non-negative.

Original: We approximate the posteriors with mean-field variational inference (MF-VI) (Gopalan et al., 2013; Blei et al., 2017).
Revised: We approximate the posteriors with mean-field variational inference (MF-VI) (Gopalan et al., 2013; Blei et al., 2017).

Original: Then, we use a Poisson likelihood to model today’s
Revised: Then, we use a Poisson likelihood to model today’s

Original: outcomes given the substitutes,
Revised: outcomes given the substitutes,

Original: [Equation]
Revised: [Equation]

Original: The model says that the variable y ik , the expected number of times person i buys item k today, is a linear function of their estimated traits, the estimated attributes of item k , and social influence from their peers.
Revised: The model says that the variable y ik , the expected number of times person i buys item k today, is a linear function of their estimated traits, the estimated attributes of item k , and social influence from their peers.

Original: We place sparse Gamma priors on the unobserved variables γ , α , β , ensuring that the term λ ik is non-negative.
Revised: We place sparse Gamma priors on the unobserved variables γ , α , β , ensuring that the term λ ik is non-negative.

Original: The following result relates the model in Eq.
Revised: The following result relates the model in Eq.

Original: (11) to average social influence.
Revised: (11) to average social influence.

Original: Proposition 3.1 If the functions of expected purchases, µ ik ( a, x ) , satisfy,
Revised: Proposition 3.1 If the functions of expected purchases, µ ik ( a, x ) , satisfy,

Original: [Equation]
Revised: [Equation]

Original: and the purchases y ik are drawn from the Poisson model in Eq.
Revised: and the purchases y ik are drawn from the Poisson model in Eq.

Original: (11) , then ψ j = β j .
Revised: (11) , then ψ j = β j .

Original: We prove this result in Section 7.2.
Revised: We prove this result in Section 7.2.

Original: Intuitively, the assumptions say that the substitutes
Revised: Intuitively, the assumptions say that the substitutes

Original: ˆ c i and ˆ w k contain the same information about the expected purchase y ik as their latent confounder counterparts.
Revised: ˆ c i and ˆ w k contain the same information about the expected purchase y ik as their latent confounder counterparts.

Original: Further, the purchase y ik is drawn from the Poisson model in Eq.
Revised: Further, the purchase y ik is drawn from the Poisson model in Eq.

Original: Under these assumptions, the average social influence ψ j of person j is equal to β j .
Revised: Under these assumptions, the average social influence ψ j of person j is equal to β j .

Original: This result holds in the context of infinite data, without guarantees on estimation quality from finite samples.
Revised: This result holds in the context of infinite data, without guarantees on estimation quality from finite samples.

Original: To estimate the average social influence ψ j , we fit the model in Eq.
Revised: To estimate the average social influence ψ j , we fit the model in Eq.

Original: (11) with posterior inference.
Revised: (11) with posterior inference.

Original: We approximate the posterior distribution p ( β 1: n , γ 1: m , α 1: n | y , x , a , ˆ c 1: n , ˆ w 1: m ) with MF-VI.
Revised: We approximate the posterior distribution p ( β 1: n , γ 1: m , α 1: n | y , x , a , ˆ c 1: n , ˆ w 1: m ) with MF-VI.

Original: We place sparse Gamma priors on the unobserved variables γ , α , β and fit the model with coordinate descent.
Revised: We place sparse Gamma priors on the unobserved variables γ , α , β and fit the model with coordinate descent.

Original: See Section 7.5 for full details.
Revised: See Section 7.5 for full details.

Original: 3.3. Limitations
Revised: 3.3. Limitations

Original: As with all methods of causal inference, PIF relies on strong assumptions, such as the ones required for estimating valid substitutes.
Revised: As with all methods of causal inference, PIF relies on strong assumptions, such as the ones required for estimating valid substitutes.

Original: In particular, we emphasize that if a trait captured by the variables ρ or τ only affects one interaction—one connection in the network, one person who purchased item k , or one item purchased by user j — then it cannot be recovered by this method for constructing substitutes.
Revised: In particular, we emphasize that if a trait captured by the variables ρ or τ only affects one interaction—one connection in the network, one person who purchased item k , or one item purchased by user j — then it cannot be recovered by this method for constructing substitutes.

Original: The assumption that variables affect multiple interactions is untestable; it has no implications for the observable data, and must instead be assessed carefully by users of the method.
Revised: The assumption that variables affect multiple interactions is untestable; it has no implications for the observable data, and must instead be assessed carefully by users of the method.

Original: We also stress that for the substitutes to be valid, the factor models must capture the empirical data distribution.
Revised: We also stress that for the substitutes to be valid, the factor models must capture the empirical data distribution.

Original: When we empirically study PIF, we evaluate the fitted factor models (see
Revised: When we empirically study PIF, we evaluate the fitted factor models (see

Original: Section 7.3 with model checks).
Revised: Section 7.3 with model checks).

Original: Finally, in practice, the factor models or the model of purchases might be misspecified, leading biased estimates of social influence.
Revised: Finally, in practice, the factor models or the model of purchases might be misspecified, leading biased estimates of social influence.

Original: We leave this exploration of estimation quality to future work.
Revised: We leave this exploration of estimation quality to future work.

Original: 4. Empirical evaluation
Revised: 4. Empirical evaluation

Original: We empirically study the performance of PIF for estimating social influence.
Revised: We empirically study the performance of PIF for estimating social influence.

Original: The challenge in addressing this empirical question is that we do not have ground truth knowledge about the social influence of each person in a known network.
Revised: The challenge in addressing this empirical question is that we do not have ground truth knowledge about the social influence of each person in a known network.

Original: Instead, our strategy will be to use a real social network and per-person covariates to simulate purchasing data.
Revised: Instead, our strategy will be to use a real social network and per-person covariates to simulate purchasing data.

Original: The purchases today will depend on both simulated influence and per-person and per-item confounders.
Revised: The purchases today will depend on both simulated influence and per-person and per-item confounders.

