Prior Specification for Exposure-based Bayesian Matrix Factorization
Authors: Zicong Zhu
Issei Sato
Venue: Accepted by TMLR ()

Abstract: The rapid development of the Internet has resulted in a surge of information, particularly with the rise of recommender systems (RSs). One of the most significant challenges facing existing RS models is data sparsity. To address problems related to sparse data, Bayesian models have been applied to RS systems because of their effectiveness with small sample sizes. However, the performance of Bayesian models is heavily influenced by the choice of prior distributions and hyperparameters. Recent research has introduced an analytical method for specifying prior distributions in generic Bayesian models. The major concept is a statistical technique called Prior Predictive Matching~(PPM), which optimizes hyperparameters by aligning virtual statistics generated by the prior with observed data. This approach aims to reduce the need for repeated and costly posterior inference and enhance overall Bayesian model performance. However, our evaluation of this theoretical method reveals considerable deviations in prior specification estimates as data sparsity increases. In this study, we present an enhanced method for specifying priors in Bayesian matrix factorization models. We improve the estimators by implementing an exposure-based model to better simulate data scarcity. Our method demonstrates significant accuracy improvements in hyperparameter estimation during synthetic experiments. We also explore the feasibility of applying this method to real-world datasets and provide insights into how the model's behavior adapts to varying levels of data sparsity.

Published in Transactions on Machine Learning Research (10/2025)
Prior Specification for Exposure-based Bayesian Matrix Fac- torization
Zicong Zhu
achhu@is...u-tokgo..o
Deportment of Computer Science The University of Tokyo
Issei Sato
vato@gg.ccc.--tokyo.
Department Computer Science The University of Tokyo
Reviewed on OpenReview: openrevvee. net/ forum? id= o6R4Hu9X9C
Abstract
The rapid development of the Internet has resulted in surge of information, particularly with the rise of recommender systems (RSs). One of the most significant challenges facing existing RS models is data sparsity. To address problenus related to sparse data, Bayesian models have been applied to RS systems because of their effectiveness with small sample sizes. However, the performance of Bayesian models heavily influenced by the choice of prior distributions and hyperparameters. Recent research has introduced an analytical method for specifying prior distributions in generic Bayesian models. The major concept is statistical technique called Prior Predictive Matching (PPM), which optimizes hyper- parameters by aligning virtual statistics generated by the prior with observed data. This approach aims to reduce the need for repeated and costly posterior inference and enhance overall Bayesian model performance. However, our evaluation this theoretical method reveals considerable deviations prior specification estimates as data sparsity increases. In this study, we present an enhanced method for specifying priors in Bayesian matrix factor- ization models. We improve the estimators by implementing exposure-based model to better simulate data scarcity. Our method demonstrates significant accuracy improvements in hyperparameter estimation during synthetic experiments. We also explore the feasibility of applying this method to real-world datasets and provide insights into how the model's behavior adapts to varying levels of data sparsity.
1 Introduction
Recommender systems are defined as information filtering and retrieval systems (Belkin & Crofi, 1992 playing a significant role in our daily lives. The goal of these systemss is to make accurate and satisfactory recommendations for users based on a limited number of observations. These observations typically consist of interactions between users and items, such as user ratings and item clicks. Over the past few decades, latent factor models have had a profound impact on the field of recommender systems. These models assume that users' preferences are determined by a set of latent factors {Ricci et al.) 2010). The common approach in these models involves representing observed ratings the inner product of user and item factor matrices. Matrix Factorization (MF) refers to group of algorithms widely used to learn these latent factor models by decomposing the obscrved matrix into the product of two matrices with lower dimensions (Koren et al. 2009). Various lypes of MF algorithms have been developed for different scenarios. For instance, Funk (2006) proposed Funk SVD, which demonstrated outstanding performance in the Netflix Prize contest. Paterek (2007) built upon Simon's work by incorporating user and item biases, achieving higher prediction accuracy the same contest. Additionally, Non-negative Matrix Factorization (NMF) has boen applied recommender systems when the non-negativity of the resulting matrices required (Luo et. al. 2014p.

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Due to the increasing demands of handling sparse and imbalanced real-world datasets, Bayesian models have attracted significant attention from researchers. Several novel models have been proposed to address the matrix factorization (MF) problem in a probabilistic manner, including Probabilistic MF (PMF) (Mnih & Salakhutdinov/ 2007), Bayesian Probabilistic MF (BPMF) (Salakhutdinov & Mnih, 2008). and Generalized Probabilistic MF (GPMF) (Shan & Banerjee 2010). Compared classical deterministic models, Bayesian models naturally integrate prior information with observed data within a rigorous framework, enabling them to effectively handle small sample sizes (Van de Schoot al. 2014). Although Bayesian models offer greater flexibility in dealing with sparse and imbalanced data, well as comprehensive expressions of uncertainty and convenient updating rules, proper prior specification remains critical issue that directly affects their performance (da Silva et al.) 2023). Unfortunately, there is no standard statistical method for selecting prior distributions or specifying priors for all unknown hyperparameters. In practical implementations, optimal hyperparameters are typically determined through Bayesian optimization (Snoek et al., 2012) or cooss-aalidttinn (Vehtari et al., 2017). However, these evaluation methods require repcatedly performing computationally expensive posterior inferences across all possible hyperparameler sets. Prior specification offers advantages, but it also faces challenges such computational efficiency and the trade-off between model complexity and the interpretability of latent factors.
Recently, da Silva et al. (2023) proposed a state-of-the-art prior specification method for Bayesian MF based on the idea of prior predictive matching. This method allows for the estimation of appropriate hyperparameters, including the number of latent factors, without requiring redundant posterior inference. In their empirical experiments using synthetic datasets, they found that the relative error between estimated hyperparameters and true settings is quite small when the dataset is nearly dense (with sparsity less than 10%). However, this error grows rapidly sparsity increases. These negative results, heavily affected by data sparsity, have hindered further research and practical applications, most real-world dalasets are more than 99% sparse. We also conducted additional experiments on the real-world datasets (Harper & Konstan, 2015) for the uncontrollable prior sectings and attached the results in Appendix C
In this study, we outline our contributions as follows:
We evaluated a state-of-the-at prior specification method for Bayesian Matrix Factorization (MF) under varying levels of data sparsity and analyzed the potential reasons for inaccurate hyperparam- eter estimation.