Original: We create several datasets by varying the strength of confounding and we evaluate the accuracy of PIF and related methods in recovering the simulated influence across these settings.
Revised: We create several datasets by varying the strength of confounding and we evaluate the accuracy of PIF and related methods in recovering the simulated influence across these settings.

Original: In the experiments, PIF and the compared methods differ only in the degree to which they adjust for confounders.
Revised: In the experiments, PIF and the compared methods differ only in the degree to which they adjust for confounders.

Original: This allows us to evaluate how well PIF adjusts for the bias due to confounding compared to related methods when estimating social influence.
Revised: This allows us to evaluate how well PIF adjusts for the bias due to confounding compared to related methods when estimating social influence.

Original: The key finding is that PIF is most accurate at recovering influence among related methods.
Revised: The key finding is that PIF is most accurate at recovering influence among related methods.

Original: Furthermore, strong empirical performance of multiple PIF variants suggests that the method is not sensitive to modeling choices.
Revised: Furthermore, strong empirical performance of multiple PIF variants suggests that the method is not sensitive to modeling choices.

Original: Moreover, fitting a joint factor model to purchases and network data to obtain per-person substitutes typically leads to better empirical performance than using substitutes obtained only from the network.
Revised: Moreover, fitting a joint factor model to purchases and network data to obtain per-person substitutes typically leads to better empirical performance than using substitutes obtained only from the network.

Original: In Section 7.6 of the appendix, we analyze the sensitivity of PIF to its assumptions.
Revised: In Section 7.6 of the appendix, we analyze the sensitivity of PIF to its assumptions.

Original: In particular, we consider the assumption that per-person and per-item confounders must affect multiple connections or purchases (Section 3) to be captured by their substitutes.
Revised: In particular, we consider the assumption that per-person and per-item confounders must affect multiple connections or purchases (Section 3) to be captured by their substitutes.

Original: We find that PIF accurately estimates social influence even under moderate violation of its assumptions.
Revised: We find that PIF accurately estimates social influence even under moderate violation of its assumptions.

Original: Finally, we apply PIF to perform exploratory data analysis on real data from Last.fm, a songsharing platform.
Revised: Finally, we apply PIF to perform exploratory data analysis on real data from Last.fm, a songsharing platform.

Original: Code and data will be publicly available.
Revised: Code and data will be publicly available.

Original: 4.1. Semi-synthetic studies
Revised: 4.1. Semi-synthetic studies

Original: The semi-simulated datasets use a real social network with 70k users and millions of connections 2 .
Revised: The semi-simulated datasets use a real social network with 70k users and millions of connections 2 .

Original: For each user, we observe the region where they live, which we found to be a strong predictor of network connections.
Revised: For each user, we observe the region where they live, which we found to be a strong predictor of network connections.

Original: Our strategy will be to simulate purchases that depend on homophily (i.e., people from the same region have similar purchasing habits), preferences and social influence.
Revised: Our strategy will be to simulate purchases that depend on homophily (i.e., people from the same region have similar purchasing habits), preferences and social influence.

Original: We will evaluate how well the compared methods recover social influence as homophily and preference increasingly affect the purchases.
Revised: We will evaluate how well the compared methods recover social influence as homophily and preference increasingly affect the purchases.

Original: Setup.
Revised: Setup.

Original: To produce each simulated dataset, we first snowball sample a subgraph of 3k users from the full network.
Revised: To produce each simulated dataset, we first snowball sample a subgraph of 3k users from the full network.

Original: Then, we simulate yesterday’s purchases across 3k items.
Revised: Then, we simulate yesterday’s purchases across 3k items.

Original: Each purchase of a hypothetical item depend on the item’s attributes and the user’s preferences for those attributes.
Revised: Each purchase of a hypothetical item depend on the item’s attributes and the user’s preferences for those attributes.

Original: We simulate each item’s attributes based on a randomly chosen region in which it is popular and a randomly drawn categorical variable, which is exogenous to the network.
Revised: We simulate each item’s attributes based on a randomly chosen region in which it is popular and a randomly drawn categorical variable, which is exogenous to the network.

Original: We also generate variables that capture a user’s preference for items that are popular in their region, and their preference for a randomly chosen categorical variable.
Revised: We also generate variables that capture a user’s preference for items that are popular in their region, and their preference for a randomly chosen categorical variable.

Original: We use the Poisson generative model of purchases (Eq.
Revised: We use the Poisson generative model of purchases (Eq.

Original: (10)) to sample yesterday’s purchases based on the per-item and per-person variables.
Revised: (10)) to sample yesterday’s purchases based on the per-item and per-person variables.

Original: To simulate today’s purchases, we first simulate each user’s social influence.
Revised: To simulate today’s purchases, we first simulate each user’s social influence.

Original: Then, we simulate today’s purchases based on the variables that generated yesterday’s purchases as well as social influence, using the Poisson model in Eq.
Revised: Then, we simulate today’s purchases based on the variables that generated yesterday’s purchases as well as social influence, using the Poisson model in Eq.

Original: We describe the data generating process in full detail in Section 7.4.
Revised: We describe the data generating process in full detail in Section 7.4.

Original: Methods compared.
Revised: Methods compared.

Original: Our goal is to evaluate methods that adjust for confounding to different extents when estimating social influence.
Revised: Our goal is to evaluate methods that adjust for confounding to different extents when estimating social influence.

Original: As a gold standard, we run a version of PIF with the known per-person variables ρ 1: n and τ 1: m instead of substitutes.
Revised: As a gold standard, we run a version of PIF with the known per-person variables ρ 1: n and τ 1: m instead of substitutes.

Original: We refer to this method as Oracle.
Revised: We refer to this method as Oracle.

Original: Then, we hide the known confounders and study two variants of PIF: 1) using the community model in Eq.
Revised: Then, we hide the known confounders and study two variants of PIF: 1) using the community model in Eq.

Original: (7) to construct per-person substitutes; 2) using the joint model in Eq.
Revised: (7) to construct per-person substitutes; 2) using the joint model in Eq.

Original: (9) to construct per-person substitutes.
Revised: (9) to construct per-person substitutes.

Original: The factor models are both fit using five components based on model checking results in Section 7.3.
Revised: The factor models are both fit using five components based on model checking results in Section 7.3.