We incorporated the concept of the exposure-based model (Liang et al. 2016 to capture the effects of data sparsity and derive improved hyperparameter estimators.
We conducted experiments on synthetic datasets, demonstrating that our new estimators outperform cxisting methods, especially as the dataset bocomes sparser.
We investigated how our estimators respond to changes in data sparsity and analyzed the features of estimators. We discussed the limitations of the conventional Probabilistic Matrix Factorization (PMF) model.
2 Preliminaries
2.1 Bayesian Matrix Factorization
Fundamental notations Let Y E RIx denote the observed interaction matrix with I users and J items, P. E R refer to the latent matrix of the user, and Q E K refer to the latent matrix of the item. Denote by K the number of latent factors, which a hyperparameter for latent factor models and holds the condition that K & min For a particular position with user and item j, P: E K denotes the user latent vector and 9; E K denotes the item latent vector. The corresponding preference score from user to item j is given by the inner product that Yi = p. qj-
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Generic Bayesian MF
A generalized template model for Bayesian MF can be described in the following
form:
Pik F(up;
Uih
K
K
(1)
Yi ~ Fy Σ Pallia o²
with E = Par
Figure 3 shows the model structure of the generic Bayesian MF. The user latent variable of the vector p follows priors ((pp.o)) the item latent variable vector 9j follows priors F(ug. and the hyperparameters can be specified separately. Fy the observation model. The inner product of the latent vectors of the user and the item is longer used directly predictions, but the expectation of the observation model with natural noise The observation model also implies the capability of the Bayesian model to provide results with uncertainty.
Particular selections of paramctric distribution Different selections the prior distribution and the observation model lead to different. Bayesian MF models. We introduce several examples:
Mnih & Salakhutdinov (2007 proposed the classic Probabilistic MF (PMF). This model selects
zero-mean Gaussian priors for latent features and a Gaussian distribution for observations:
F(jp; og)) o.)) N = oz), F(Jy o NN 0, og), Fy =N.
(2)
Both Cemgil (2009) and Gopalan al. 2014 contributed to Poisson MF. This model selects
Gamma priors for latent features and Poisson distribution for observations:
F(u of)) Gamma(a,b), F(ug: o Gamma(c, Fy Poisson.
a/c are the shape parameters and b/d are the rate parameters of Gamma distributions. Therefore have the following equalities:
c Mg d' o de
Up
0
(2008) proposed the Bayesian Probabilistic MF (BPMF). This model selects Gaussian- Wishart priors for the prior distribution F. The hyperparameters ppp Opt Pas dg are controlled by additional (higher-level) priors. This type of model is known hierarchical Bayesian model.
Salakhutdinov & Mnih
2.2 Exposure-base based Model
Miising-Nott-t-Rannee (MNAR) In this research, we treat the term dala sparsily problem as a synonym for the term missing data problem. According to Rubin (1976), missing data problems can classified into three categories: Missing-Completeen Completely-A Completely-At-Randdee At-Random (MCAR), -tssinog-At-aandom (MAR), and (MNAR). For example, da Silva et. al. (2023) address the data sparsity of synthetic dataset by using Bernoulli random variable mask. This typically represents an MNAR case, indicating that the probability of data being missing depends on unknown factors. Additionally, making MNAR assumptions helps us gain insights for modeling data sparsity.
Exposure MF (ExpoMF) The positive-unlabele problem refers to the trivial consideration of missing implicit feodback as a nogative response (Jannach et al.) 2018).. This treatment can be inappropriate because the items are possibly not exposed instead of disliked by the users. Liang al.) (2016) proposed this probabilistie model with exposure variable to solve this problem. The exposure variable modernizes the
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missing pattern of data in not-random manner. The ExpoMF is defined follows:
Latent Factors:
Ps NV ofJ,, qi ~~N(00,f1))
Exposure Mechanism:
Oi ~ Bern((ly).
(4)
Observed Feedback:
Yy =
Yu 10, =0~ 60- NN ~
Yú E [0,1], denotes the implicit feedback, and 8, denotes a point mass 0, for example, Pr(Y;; 0
0) - 1. Ozg the prior probability of exposure controlling the exposure variable Oig by the Bernoulli exposure model.
2.3 Prior Predictive Matching (PPM)
Prior Predictive Matching (PPM) is technique in Bayesian statistics that aims to specify prior distributions that incorporate the predictive features of observed data. The primary objective of PPM is to ensure that the selected prior information aligns with the pallerns anticipated in the data. PPM operates by utilizing the prior predictive distribution to estimate certain meaningful virtual statistics that are inferred from the prior. Subsequently, it learns the optimal hyperparameters by aligning virtual statistics with the target value. In essence, PPM involves an iterative process where the prior distributions are adjusted based on the behavior of observed data to refine the Bayesian model for coherence and informativeness. Researchers apply the PPM method to probabilistic models, which can lead to more accurate inferences and predictions. In the context of PMF models, PPM helps in selecting prior hyperparameters that directly reflect the distributions of the observed data. For instance, da Silva et al. (2023) introduced PPM to set hyperparameters in PMF models, ensuring that the priors are well-grounded in empirical observations. This approach not only enhances the robustness of the model but also improves the interpretability of the results by aligning the prior beliefs with limited observed data.
3 Proposed Method
Most of the PMF models randomly initialize the prior distributions' hyperparameters and repeatedly search for the optimal values. The quality of postcrior inference can be significantly inffuenced by the choice of the prior distribution and its hyperparameters. While Silva el al. (2023) proposed a prior specification method using Prior Predictive Matching (PPM) to estimate the hyperparameters instead of grid search, we observed that the relative error of estimation increases substantially as the data becomes sparser. This is because the statistics observed from sparse datasets are inaccurate without taking into account the data sparsity in the model. Inspired by the concept of exposure introduced by Liang et al. (2016), we integrate the generic PMF with an additional exposure component to simulate the scenario of missing data. The model structure we propose, in contrast to the established exposure- basoedd model detailod in Equation (4). is more generalizod and accommodates varying non-zero prior hyperparameters and non- non-Gaussian observation distributions. This generic model framework beneficial for our further analysis and inference of the estimator closed-form with the PPM method. Consequently, our proposed estimator derived from the exposure-based model is more flexible to handle sparse data and achieves lower relative error in experiments.