Original: Both variants adjust for per-person and per-item confounders.
Revised: Both variants adjust for per-person and per-item confounders.

Original: We refer them as PIF-Net and PIF-Joint respectively.
Revised: We refer them as PIF-Net and PIF-Joint respectively.

Original: Next, we study two methods that are closely related to those proposed in Shalizi and McFowland
Revised: Next, we study two methods that are closely related to those proposed in Shalizi and McFowland

Original: III (2016) and Chaney et al. (2015).
Revised: III (2016) and Chaney et al. (2015).

Original: The first is a modified version of PIF that does not adjust for per-item substitutes in the Poisson model of today’s purchases in Eq.
Revised: The first is a modified version of PIF that does not adjust for per-item substitutes in the Poisson model of today’s purchases in Eq.

Original: We refer to this method as Network-Only.
Revised: We refer to this method as Network-Only.

Original: The idea is to only adjust for per-person confounders but not per-item ones.
Revised: The idea is to only adjust for per-person confounders but not per-item ones.

Original: This is similar to the algorithm proposed by Shalizi and McFowland III (2016).
Revised: This is similar to the algorithm proposed by Shalizi and McFowland III (2016).

Original: The second method relates to Social Poisson Factorization (SPF) (Chaney et al., 2015),
Revised: The second method relates to Social Poisson Factorization (SPF) (Chaney et al., 2015),

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: To make a fair comparison to PIF
Revised: To make a fair comparison to PIF

Original: [Table]
Revised: [Table]

Original: we modify SPF to condition on yesterday’s purchases x k and fit per-person parameters β j , referring to it as modified SPF (mSPF).
Revised: we modify SPF to condition on yesterday’s purchases x k and fit per-person parameters β j , referring to it as modified SPF (mSPF).

Original: In contrast to PIF, SPF is fit jointly
Revised: In contrast to PIF, SPF is fit jointly

Original: [Table]
Revised: [Table]

Original: without inferring the per-person variables z i from a community model.
Revised: without inferring the per-person variables z i from a community model.

Original: This means that mSPF may not adjust for all the information in both the per-person and per-item confounders.
Revised: This means that mSPF may not adjust for all the information in both the per-person and per-item confounders.

Original: Finally, we analyze the data without adjusting for any confounding by fitting Eq.
Revised: Finally, we analyze the data without adjusting for any confounding by fitting Eq.

Original: (11) with only the final social influence term.
Revised: (11) with only the final social influence term.

Original: We refer to this method as Unadjusted.
Revised: We refer to this method as Unadjusted.

Original: Crucially, all the compared methods only differ in the degree to which they adjust for confounding; other modeling choices are held fixed, allowing for a direct comparison of the methods for estimating influence.
Revised: Crucially, all the compared methods only differ in the degree to which they adjust for confounding; other modeling choices are held fixed, allowing for a direct comparison of the methods for estimating influence.

Original: Experiment.
Revised: Experiment.

Original: The goal of the empirical study is to induce varying amounts of confounding (low, medium, and high), and evaluate the accuracy of the compared methods for estimating social influence.
Revised: The goal of the empirical study is to induce varying amounts of confounding (low, medium, and high), and evaluate the accuracy of the compared methods for estimating social influence.

Original: We report the mean squared error (MSE) compared to the ground truth social influence values that we simulated.
Revised: We report the mean squared error (MSE) compared to the ground truth social influence values that we simulated.

Original: There are two potential sources of confounding: per-person variables that drive both connections and purchases, and per-item variables that drive purchases.
Revised: There are two potential sources of confounding: per-person variables that drive both connections and purchases, and per-item variables that drive purchases.

Original: In the simulated dataset, the effect of per-person confounders on purchases can be controlled by making users prefer items from their region more strongly than those from other regions.
Revised: In the simulated dataset, the effect of per-person confounders on purchases can be controlled by making users prefer items from their region more strongly than those from other regions.

Original: The effect of per-item confounders on purchases can similarly be controlled by making users prefer items from their randomly chosen group (categorical variable) than those from other groups.
Revised: The effect of per-item confounders on purchases can similarly be controlled by making users prefer items from their randomly chosen group (categorical variable) than those from other groups.

Original: We separately study confounding in three settings: 1) only from per-person confounders (Homophily), 2) only from per-item confounders (Item), and 3) from both types of confounders (Both).
Revised: We separately study confounding in three settings: 1) only from per-person confounders (Homophily), 2) only from per-item confounders (Item), and 3) from both types of confounders (Both).

Original: Section 7. describes the full details of this experiment.
Revised: Section 7. describes the full details of this experiment.

Original: Table 1 summarizes the MSE across these settings.
Revised: Table 1 summarizes the MSE across these settings.

Original: PIF variants are the most accurate at recovering influence in all cases.
Revised: PIF variants are the most accurate at recovering influence in all cases.

Original: The poor performance of Unadjusted overall demonstrates the harms of confounding bias.
Revised: The poor performance of Unadjusted overall demonstrates the harms of confounding bias.

Original: The mSPF method has higher variance and is less accurate than PIF in all cases, suggesting that the two-stage approach used by PIF to recover substitutes and then perform adjustment is important for empirical performance.
Revised: The mSPF method has higher variance and is less accurate than PIF in all cases, suggesting that the two-stage approach used by PIF to recover substitutes and then perform adjustment is important for empirical performance.

Original: Among the PIF variants, PIF-Joint performs best, suggesting that joint inference of the signal from item purchases together with network in- formation, is useful when constructing substitute confounders and leads to more accurate influence estimates.
Revised: Among the PIF variants, PIF-Joint performs best, suggesting that joint inference of the signal from item purchases together with network in- formation, is useful when constructing substitute confounders and leads to more accurate influence estimates.

Original: This strong empirical performance suggests that the PIF method is capable of effectively combine the information contained in the network and in the purchases to disentangle the effects of homophily and item attributes in measuring influence between users.
Revised: This strong empirical performance suggests that the PIF method is capable of effectively combine the information contained in the network and in the purchases to disentangle the effects of homophily and item attributes in measuring influence between users.