3.1 Model Definition
We define an exposure-bass based generic Bayesian MF model that incorporates data sparsity derive improved prior specification estimators:
Pik F(up:
4ik F(ug: o7):
Rij ~ FR Σ Em
with E
Σ
Paljkk k=l
K
(5)
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Then we define an exposure random variable Ogy and the corresponding observation as:
Oig FEapos Yi- = Riy
(6)
The Ot follows the exposure distribution FEapo In this research, the Bernoulli distribution if there is no extra explanation, Og ~ Bern(O0g) The Figure shows the complete structure of the model.
The relationship between variable Yig and Rag.
K 02
YnylO = =1~ Fr.
Σ k=
Yi = Raj.
(7)
13 = ~ d0
P(Y., - 010;; =0)
3.2 Common Assumptions
We describe common assumptions which will be integrated in the specific model as combinations. Assumption (A Referring to Equation assume the exposure variable Oiş is independent of the Irue relevance Raj, then have:
[] = S[Or,Rub) = 0, B R
RR, = 0.j E[y];:
(8)
v[Yy] VlO., = [|o, + E[R,s +[0,|[n))))
V[Ra]] = /lr,] 0z (0.g 1)
(9)
This assumption ensures that the exposure process does not influence the expected value of R. and allows for straightforward factorization in expectation and variance calculations. The independence assumption simplifies statistical analysis by ensuring that O,ş does not introduce additional variability or structure into Raj-
Assumption 2 (A2). With reference to Equation assume the observation model FR is Poisson distri-
bution, then we have:
B[[[Ry) P.Q] = -[]g
(10)
The Poisson assumption is commonly used in count-based models, where the mumber of events follows a discrete stochastic process. This is natural situations where the relevance score is derived from user interactions, such clicks purchase counts, which often exhibit Poisson-like behavior, especially when event occurrences are independent and event rates are approximately constant.
This assumption leverages the key property of Poisson distributions, where the variance equals the mean. By assuming a Poisson observation model, the expected conditional variance of Raj given P and Q is directly linked to its mean. This facilitates inference and simplifies the characterization of uncertainty in Raj-
We also make two assumptions about the relationship between different exposure random variables. Assumption 3 (A3). Assume that random variable Oiş and On follows exactly the same distribution Oij, On Bern((), then we have:
CovlO Ou] Var(O) = 0(1 0).
(11)
This assumption introduces dependence between exposure variables, meaning that observing one exposure event influence each other. In practical situations, it's a natural assumption that exposure variables are correlated with each other, depending on underlying latent factors. For instance, if one item has high exposure probability, the others should have lower probabilities because of the limited total number of allowed exposure events, such as the number of items displayed single web page.
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Assumption 4 (A Assume that random variable Osg and Ou are independent and identically distributed, Oij, On iid Bern(O), then we have:
Cov|Oiy, Oul = 0.
(12)
This assumption introduces independence between exposure variables, meaning that observing exposure event does not influence another. This is common assumption in probabilistic modeling when cach ob- servalion is independently drawn. The independence simplifies variance calculations and removes potential correlations the exposure process, making statistical inference more tractable.
3.3 Model Interpretation
True relevance Our proposed model differs from the original ExpoMF discussed in Section 2.2 We introduce an intermediate variable, Ruj; to distinguish it from the variable Yjj Saito et al. (2020) interprets Yú as observations and Rsy as true relevance, which may reflect actual preferences. Therefore, high-quality recommendations should be based the true relevance Rıy rather than solely observations.
Data sparsity simulation Our proposed model incorporates the idea of data exposure to simulate the data sparsity issues in synthetic datasets. By strategically adjusting the level of exposure, we can create various scenarios that simulate real-world conditions where data may be limited or unbalancodly distributed. This approach enables us to rigorously test the performance of estimators in a wider range sparsity levels. In real-world scenarios, the reasons for missing data are intricate and unpredictable. For instance, users might not provide exposed item ratings because of their personality. While the concept of sparsity simulation is inspired by the exposure-based model, we hypothesize that it possesses remarkable capabilities to handle complex situations beyond the limitations of exposure alone.
Exposure model selection In the Exposure MF framework Liang et al. (016)), the absence of user ratings doesn't necessarily indicate a user's disinterest in a particular item. The likelihood of a rating being missing is signíficantly influenced by the extent to which the item has been exposed the user. [Liang et al.. (2016) proposed
types of exposure models that determine the probability of item exposure. The first model is based on ilem popularity, suggesting that popular ilems The second model incorporates additional information, such as text topics user locations, exposure invariants. In this rescarch, we treat the exposure variable as a Bernoulli variable. In real-world scenarios, we can replace the exposure variable with specific exposure models.
more likely to be exposed users.
3.4 Generic Propositions
In our proposed model, we concentrate on the actual relevance Ry rather than the observed values Yij. We begin by deriving fundamental principles in a general form, which we later use to derive practical results with specific assumptions. The derivation relies on the model definition provided by Equation (5) and the fundamental laws of total expectation, variance, and covariance.
To begin, we establish key statistical properties of the relevance values Raj, including their expected value and variance. This serves as the foundation for understanding how the structure of Rij influences the relationships between different entries.
Proposition 1. For any entry of the true relevance matris Rij E RIxJ the mean and voriance in generic form is given as follows:
B[ayy] = Kpppag
(13)
V(Ray] K((uooy)² + (m,0r)2
P.Q]]-
(14)
These fundamental statistical properties provide insight into how individual entries of Ray behave under the given model assumptions. However, in many applications, understanding the correlation structure between

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different entries is also crucial. We now analyze the prior predictive correlation between pairs of entries. To be clarified, the following p denotes the original correlation coefficients sampled from the dataset. The p denotes the modified p in our different models. The pi and p denote the raw/colum p' respectively. Proposition 2. For any pair of entries Ray and Rui, the prior predictive correlation given as follows:
0,
& j if i #
1,
if i= & j=l;
p(Rij, Ruj =
Pi
K(ugon]
if i= & j j#1; if i j=l.
(15)
K(ug00) V[Rss]
P2
This result. characterizes how the dependencies between different entries of Raj are structured within the model. In particular, it shows that direct dependencies exist when either the row index or the column index is shared, with specific correlation coefficients.
Given this structure, now extend our analysis to the observed values Yúj, which depend not only on Raj but also on the observation process itself. We derive the covariance structure of Yij in the following proposition.