Original: We report the average
Revised: We report the average

Original: ( × consider simulation
Revised: ( × consider simulation

Original: (“Both”
Revised: (“Both”

Original: Table
Revised: Table

Original: [Table]
Revised: [Table]

Original: Bias in a zero-influence setting.
Revised: Bias in a zero-influence setting.

Original: To further study the estimation error of the compared methods, we consider semi-synthetic datasets drawn from the “Both” setting described in the previous experiment (in Table 1) but with all ground truth influence values set to 0, i.e., β j = 0 for each Pokec user j .
Revised: To further study the estimation error of the compared methods, we consider semi-synthetic datasets drawn from the “Both” setting described in the previous experiment (in Table 1) but with all ground truth influence values set to 0, i.e., β j = 0 for each Pokec user j .

Original: Table 2 summarizes the MSE ( × 10 5 ) of estimating influence averaged overruns.
Revised: Table 2 summarizes the MSE ( × 10 5 ) of estimating influence averaged overruns.

Original: Although all methods demonstrate some estimation error in this setting, PIF has the lowest MSE across low, medium and high degrees of confounding.
Revised: Although all methods demonstrate some estimation error in this setting, PIF has the lowest MSE across low, medium and high degrees of confounding.

Original: 4.2. Real-world study
Revised: 4.2. Real-world study

Original: Having shown how the model works on semisynthetic data, we now showcase potential applications on real datasets.
Revised: Having shown how the model works on semisynthetic data, we now showcase potential applications on real datasets.

Original: We applied PIF and related methods to a real social network and conducted exploratory analysis.
Revised: We applied PIF and related methods to a real social network and conducted exploratory analysis.

Original: We used data collected from the song-sharing platform Last.fm Sharma et al.
Revised: We used data collected from the song-sharing platform Last.fm Sharma et al.

Original: The complete dataset consists of about 100k users and 4m songs that users listen to, with the listening activities timestamped.
Revised: The complete dataset consists of about 100k users and 4m songs that users listen to, with the listening activities timestamped.

Original: The platform allows users to form social connections.
Revised: The platform allows users to form social connections.

Original: We processed the data by filtering down to the 10k users that had the most network ties (in- and out-edges).
Revised: We processed the data by filtering down to the 10k users that had the most network ties (in- and out-edges).

Original: For these users, we collected the 6k most frequently listened to songs and divided the timestamps into three time periods (bounded above by the first and third quartiles, respectively).
Revised: For these users, we collected the 6k most frequently listened to songs and divided the timestamps into three time periods (bounded above by the first and third quartiles, respectively).

Original: We discarded users that did not listen to any of selected songs.
Revised: We discarded users that did not listen to any of selected songs.

Original: We were left with a dataset of approximately 4k users and 4k songs.
Revised: We were left with a dataset of approximately 4k users and 4k songs.

Original: We considered activities from the first time period as the matrix x , those from second time period as the matrix y and reserved those from the third period to perform hold out evaluation.
Revised: We considered activities from the first time period as the matrix x , those from second time period as the matrix y and reserved those from the third period to perform hold out evaluation.

Original: We stress that it is only possible to conduct an exploratory data analysis with this data since there are no ground truth values of influence for the Last.fm users.
Revised: We stress that it is only possible to conduct an exploratory data analysis with this data since there are no ground truth values of influence for the Last.fm users.

Original: Our strategy will be to evaluate how well the methods can predict the held-out data, and examine the average influence across users determined by each method.
Revised: Our strategy will be to evaluate how well the methods can predict the held-out data, and examine the average influence across users determined by each method.

Original: We might hypothesize that a model that captures the causal process of the data will generalize better to held-out data from an altogether new time period.
Revised: We might hypothesize that a model that captures the causal process of the data will generalize better to held-out data from an altogether new time period.

Original: Results.
Revised: Results.

Original: Table 3 summarizes the findings from the real-world study.
Revised: Table 3 summarizes the findings from the real-world study.

Original: First, we study the average influence across all users that were found by each method.
Revised: First, we study the average influence across all users that were found by each method.

Original: The average influence recovered by the
Revised: The average influence recovered by the

Original: Unadjusted method reveals that there are correlated patterns of song-listening behavior between friends.
Revised: Unadjusted method reveals that there are correlated patterns of song-listening behavior between friends.

Original: However, the Unadjusted simply attributes all correlated behavior between friends to the influence of friends’ listening patterns from the previous time period.
Revised: However, the Unadjusted simply attributes all correlated behavior between friends to the influence of friends’ listening patterns from the previous time period.

Original: As expected, the remaining methods, which perform some form of adjustment, all estimate the average influence across users in the network to be lower.
Revised: As expected, the remaining methods, which perform some form of adjustment, all estimate the average influence across users in the network to be lower.

Original: We see that the estimated average influence from PIF is almost an order of magnitude smaller than reported by the Unadjusted method.
Revised: We see that the estimated average influence from PIF is almost an order of magnitude smaller than reported by the Unadjusted method.

Original: [Table]
Revised: [Table]

Original: Next, we consider the average
Revised: Next, we consider the average

Original: Poisson log likelihood of the held-out data under the fitted models (higher is better).
Revised: Poisson log likelihood of the held-out data under the fitted models (higher is better).

Original: As a baseline, we implemented a Poisson model that, for each held-out listen y ik , predicts with the Poisson rate 1 m i , where m i is the number of songs that user i listened to in the previous time period.
Revised: As a baseline, we implemented a Poisson model that, for each held-out listen y ik , predicts with the Poisson rate 1 m i , where m i is the number of songs that user i listened to in the previous time period.

Original: The held-out data has an average Poisson log likelihood of -317.8 under this baseline model.
Revised: The held-out data has an average Poisson log likelihood of -317.8 under this baseline model.

Original: Table 3 shows that most of the compared methods perform better than the baseline, but PIF achieves the highest held-out log likelihood.
Revised: Table 3 shows that most of the compared methods perform better than the baseline, but PIF achieves the highest held-out log likelihood.

Original: Finally, we also study the area under the receiver-operator curve (AUC), a classification metric that ranks the predictions based on a score.
Revised: Finally, we also study the area under the receiver-operator curve (AUC), a classification metric that ranks the predictions based on a score.

Original: Here, we use the rate of the fitted Poisson models as a prediction score to compute the AUC for predicting whether a held-out song is listened to by a user or not.
Revised: Here, we use the rate of the fitted Poisson models as a prediction score to compute the AUC for predicting whether a held-out song is listened to by a user or not.

Original: Table 3 shows that PIF achieves the highest AUC in this setting.
Revised: Table 3 shows that PIF achieves the highest AUC in this setting.

Original: 5. Discussion
Revised: 5. Discussion

Original: Poisson influence factorization (PIF) estimates social influence from observed social network data and and data from a behavior of interest, such as users purchasing items.
Revised: Poisson influence factorization (PIF) estimates social influence from observed social network data and and data from a behavior of interest, such as users purchasing items.

Original: PIF uses latent variable models of such data to construct substitutes for unobserved confounders: per-person variables that affect both connections and purchases, and per-item variables that drive purchases.
Revised: PIF uses latent variable models of such data to construct substitutes for unobserved confounders: per-person variables that affect both connections and purchases, and per-item variables that drive purchases.

Original: PIF then uses substitutes as drop-in replacements when estimating social influence.
Revised: PIF then uses substitutes as drop-in replacements when estimating social influence.

Original: We demonstrated the accuracy of PIF for estimating social influence on semi-synthetic datasets.
Revised: We demonstrated the accuracy of PIF for estimating social influence on semi-synthetic datasets.

Original: In Section 7.6, we analyzed the sensitivity of PIF to assumption violations.
Revised: In Section 7.6, we analyzed the sensitivity of PIF to assumption violations.

Original: Finally, we applied PIF to study real data from Last.fm.
Revised: Finally, we applied PIF to study real data from Last.fm.

Original: Future work.
Revised: Future work.

Original: In this paper, we made the assumption that influence only spreads behavior in a network from one time period to the next.
Revised: In this paper, we made the assumption that influence only spreads behavior in a network from one time period to the next.

Original: One avenue of future work is modeling time-series influence.
Revised: One avenue of future work is modeling time-series influence.

Original: We also focused on Poisson factor models and generalized linear model of outcomes.
Revised: We also focused on Poisson factor models and generalized linear model of outcomes.

Original: Another avenue of future work is developing more flexible models of outcomes, networks and purchases.
Revised: Another avenue of future work is developing more flexible models of outcomes, networks and purchases.

Original: For example, it is worthwhile to study heterogeneous influence effects with outcome models that capture this variation.
Revised: For example, it is worthwhile to study heterogeneous influence effects with outcome models that capture this variation.

Original: Finally, we only studied the estimation error of PIF empirically.
Revised: Finally, we only studied the estimation error of PIF empirically.

Original: It is important future work to establish technical results about estimation with finite samples and potentially mis-measured substitutes.
Revised: It is important future work to establish technical results about estimation with finite samples and potentially mis-measured substitutes.

Original: 6. Acknowledgements
Revised: 6. Acknowledgements

Original: This work is supported by the Simon Foundation, the Sloan Foundation, ONR grants N00014-171-2131 and N00014-15-1-2209, and NSF grant IIS 2127869.
Revised: This work is supported by the Simon Foundation, the Sloan Foundation, ONR grants N00014-171-2131 and N00014-15-1-2209, and NSF grant IIS 2127869.

Original: 7. Appendix 7.1. Substitutes for confounders
Revised: 7. Appendix 7.1. Substitutes for confounders

Original: In this section we will formally state a result about valid substitutes for per-person and per-item confounders.
Revised: In this section we will formally state a result about valid substitutes for per-person and per-item confounders.

Original: Proposition 7.1 Let u j ⊂ { ρ j , z j } be the traits of person j that affect more than one connection in a j .
Revised: Proposition 7.1 Let u j ⊂ { ρ j , z j } be the traits of person j that affect more than one connection in a j .

Original: Suppose there exists a variable c j so that the distribution p ( a j | c j ) factorizes as,
Revised: Suppose there exists a variable c j so that the distribution p ( a j | c j ) factorizes as,

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

Original: for parameters
Revised: for parameters

Original: η a j .
Revised: η a j .

Original: Then, the variable c j contains the information that is contained in the set of user traits u j .
Revised: Then, the variable c j contains the information that is contained in the set of user traits u j .

Original: Let τ k be the attributes of item k that affect more than one purchase in x k .
Revised: Let τ k be the attributes of item k that affect more than one purchase in x k .

Original: Suppose there exists a variable w k so that the distribution p ( x k | w k ) factorizes as,
Revised: Suppose there exists a variable w k so that the distribution p ( x k | w k ) factorizes as,

Original: [Equation]
Revised: [Equation]

Original: 15) for parameters η x k .
Revised: 15) for parameters η x k .

Original: Then, the variable w k contains the information contained in the variable τ k .
Revised: Then, the variable w k contains the information contained in the variable τ k .

Original: Proof Assume that a variable w k which satisfies the factorization above exists.
Revised: Proof Assume that a variable w k which satisfies the factorization above exists.

Original: Then, by definition, each pair of variables ( x ik , x i ′ k ) in the set x k are d-separated by w k .
Revised: Then, by definition, each pair of variables ( x ik , x i ′ k ) in the set x k are d-separated by w k .

Original: Suppose that w k does not contain all the information contained in the confounders τ k , common causes of multiple variables in x k .
Revised: Suppose that w k does not contain all the information contained in the confounders τ k , common causes of multiple variables in x k .

Original: That means, there exists some subset U ⊂ τ k
Revised: That means, there exists some subset U ⊂ τ k

Original: that is not contained in w k but which are parents of at least two variables, ( x ak , x bk ) in x k .
Revised: that is not contained in w k but which are parents of at least two variables, ( x ak , x bk ) in x k .

Original: But, we assumed that w k d-seperates all pairs of variables in x k .
Revised: But, we assumed that w k d-seperates all pairs of variables in x k .