Proposition 3. For any pair of entries Yig and Ya, the generic expansion of the two entries is given as follows:
Cov[Yig, Yu] = Cov[Oi; OnBoal
Ou]] Ray, Ru] [[d Oul B[O,)][[Os]Cor RRij. Ra]. (16)
This result shows how the covariance of the observed values Yij is influenced by both the underlying relevance structure and the observation process.
3.5 Particular Models
We describe two particular models to derive practical results. Refer to Appendix A for proof details. In this stage, we focus the estimator of the number of latent factors.
Model 1 (M
With generic model defined in Section 3. and
we can solve that:
V(Yylp - 0(1 O)E[
(17)
OV(R.)]
And the new estimalor K is wrillen as:
K (1 (pi + (Ro) E[R
E[ ([;;;)
2
(18)
PiP5
Model 2 (M With generic model defined in Section 3.1 and
4
we can solue that:
0
(19)
And the new estimator K written as:
(pi
(5) ((Re) B[Rgg]
E[ v[R.g]
2
K
(20)
PiP
The estimators of K can be represented by five statistics including E[y,), V[Yy), the row correlation coefficients p1/p2 and parameter which sampled from the observation matrix Y.

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4 Experiments
In this section, we reproduce the experiment conducted in the original work (da Silva al. 2023 in advance. To conduct practical experiment, need to determine the prior distribution beforehand. For simplicity and demonstration purposes, we use the probability mass function (PMF) for two reasons: (1) The gamma
prior is conjugate to the Poisson likelihood; (2) The Poisson distribution has the same mean and variance (described in Equation (46
4.1 Hyperparameters specification
Referring to the original work, we consider initializing the hyperparameters with the following specifications (a/c e are the shape parameters and b/d are the rale parameters of Gamma distributions):
Table 1: Hyperparamoters Initialization for Different Specifications
Spec.
a
c
d
Pp
Op
0g
E|
V R
10
10
1
10.0
3.16
10.0
3.16
2500.00
55000.00
D
0.1
0.1
1
0.1
0.32
0.1
0.32
0.25
0.55
0.1
0.1
1.0
1.0
1.0
3.16
25.00
550.00
We conduct the experiments with specifications A, D, and F because they are distinct from each other. The full specification sctup defined by da Silva et al. (2023) described in Table 4 In specification A, matrices P and Q share the same prior paramelers, but their shape parameters are 10 Limes larger than their rale parameters. In specification D, matrices and Q also share the same prior parameters, but their shape parameters are 10 times smaller than their rate parameters. In specification F, the shape parameters are the same for both distributions, but the prior parameters of matrix P are numerically 10 times larger than those of matrix Q.
4.2 Experimental Steps
We first generate the synthetic data with the following 3 steps repeatedly: (1) We sample the matrix P and Q with the prior hyperparameters for particular specifications; (2) We recover the fully dense matrix R by the product of PT and Q; (3) We sample the Bernoulli variables Oi with different sparsity levels and multiply them with each entry of the dense matrix R obtain the sparse observation matrix Y.
Table 2 outlines the three variables we modify to up our experiments. We employ prior specifications A, D, and F, as outlined in Table D We alter the true value of K, representing the number of latent factors, from 25 to 150. Additionally, we change the parameter Pobs of the Bernoulli distribution into three magnitude groups. Group 1, the only setup used in the original work, produces excessively dense observations. To enhance the realism of our simulations, we introduce Groups and 3, which generate more sparse observation matrices Y. Consoquently, we generate 30 records for each setup, resulting in a total of 8,640 data records.
Table 2:
Variables of Experiment Setups
Prior Spec.
K (Num. of Latent Factors)
Pobs. (Parameter of Bernoulli distribution)
Group [1.0, 0.98, 0.96, 0.94, 0.92, 0.90
[A, F]
[25, 50, 75, 100, 125, 150)
Group 2: [0.5, 0.4, 0.3, 0.2, 0.1]
Group 3: [0.05, 0.04, 0.03, 0.02, 0.01]
Then we continue our experiment with the following steps: (4) We sample the statistics from the observation matrix Y including E[Yy), V[Yg] and the row/column correlation coefficients p1 (5) We calculate the
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estimations of K (K) with different models; (6) We measure the model performance by the relative error between the true K and the estimated K K
( %). %).
4.3 Results
The figures presented this section, Figures and 3 illustrate the outcomes of the experiments conducted using the previously proposed method. These figures are quantile plots that represent the relative error between K and K for each model, with Pobs ranging from 0.01 to 1.0. The cross-over points indicate the median values. Due to the large and dispersed results with 95% percentile, we plot with 50% percentile error bars for better visualization. Table 3 presents the median values for each specification, and the lowest absolute values among model comparisons are highlighted. We will state our observations and discuss the results in the next section.
crana
aigeal
Hodel
0.-
6.055 6130
2030
000
010
2.00
Figure 1: Results of Spec. A with 50% percentile values.
onggral
0:30
0003
8.22
2059
0392
0d5
0.50
620
025
942
2.6
0147
0.95
1.50
Asi
Figure 2: Results Spec. with 50% percentile values.
Part
ongiie
335
$2.00 DN)
0:5
1.0
-099
0003
2033 0.00
0.10 0.20
4.5
2.50
Figure 3: Results of Spec. with 50% percentile values.
Table 3: The Median Value of Relative Errors for All Experiments
Puile
0.01
0.05
0.04
0.06
0.5
0.92
I.00
Original
96.17
90.992
77..65
227.66
202.57
47040.06
43297.74
Sper.
Minis
19:56
-75.5
-09.10
-00.8
-39.29
-0.16
Medion Volue
Model 2
45.89
-615
7007
77.29
23553
209.20
065.1
77365
983.1
274034
S007.24
4778.88
2513.0
1242
142,51
.0.16
Original
G1.01
20.30
20.30
41.05
06.02
02.72
02.10
2002
30.70
23.01
10.25
12.2
3.74
4.37
40.1
-1.22
Spoc.
Model
24.55
40.01
20.00
73.85
77.42
92.77
99.02
96.3
33.56
S8
775.32
1264.5
565.97
150.50
50.89
-442
Relatine
Midiel 2
-18.05
-02.8
43.20
-72.26
-74.3
-74.32
-71.0
-7688
-82.55
-76.98
-3.53
+3.09
-3.00
-1.88
-5.25
-1.12
Broor (5)
Original
20.43
10000
11201
45.05
-31.05
208-27
89.22
70.15
43.50
-100
10.00
-1.24
03.55
53.00
-1.00
Midel

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5 Discussions
5.1
Observations of Experiment Results
We evaluate the ratio between our estimated K and the true K:
(1- V R(R.