Original: By contradiction, w k contains all the information contained in τ k .
Revised: By contradiction, w k contains all the information contained in τ k .

Original: The same proof applies for the variable c j .
Revised: The same proof applies for the variable c j .

Original: 7.2. Proof of Proposition 3. 
Revised: 7.2. Proof of Proposition 3.

Original: We give the proof for Proposition 3.1.
Revised: We give the proof for Proposition 3.1.

Original: Proof Recall that a i, \ j = a ′ i and x k, \ j = x ′ k .
Revised: Proof Recall that a i, \ j = a ′ i and x k, \ j = x ′ k .

Original: Define ξ ik = E
Revised: Define ξ ik = E

Original: [ y ik | a ij = a, x jk = x, a i, \ j , x k, \ j , ˆ c
Revised: [ y ik | a ij = a, x jk = x, a i, \ j , x k, \ j , ˆ c

Original: i , ˆ w k ] .
Revised: i , ˆ w k ] .

Original: Recall from Eq.
Revised: Recall from Eq.

Original: [Equation]
Revised: [Equation]

Original: Consider the Poisson likelihood of today’s purchases y ik in Eq.
Revised: Consider the Poisson likelihood of today’s purchases y ik in Eq.

Original: Using the fact that E [ Y ] = λ for a random variable Y ∼ Pois ( λ ) , the fitted rate λ ik for each purchase y
Revised: Using the fact that E [ Y ] = λ for a random variable Y ∼ Pois ( λ ) , the fitted rate λ ik for each purchase y

Original: ik gives us ξ ik ( a, x ) when we plug in the values a and x for a ij and x jk in the expression for λ ik .
Revised: ik gives us ξ ik ( a, x ) when we plug in the values a and x for a ij and x jk in the expression for λ ik .

Original: Substituting λ ik for µ ik ( a, x ) and by assumption that µ ik ( a, x ) =
Revised: Substituting λ ik for µ ik ( a, x ) and by assumption that µ ik ( a, x ) =

Original: ξ ik ( a, x ) , we write
Revised: ξ ik ( a, x ) , we write

Original: [Equation]
Revised: [Equation]

Original: We separated the sum over people l into the term that corresponds to person j plus the remaining terms.
Revised: We separated the sum over people l into the term that corresponds to person j plus the remaining terms.

Original: Finally, substituting the above equation into the definition of ψ j ,
Revised: Finally, substituting the above equation into the definition of ψ j ,

Original: [Equation]
Revised: [Equation]

Original: 7.3. Model Checking.
Revised: 7.3. Model Checking.

Original: Section 3 emphasizes that factor models provide substitute confounders only when the fitted models capture the empirical data.
Revised: Section 3 emphasizes that factor models provide substitute confounders only when the fitted models capture the empirical data.

Original: Thus, we must always perform model checks to license the use of substitute confounders.
Revised: Thus, we must always perform model checks to license the use of substitute confounders.

Original: Here, we perform posterior predictive checks (PPC), one form of model checking.
Revised: Here, we perform posterior predictive checks (PPC), one form of model checking.

Original: We describe the PPC procedure using the factor model of the network in Eq.
Revised: We describe the PPC procedure using the factor model of the network in Eq.

Original: (7) as an example.
Revised: (7) as an example.

Original: Create a matrix of held out data a heldout by randomly holding out network connections for each user.
Revised: Create a matrix of held out data a heldout by randomly holding out network connections for each user.

Original: Create a replicated dataset a rep where for each user
Revised: Create a replicated dataset a rep where for each user

Original: i we draw s samples from the posterior predictive,
Revised: i we draw s samples from the posterior predictive,

Original: [Equation]
Revised: [Equation]

Original: We approximated this by sampling from p ( a rep ij | ¯ z i , ¯ z j ) where ¯ z = E [ p ( z | a )] is the posterior mean calculated from the observed dataset. 3.
Revised: We approximated this by sampling from p ( a rep ij | ¯ z i , ¯ z j ) where ¯ z = E [ p ( z | a )] is the posterior mean calculated from the observed dataset. 3.

Original: For a chosen discrepancy function D ( a ) (a common choice is log likelihood), calculate its value D ( a rep ) on the replicated dataset.
Revised: For a chosen discrepancy function D ( a ) (a common choice is log likelihood), calculate its value D ( a rep ) on the replicated dataset.

Original: Then calculate its value D ( a heldout ) on the held out dataset. 4.
Revised: Then calculate its value D ( a heldout ) on the held out dataset. 4.

Original: The posterior predictive p-value is p ( D ( a rep ) > D ( a heldout )) .
Revised: The posterior predictive p-value is p ( D ( a rep ) > D ( a heldout )) .

Original: It can be estimated empirically by sampling m different replicated datasets and producing the ratio m R m where m R is the number of datasets in which D ( a rep ) > D ( a heldout ) .
Revised: It can be estimated empirically by sampling m different replicated datasets and producing the ratio m R m where m R is the number of datasets in which D ( a rep ) > D ( a heldout ) .

Original: P-values that are close to 0.5 suggest that the model explains the replicated data as well as it explains the heldout data; this is ideal.
Revised: P-values that are close to 0.5 suggest that the model explains the replicated data as well as it explains the heldout data; this is ideal.

Original: [Table]
Revised: [Table]

Original: We performed PPCs on the real social network, and the simulated item purchasing data.
Revised: We performed PPCs on the real social network, and the simulated item purchasing data.

Original: We simulated item purchases using the setting where both homophily and confounding are present, and set all the parameters denoted by s to be 50.
Revised: We simulated item purchases using the setting where both homophily and confounding are present, and set all the parameters denoted by s to be 50.

Original: We fit the Poisson community model Eq.
Revised: We fit the Poisson community model Eq.

Original: (9) and Poisson matrix factorization Eq.
Revised: (9) and Poisson matrix factorization Eq.

Original: (10) to the empirical network and purchasing data.
Revised: (10) to the empirical network and purchasing data.

Original: We varied the number of components used to fit the factor models of item purchases x and the network a across 3 , 5 , 8 and 10.
Revised: We varied the number of components used to fit the factor models of item purchases x and the network a across 3 , 5 , 8 and 10.

Original: We simulated 20 datasets for each setting and averaged the PPC p-values across these experiments.
Revised: We simulated 20 datasets for each setting and averaged the PPC p-values across these experiments.

Original: To estimate the PPC p-values, we used 100 replicated datasets within each experiment.
Revised: To estimate the PPC p-values, we used 100 replicated datasets within each experiment.

Original: Table 4 summarizes the results from these PPCs.
Revised: Table 4 summarizes the results from these PPCs.

Original: It is not surprising that the simulated item purchases are well-explained by Poisson matrix factorization, but interestingly, even the real social network is fit well with a Poisson community model.
Revised: It is not surprising that the simulated item purchases are well-explained by Poisson matrix factorization, but interestingly, even the real social network is fit well with a Poisson community model.

Original: Based on these results, we used 5 components to fit both factor models.
Revised: Based on these results, we used 5 components to fit both factor models.

Original: 7.4. Empirical data description
Revised: 7.4. Empirical data description

Original: Users’ and items’ region are given by the one-hot encoding matrix r .
Revised: Users’ and items’ region are given by the one-hot encoding matrix r .

Original: To simulate other item attributes that do not depend on region, each item and user are associated with a randomly drawn categorical variable, given by the matrix v .
Revised: To simulate other item attributes that do not depend on region, each item and user are associated with a randomly drawn categorical variable, given by the matrix v .

Original: The simulation is,
Revised: The simulation is,

Original: [Equation]
Revised: [Equation]

Original: In words, users have preferences ρ and α based on their region and the other group with which they are associated.
Revised: In words, users have preferences ρ and α based on their region and the other group with which they are associated.

Original: Preferences are mixtures over these groups, with proportions controlled by s ρ and s α .
Revised: Preferences are mixtures over these groups, with proportions controlled by s ρ and s α .

Original: For example, when s ρ is big, users’ preferences ρ are determined mostly by the region to which they belong.
Revised: For example, when s ρ is big, users’ preferences ρ are determined mostly by the region to which they belong.

Original: Item have attributes τ and γ based on which groups of people they appeal to.
Revised: Item have attributes τ and γ based on which groups of people they appeal to.

Original: These attributes are also mixtures over groups, and their proportions are controlled by s τ and s γ .
Revised: These attributes are also mixtures over groups, and their proportions are controlled by s τ and s γ .

Original: We create co-purchasing patterns by simulating influence β and by correlating yesterday and today’s purchases with preferences ( ρ and α ) and attributes ( τ and γ ).
Revised: We create co-purchasing patterns by simulating influence β and by correlating yesterday and today’s purchases with preferences ( ρ and α ) and attributes ( τ and γ ).

Original: This correlation creates confounding due to the item attributes and confounding due to homophily, because a person’s region affects her preferences.
Revised: This correlation creates confounding due to the item attributes and confounding due to homophily, because a person’s region affects her preferences.

Original: We can control the strength of confounding by varying the parameters s .
Revised: We can control the strength of confounding by varying the parameters s .

Original: To simulate varying amounts of confounding, the parameters s ρ and s τ are set to 50 but the parameters s γ and s α are varied across (10 , 50 , 100) , corresponding to low, medium and high amounts of confounding.
Revised: To simulate varying amounts of confounding, the parameters s ρ and s τ are set to 50 but the parameters s γ and s α are varied across (10 , 50 , 100) , corresponding to low, medium and high amounts of confounding.

Original: To simulate the influence of each user, we sample from a sparse Gamma distribution.
Revised: To simulate the influence of each user, we sample from a sparse Gamma distribution.

Original: The outcome model given in Eq.
Revised: The outcome model given in Eq.

Original: (11) places a Gamma prior of Gam(0.01, 10) on the variables α and τ , and a Gamma prior of Gam(0.1, 0.1) on the influence variable β .
Revised: (11) places a Gamma prior of Gam(0.01, 10) on the variables α and τ , and a Gamma prior of Gam(0.1, 0.1) on the influence variable β .

Original: We noted that some users did not have any friends in the social network.
Revised: We noted that some users did not have any friends in the social network.

Original: For these users, there is no data for PIF to learn influence and so we omitted such users.
Revised: For these users, there is no data for PIF to learn influence and so we omitted such users.

Original: All experiments were run on a CPU only.
Revised: All experiments were run on a CPU only.

Original: Each single experiment with 3k users and items took less than 10 seconds on average to complete.
Revised: Each single experiment with 3k users and items took less than 10 seconds on average to complete.

Original: 7.5. Variational updates for PIF
Revised: 7.5. Variational updates for PIF

Original: The PIF method is fitted using mean-field variational inference.
Revised: The PIF method is fitted using mean-field variational inference.

Original: Given an observed matrix of today’s item purchases, y , we would like to calculate the posterior distribution of the user preferences α , item attributes γ and user influence β , p ( α, γ, β | y ) .
Revised: Given an observed matrix of today’s item purchases, y , we would like to calculate the posterior distribution of the user preferences α , item attributes γ and user influence β , p ( α, γ, β | y ) .

Original: Variational inference approximates the posterior distribution using optimization: it finds a variational distribution of the latent variables that is closest in Kullback-Liebler (KL) divergence to the true posterior distribution.
Revised: Variational inference approximates the posterior distribution using optimization: it finds a variational distribution of the latent variables that is closest in Kullback-Liebler (KL) divergence to the true posterior distribution.

Original: Mean-field variational inference uses a fully factorized variational family of distributions.
Revised: Mean-field variational inference uses a fully factorized variational family of distributions.

Original: The optimal variational distribution for each latent variable depends on its complete conditional, the distribution of a single latent variable conditioned on all other latent and observed variables.
Revised: The optimal variational distribution for each latent variable depends on its complete conditional, the distribution of a single latent variable conditioned on all other latent and observed variables.