P1P2
(21)
K
PiPE
E (
(1- (2)) V[. E[Y,]]
Then we separately analyze the following term for the different models.
Model
We can write p* as:
p+ (1---~)
8²
(22)
1+ (1-10--)) (
We denote the denominator + (1 (1- 0-1)
F(y
V/rT as T for simplicity, then Equation 21
can be rewritten as:
K = K (p1 1 - - (1- (p1 + p2)) V(V]
(2 T (p1 + P2)) V
p1p2
(23)
BY
There are restrictions in the above equations to avoid zero denominators:
1. In the formula of p", the root of 7 = 0 always exists. For simplicity of analysis, we assume the term BYY
V as constant once it is sampled. The value of 7 is inversely proportional to the value of 0 with
upper bound of
23 the denominators (pi- (01-1+7) and (p2 (p2-1+7) 1+) are possible to be zero. Because the two terms have equal forms, we use P1 for discuss. The existence of the root of equation (pI - -++) depends the sampled value of P1. We assume the root exists, then the closed form can be written as:
In Equation
0g
(24)
Pm
We use the true expectations and variances for visualization. Because the sampled value of Pı is difficult to predict, we calculate the On according to the theoretical range [-1, 1] of the correlation coefficient.
0.99
10
Spec.
Spec.
0.8
Spec.
Value of Bo
0.53
0.4
0.2
0.1
0.0
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
Value of
Figure 4: The relationship between p and 0, with different specifications
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The Figure 4 shows the variation of the root dn corresponding to different values. Because the valid range of 0g is [0, 1], we discard the invalid values in the graphs. For Spec. A, the value of do remains around for all values. This indicates that Model 1 performs best in Spec. A because the minimal value of its 0g is larger than 0.99. However, for any valid value of fu, there cxists at least one corresponding P causing a zero denominator Spec. D. For Spec. F, when Og smaller than 0.53 in Spec.5, both of the the denominators (p1 + 7) and (p2 + r) are impossible be zero. When On > 0.53, Model performs better in Spec. F than Spec. D because there are two corresponding P values causing zero denominator.
In Equation 230, the last denominator (1 - (01 + p2)) independent of the variable 8.
is possible to be zero, which is V
BY
Refer to the terms in Equation 23 the primary factors affecting the relative error are the variable and particular scenarios that result in denominator. Essentially, the ratio increases as 0 decreases until zero denominator arises. After this, the ratio falls to the limitation as 0 approaches zero, indicating that the k also reaches to the the lowest valid value for estimation.
R
PIP2
V
lim 0
1-
(25)
K
(01 +
BY ক.V
(02
+
3MP
(1- (p) + p2)) - BYY
Model 2
We can write p² as:
p-1p
p¹
(26)
1+ (1 11-10-000 0- BYT
V
We denote the denominator + +(1 (1- 1-)
F(y V/T as T for simplicity, then Equation (1) can be rewritten as:
K (T- (pi p2)) FY ViT
(27)
K (1 (1- (p1 + p2))
Y V
Different from the ratio form of Model 1, Equation 27 only has one denominator independent of variable 0, which means Model 2 is less likely to have a zero denominator.
5.2 Analysis of Correlation Coefficients
The Pearson correlation coefficient, employed in the derivation, ranges from -1 to +1. In the matrix factor- ization problem, we calculate the correlation coefficients between two entrics, specified in Equation ( under four distinct scenarios. Consequently, the general matrix factorization model presented in Equation 5 In the intermediate proof showed in Equa- tion 42 the covariance when i = is that: Cov [pagdjk Ptt = 2ă Because the generic model assumes that the mean values Pp and Pg are constant, the equation E[gu]= B[qu] implics that the covariance matrix is non-negative. However, practice, possible to obtain negative sampled correlation coefficients from sparse datascts. In real-world applications, negative correlation cocfficients are more rcalistic. For exam- ple, in user-item recommendation scenarios, we interpret Pi as the correlation between rating scores from different items for the same user. This conflict between the assumption of constant mean values and the possibility of negative correlation coefficients can lead to ambignous estimation of hyperparametors, such as a negative value for K.
obtain non-negative values for P1 and P2 in accordance with
This section highlights a limitation the generic matrix factorization model: it lacks the ability to capture the inverse relationship between latent features. Naturally, we set a hyper-prior for the hyperparameters Mo and My instead of a constant. For the random variable qyk ~ F(Pg. oz), where F is an unknown distribution and Hy G(2g, 72), where G is also unknown distribution, we assume that both distributions unknown.
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We denote the resulting distribution H, which is compound distribution of F and G. To integrate out the unknown parameter X, we write the probability density function follows:
/
PH (9ja)
pr pr (p
(28)
According to the law of total expectation:
Eu En(μιk] = (Er [g
(29)
If Ma. the mean of F is distributed as Xy, the mean of G, the above equation informs that the unconditional expectation Eh [x[g Ag. and Eslgin = Ag. However, the above covariance remains non-negative in quadratic form equal to o
5.3 Correlated Latent Features
To overcome the limitation of non-negative correlation coefficients in the generic models, we propose an essential solution with the following new assumption: for any valid combinations indered by i,j,4, any two random variables from the same latent feature vector are dependent. Otherwise, they are independent. Assune latent factors follow Multivariate Distribution denoted as Fu:
Pt Fullyp Ep);
9j Fu(Ug: Eg),
(30)
finally, we obtain the closed form of the covariance,
Cov(Rij, Ral = E,
The above notation refers the all-oncs vector. However, the covariance matrix is always positive semi- definite, and the covariance of any two entries always non-negalive. Details of derivation are described in A.7
6 Conclusion
In conclusion, the exploration of Bayesian Matrix Factorization (MF) models highlights the versatility and adaptability of the framework in addressing various data-related challenges, particularly in the context of missing data. The distinct models presented, including conventional Probabilistic MF, Poisson MF, and Bayesian Probabilistic MF, illustrate how different prior choices and observation models can effectively capture the latent patterns and uncertainties inherent in user-item interactions.
The introduction of the Exposure-based Model (FxpoMF) further enriches this discussion by addressing the complexities of data sparsity and the positive-unlabeled problem, providing a nuanced understanding of implicit feedback mechanisms. Adopting an exposure variable allows for a more accurate representation of user preferences and item visibility, enhancing recommendation systems.