Original: If a model is conditionally conjugate, i.e., the posterior distribution of the latent variables is in the same family as the prior distribution of latent variables, then complete conditionals are in the exponential family.
Revised: If a model is conditionally conjugate, i.e., the posterior distribution of the latent variables is in the same family as the prior distribution of latent variables, then complete conditionals are in the exponential family.

Original: This leads to coordinate ascent algorithms where we update the variational parameters for each latent variable in turn, holding the others fixed
Revised: This leads to coordinate ascent algorithms where we update the variational parameters for each latent variable in turn, holding the others fixed

Original: [Equation]
Revised: [Equation]

Original: To make the outcome model for PIF conditionally conjugate
Revised: To make the outcome model for PIF conditionally conjugate

Original: [Equation]
Revised: [Equation]

Original: we introduce auxiliary latent variables, ψ ikl , ξ ikp and τ ikq ,
Revised: we introduce auxiliary latent variables, ψ ikl , ξ ikp and τ ikq ,

Original: [Equation]
Revised: [Equation]

Original: With these auxiliary variables in hand
Revised: With these auxiliary variables in hand

Original: [Equation]
Revised: [Equation]

Original: the complete conditional for each latent variable is,
Revised: the complete conditional for each latent variable is,

Original: [Equation]
Revised: [Equation]

Original: where the scalar values a, b, c and d are prior shape and rate parameters for the corresponding latent variables.
Revised: where the scalar values a, b, c and d are prior shape and rate parameters for the corresponding latent variables.

Original: The variational distribution for each latent variable is in the same family as its complete conditional.
Revised: The variational distribution for each latent variable is in the same family as its complete conditional.

Original: Let κ αip , κ γkq , κ βl be the shape parameter of the variational distribution for each corresponding latent variable, and let ν αip ,
Revised: Let κ αip , κ γkq , κ βl be the shape parameter of the variational distribution for each corresponding latent variable, and let ν αip ,

Original: ν γkq , ν βl be the rates defined in the same way.
Revised: ν γkq , ν βl be the rates defined in the same way.

Original: Let ϕ τ
Revised: Let ϕ τ

Original: ik , ϕ ξik , ϕ ψik be the variational parameters for the auxiliary latent variables.
Revised: ik , ϕ ξik , ϕ ψik be the variational parameters for the auxiliary latent variables.

Original: Note that the rate parameter of each complete conditional for the latent variables α, γ and β involves variables that are observed (from the first stage of fitting substitute confounders).
Revised: Note that the rate parameter of each complete conditional for the latent variables α, γ and β involves variables that are observed (from the first stage of fitting substitute confounders).

Original: As such, we can set the rate variational parameters for these latent variables without the need for updates,
Revised: As such, we can set the rate variational parameters for these latent variables without the need for updates,

Original: [Equation]
Revised: [Equation]

Original: The coordinate ascent algorithm applies the following updates to the remaining variational parameters in each iteration,
Revised: The coordinate ascent algorithm applies the following updates to the remaining variational parameters in each iteration,

Original: [Equation]
Revised: [Equation]

Original: where Ψ ( · ) is the digamma function.
Revised: where Ψ ( · ) is the digamma function.

Original: 7.6. Sensitivity analysis
Revised: 7.6. Sensitivity analysis

Original: [Figure]
Revised: [Figure]

Original: How sensitive are PIF’s results to its assumptions being violated?
Revised: How sensitive are PIF’s results to its assumptions being violated?

Original: PIF relies on the constructed per-item and per-user substitute confounders to adjust for confounding effects when estimating influence.
Revised: PIF relies on the constructed per-item and per-user substitute confounders to adjust for confounding effects when estimating influence.

Original: Substitute confounders produced by factor models can only capture variables that affect multiple interactions, e.g., a latent item attribute that affects multiple purchases that a user makes.
Revised: Substitute confounders produced by factor models can only capture variables that affect multiple interactions, e.g., a latent item attribute that affects multiple purchases that a user makes.

Original: We evaluate the accuracy of PIF’s influence estimates as this assumption about substitutes is increasingly violated.
Revised: We evaluate the accuracy of PIF’s influence estimates as this assumption about substitutes is increasingly violated.

Original: We construct assumption violations in semi-simulated datasets.
Revised: We construct assumption violations in semi-simulated datasets.

Original: We randomly choose 30% of all friends to share a preference for about 1k randomly chosen items.
Revised: We randomly choose 30% of all friends to share a preference for about 1k randomly chosen items.

Original: Notice that because friends share a preference for items in yesterday and today’s purchases, it will appear as though a user influences her friend.
Revised: Notice that because friends share a preference for items in yesterday and today’s purchases, it will appear as though a user influences her friend.

Original: Moreover, because they randomly select items to like, the preference is not shared across multiple purchases of a user, violating the assumption for substitute confounders.
Revised: Moreover, because they randomly select items to like, the preference is not shared across multiple purchases of a user, violating the assumption for substitute confounders.

Original: Factor models fit to these purchases cannot recover these random preferences.
Revised: Factor models fit to these purchases cannot recover these random preferences.

Original: Findings.
Revised: Findings.

Original: We increase the strength of friends’ random preferences for items relative to their other preferences (based on region and other covariates).
Revised: We increase the strength of friends’ random preferences for items relative to their other preferences (based on region and other covariates).

Original: As the relative strength increases, the substitute confounder assumptions are increasingly violated.
Revised: As the relative strength increases, the substitute confounder assumptions are increasingly violated.

Original: Figure 3 shows the MSE of PIF’s influence estimates as this relative strength of preferences varies.
Revised: Figure 3 shows the MSE of PIF’s influence estimates as this relative strength of preferences varies.

Original: The plot shows that PIF’s accuracy only worsens drastically when the assumption violation is dramatic: 30% of friends have shared random preferences for nearly 1k items that are twice as strong as their other preferences.
Revised: The plot shows that PIF’s accuracy only worsens drastically when the assumption violation is dramatic: 30% of friends have shared random preferences for nearly 1k items that are twice as strong as their other preferences.

Original: Analysts can assess if this degree of assumption violation is realistic or not when applying PIF.
Revised: Analysts can assess if this degree of assumption violation is realistic or not when applying PIF.