The Prior Predictive Matching (PPM) technique serves as a critical tool for ensuring that the prior distribu- tions aligu with observed data patterns, ultimately leading to more robust and interpretable models. This iterative approach emphasizes the importance of grounding Bayesian models in empirical evidence, paving the way for improved predictive performance and insights. Collectively, these advancements contribute to a deeper understanding of Bayesian MF and its applications in modern recommendation systems, setting a solid foundation for future research and development in this field.
Overall, these advancements not only enrich our comprehension of Bayesian MF but also lay a strong groundwork for ongoing innovation within recommendation system research, fostering future developments that could further refine the intersection of data analysis and user experience.
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References
Nicholas J Belkin and W Bruce Croft. Information filtering and information retrieval: Two sides of the same coin? Communications of the ACM, 35(12):29-35 -2:3, 1992.
Ali Taylan Cemgil. Bayesian inference for nonnegative matrix factorisation models. Computational intelli- gence and neuroscience, 2009, 2009.
Eliezer de Souza da Silva, Tomasz Kul, Marcelo Harlmann, Arto Klami, el al. Prior specification for bayesian matrix factorization via prior predictive matching. Journal of Machine Learning Research, 24 (67):1 51, 2023.
Simon Funk. Netflix update: Try this home, 2006.
Prem K Gopalan, Laurent Charlin, and David Blei. Content-based recommendations with poisson factor- ization. Advances in neural information processing systems, 27, 2014.
F Maxwell Harper and Joseph A Konstan. The movielens datasets: History and context. Acm transactions on interactive intelligent systems (tiis), 5(4):1-19, 2015.
Dictmar Jannach, Lukas Lerche, and Markus Zanker. Recommending based on implicit feedback. In Social information access: syslems and technologies, pp. 510 569. Springer, 2018.
Yehuda Koren, Robert Bell, and Chris Volinsky. Matrix factorization techniques for recommender systems. Computer, 42(3):30 37, 2009.
Dawen Liang Laurent Charlin, James Mcinerney, and David M Blei. Modeling user exposure in recommen- dation. In Proceedings of the 25th international conference on World Wide Web, pp. 951-961, 2016.
Xin Luo, Mengehu Zhou, Yunni Xia, and Qingsheng Zhu. An efficient non-negative matrix-factorizan based approach to collaborative filtering for recommender systems. IEEE Transactions on Industrial Informatics, 10(2):1273-1288 2014.
Andriy Mnih and Russ R Salakhutdinov. Probabilistic matrix factorization. Advances in neural information processing systems, 20, 2007.
Arkadiusz Paterek. Improving regularized singular value decomposition for collaborative filtering. In Pro- ceedings of KDD cup and workshop, volume 2007, pp. 5 8, 2007.
Francesco Ricci, Lior Rokach, and Bracha Shapira. Introduction to recommender systems handbook. Tn Recommender systems handbook, pp. 1-35. Springer, 2010.
Donald B Rubin. Inference and missing data. Biometrika, 63(3))581-58) 1976.
Yuta Saito, Suguru Yaginuma, Yuta Nishino, Hayato Sakata, and Kazuhide Nakata. Unbiased recommender learning from missing--ol-all-aaddon implicit feedback. In Proceedings of the 13th International Conference on Web Search and Data Mining, pp. 501-500000000 509, 2020.
Ruslan Salakhutdinov and Andriy Mnih. Bayesian probabilistic matrix factorization using markov chain monte carlo. In Proceedings of the 25th international conference on Machine learning, pp. 880-881 2008,
Hanhuai Shan and Arindam Banerjec. Generalized probabilistic matrix factorizations for collaborative fil- tering. In 2010 IEEE international conference on dala mining, pp. 1025-1032 1030. IEEE, 2010.
Jasper Snock, Hugo Larochelle, and Ryan P Adams. Practical bayesian optimization of machine learning algorithms. Advances in neural information processing systems, 25, 2012.
Rens Van de Schoot, David Kaplan, Jaap Denissen, Jens B Asendorpf, Franz Neyer, and Marcel AG Van Aken. gentle introduction to bayesian analysis: Applications to developmental research. Child development, 85(3):842 860, 2014.
Aki Vehtari, Andrew Gelman, and Jonah Gabry. Practical bayesian model evaluation using leave-one-out cross-vale validation and waic. Statistics and computing, 27:1413 1432, 2017.
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A Proof
A.1 Preliminaries
In this section, we introduce the preliminaries for proof including the Law of Total Expectation, the Law of Total Variance, and the Law of Total Covariance, For two random variable X and Y, we know that:
B|Y] = ||
(31)
V[Y] EVV X + A[i[
(32)
CoulX [X, Y] = [E[Cou [X, Y|| + [,1
(33)
E[X - V[X] + E[X]
(34)
V
Σ k
X
v(X + Σ Cou
(35)
If and are independent:
V[XY] = EXX
(36)
=
(37)
For four random variable A, B, X and Y:
Cov[AX, BY] Cov(A, (1 Y] [[x])E B).
(38)
A.2 Intermediate propositions
Proposition 4. For any valid combinalions indered by i,j,h,, if the latent. indexes & + k', then Cookpaagj. P 0.
Proof. By definition of covariance:
Cov|pa- Pix' 9 BppsagppP me/ E (ppagnBBBpp 4
Bocause when + k,, B[parajsPP = E| ElP Iwc then we have Coc Pik Puk qu/ = 0.
Proposition 5. For any valid combinations indezed by i,j,t, the following equations holds:
E0
Piaqja] KHpHa
(39)
Pek K((upoy)² + ((g00)²² ()
(40)
Cov
Σ Pikqyke
Σ
PeR
(41)
Proof.
Because in model, Pak and qzk independent, then E Purqgk] E. Eppu E[gat]
Kup)'gi
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Referring to Equation
35 we know that:
V
Pin
Σ Vipagal +
Σ
PorlpnsiissPos 9uc1
And referring to Proposition we have:
we know that the second term of above equation will be zero. Then
V Σ Pı
Σ
V[puqul.
Because Pik and dis independent, recall the Equation
37
know that:
Vipagall |Bppu] ( + [Vpix] + V[pa]Vlaal
(upoa)² + (2 + (o,or)?
Thus we obtain:
V
Σ
Pak
= k((ppog)) (2g00)2 + (0p0g)²1:
Recall the definition of covariance:
Cov
Σ Epagghh Σ PL
E
Σ Ps QaReeeP Onk
E Σ Pik E
Σ Pexe Yer
E Ei 9
Elpra E I9gn Pew [E]a
k,k
Covlpu-0sk-Pe' Per Iw/
Referring to Proposition all the terms with different k are zero, thus:
Cov Σ Pa
(42)
Then for the same latent index k, there will be four different cases to calculate Coufpieejs, Peaqul: 1. if it & j + due the independence of these variables, we know that
Corlpuaj = 0;
2. if i= & j+l:
Covlpa Peasul = (wwppacheeePPasu
E Elpiag
R Bppa. PTE [q [[[ux]
E Elpa. EE |
E E(qu
Elpes (E(p)) [1(
E | (Eppa]
E[pax
=E [E]gn]E[qqua [ppa]
3. if i i & j= similarly to above steps, obtain
Coulpusht, PIR 1
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4. if i=t & j= referring to proof in last steps:
Cov(pud3ke Pusuul = Cor(puekk Pugy,k
V(puqgak
(ppog)² + (Mg0p)² + (opog)²
For simplicity of presentation, the four cases can be written in one closed form with Kronecker delta (if i = j, dig = 1. Otherwise, dij = 0):
Covlpaqj.. Puaqu] dzo(u)a)) + )zu(1)0p)² + )
Finally, the last equation of this proposition holds:
Cov Σ Pak
E Poaau K(ōu(up0a) + +
A.3 Expected values and variance of true relevance
Now we calculate the prior predictive expected value and variance of true relevance Rgg. This part of model definition is described in Equation (5) And this part of proof corresponds to Proposition ]
Proposition 6. For any valid combinations indered by i,j.t, 1, the following equations holds:
Rayl K KppHa
(43)
V Rey, = K((poo)) + (# (o,0,) P.Q.I.
(44)
Specifically for the PMF model described Equation 3 the variance of relevance can simplified as:
v[R,y] k((ppog)) (Mg0p)2 + (opo,)² uppyl.
(45)
Proof.
Because of the law of total expectation and results in Equation (39
Rayl
E
E
Rij
Σ
Pagsk
k
R
Pik
K Kpllai
Because of the law of lotal variance (Equation
32
have:
V[R
=
E
R
Pik Ojk
V
R
Σ Piudik
k
E
W
Rig
PiR
+ V
K
Denote the latent parameters as P, Qu we finally obtain:
V(Ryj k((u))a)) (2 + [P.Q]:
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For experiments, should determine the specific prior distribution. In the PMF model with Poisson distribution as observations, we have that:
V
Raj
Σ
Pik
= E
R
Σ
PiR
=
>
= HpH4
(46)
Then we obtain the particular form for PMF:
V(Ray = k((pp0g) + (1g0p)2 + (0pog)² 2 HyHpy.
A.4 Covariance and correlation of true relevance
Now we calculate the covariance and correlation for the prior predictive distribution of true relevance Ri-- This part of the model definition is described in Equation And this part of the proof corresponds to Proposition 2
Proposition 7. The prior predictive connivance of true relevance for the generic model is given by:
Cov(Rij. Ral 6,o, ((ō + 6 )u051(0700)))
(47)
In the PMF model with Poisson distribution observations, obtain the particular form of covoriance
Cov(Rig, Ra] K (0 6z((u0p)) ) + HpHlg]]
(48)
Proof.
Recall the total law of covariance (Equation
33
Cov[Rig, Ra] = E[Cou[ RRy. Ra le, tb Pt. ai] + (Riyle 4il, B[Rudpe a!]
(Reylop 4ill + Cov
Σ
Pisdliks Σ PeR-u
K
= BV Q] + (o0n(1,, + )5((110p)) + )a03((0,0))
Referring Lo Equation
46
we know that for Poisson distribution B[V[R, Q] B[Rgg) Therefore
Equation 48 holds.
Proposition 8. The correlation of true relevance with four different cases:
0,
i t & j
i=t & j=1
B[aaj Ru]
K(ngop)²
(49)
P
i=t & 5
)(
15
V[Rss
i t & j=1
o and p refer to the correlation of true relevance R, in comparison with pi and p2 that refer to observation Y.
Proof. We know the relationship between the covariance and correlation R( Ra] Roy and we [VRR,)](Ra)
analyze the four different cases that similar to Proposition 5 the above proposition holds:
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1. if itt & j due to the independence of these variables, we know that:
p| Rul = 0;
2. if == & j=
R.R, Raj = p Rij. R,yl Cov(Rig. Ray 1;
VV[R,yy
3. if ==t t & j#l: this case indicates the row correlation
Cov Rag, Ral
k(
p|R,y, Ral = Pl Rul
(
V(Ris]
4. is itl & j= 1: this case indicates the column correlation
RR., Ral = p[Rsy. Ray = Cou( Rajl
k(up0,))
V
V(Ray]
A.5 Determine the hyperparameters without exposure
In this section, we show how to determine the hyperparameters for generic model (Equation 5 without exposure. In this case, assume that no entries in the relevance matrix R missing. Proposition 9. Assume that the relevance matriz R is dense. Given that we know E[Rg V(Rgg of and p2. we can obtain the closed form for the true number of latent factor K denoted as K* as:
K (1 (1- + -
E[R
(50)
0905
V (.,
Proof. In Proposition & K(Hpo,))² = p2 (R And we can further multiply the two terms:
we know that pi = k( thus K(Hgo,)2 = R.y) Similarly we have v[R.]]
Pip5V[R,g]2 = K2 (H E(R,y =E[R,)((opouu)
V[Rs E[Ray]
2
#0,0o)) = PiP
Recall the variance of true relevance (Equation
44 ):
V R.y] = K((1p9g) + (Hg0p)² + B
V[Raz]
= (pi + ( + Kpipi
+ E[
R(:)
Solve the above equation for K, obtain the Equation 50h_
A.6 Determine the hyperparameters with exposure
In this section, show that how determine the hyperparameters for exposure-basod model. The complete definition is described in Section 3.1
Proposition 10. For any pair of entries Yi and Ya, the generic exponsion of the two entries is given as follows:
Cov(Yij, Yul = Cov(Oin Ra]
= Cov(Oir Oul Cov Raj, Ra] + Ra Ou) + E[Or)][(Onl Covl Cov [ov Rul (51)
Proof. By the Equation
38
the above equation holds.
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A.6.1 Constant exposure hyperparameter
In this case we consider the hyperparameter the Bernoulli variable Ois as constant: Oi ~ (ern(f).
We first derive the estimator for Model [
Proposition 11. Assume that the observation matriz Y is sparse and random variable Oig and Ou follow eractly the same distribution Oıj. Ou Bern(o). Given that we obtain sample of E[y,),, V[Yi,, Pi and p2. we can oblain the estimator of number of latent. factor K denoted as K as:
k (1- + (b)) - ( P.Q]
E[R 2
(52)
PiP5
where,
E[R,y] 8 [Y
(53)
Ri = 8 /V[Y,l + 0 ( 1)1
(54)
p virolp- -0(1 B)E R,yl?
(55)
(V
Proof. Because Oig and Ou follow exactly the same distribution, thus:
B[O,] E[O.] 0, Cov (ijj Ou) = Var(O) = 0(1 -0)-
Recall Equation 10
we have:
Cov[Yij, Yu] 0(1 O)Cov((oej)) Ra] + 0(1 O, + Cov[[Rig, Ru] 0Cov [Rij, Ru] + 0(1 [);[,,
By the definition covariance, we know that Cov(Yij, Yul = V[Kaap and Cov[Rig, Ral = V[Riylp', and we solve p" as Equation (5
We then derive the estimator for Model Proposition 12. Assume that the obsercation malric is sporse and the random variable Oiğ and On are lid independent and identically distributed Oij, Ou ~ Bern(O). Given that we oblain sample of E(yis, V[Yg), Pr and P2, we can obtain the estimator of number of latent factor K denoted as K as:
k (1 (pi (R,) EV PQQ]
E R 2
(56)
PIPS
V R.y
where,
ElR = 8 -
(57)
V[Ry] = 8 - + 8-2
1
(58)
p² - v[Y p.
(59)
V[RAa]
Proof. Because Og and Ou are independent and identically distributed, thus:
B[O,₁] = E[O.i 8,
Cov Onl O.
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Recall Equation 10
we have:
Cov(Yij, Yu) = Cov [Raj, Ru]-
By the definition of covariance, we know that Cov( Yul Viya|p and Cov(R. Ral V(Rysp",, and we
solve p" as Equation 59
A.7 Correlated Latent Features
In this section, we derive the results described in Section
5.3
We know that:
Cov[Rig, Red = Cov
Σ Pikliik, Eperq
Covlpindiks Pex P
R.K
For the covariance in the same row that if i=t & i it::
Σ k.
Covlpagji Pix qudl = Σ Covlpix Puc qu/
Σ Porlpx
Σ
Coulpadjj: Psw Iwc)
We know the result of the first term when ls k', now we calculate the second term when k'. We first caleulate the inside term:
Covlpukgjk Puk qu
ElpikPed 93k qp [P 4xx
ElpaxPes B[qju< - E P B [a E
|E(
[1[ppL]B[PPa + Cov Pax Derolpueaaaa)D/a/a/aaaa [B( + Covlgja, mm)l)
|[ 1) Elq,x] Biq Cov [Psk 1
Then we have:
Cov Cov(Rij, RRij Rı]
Σ Elqu)]E[qul/Vpaa + Σ E Elaja Blaw [Cow[pak. Pix) )-
R
Assume latent factors follow Multivariate Distribution denoted as Fx:
pt Fellpp qj Fully: Eg),
(60)
finally, we obtain the closed form of the covariance,
Cov Roy, Rad =
Elasa Flgu
E. Elgaa [[qu:]Cov (piks Psk)
Σ
Cov Paol Kg The element-wise of E,
Vppl
= 2 7 E
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The above notation refers to the all-ones vector. However, the covariance matrix is always positive semi- definite, and the covariance of any two entries is always non-negative.
Additional Figures & Tables
The following tow figures are illustrations of the two different generic Bayesian MF models.
Pp
Hg
pi
qi
YU
Figure Illustration of Generic Bayesian MF
Op
Up
Ps
0ig
Osi
Rij
Y!
Figure 6: Illustration of Generic Exposure-based Bayesian MF
The following table shows all of the hyperparameters initialization described in the previous research (da Silva
et al. 2023
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Table 4: Hyperparameters Initialization for Different Specifications
Spec.
a
c
Hp
Op
Pg
0g
E[R]
V[R]
A
10
10
1
10.0
3.16
10.0
3.16
2500.00
55000.00
B
10
2
10
2
5.0
1.58
5.0
1.58
625.00
3906.25
C
0.001
0.01
0.01
0.1
0.1
3.16
0.1
1.0
0.25
253.00
D
0.1
0.1
1
0.1
0.32
0.1
0.32
0.25
0.55
E
0.1
0.1
0.1
0.1
1.0
3.16
1.0
3.16
25.00
3025.00
F
1
0.1
0.1
1.0
1.0
1.0
3.16
25.00
550.00
G
1000
1000
1000
1000
1.0
0.03
1.0
0.03
25.00
25.05
C Additional Experiments on Real-World Datasets
We conducted additional experiments on real-world datasets MovieLens (Harper & Konstan 2015 which has been widely studied for recommender systems. The datasets contain users" ratings of different movies on a 5-star scale, with half-star increments (0.5 stars 5.0 stars). We selected three MovieLens datasets with different sizes, from 100k records to 10m records. The following table shows the basic information and corresponding results of each dataset.
Table 5: Information of each MovieLens datasets and the corresponding estimation of the number latent factor K by the different models
dataset
users
7111777
mories
num ratings
sparsity
mean
vor
K
m1 K
m.2 K
ml-100k
943
1682
100k
0.94
3.53
1.27
389.81
36.95
24224.29
ml-1m
6040
3706
Im
0.96
3.58
1.25
504.39
39.58
37499.15
ml-10m
69878
10677
10m
0.99
3.51
1.12
614.01
45.17
58315.81
The num users and num aum_movies denote the total number of users and movies in each dataset. The num_ num_ratings represents the total number of ratings or records in cach datasct, ranging from 100,000 to 10 million. The sparsity refers to the ratio of the total number of ratings to the total number of entries in the rating matrix. In other words, the parameter of the Bernoulli variable used our paper, denoted as 8 (Pobs), oquals 1 sparsity. The mean and var signify the mean and variance sampled from the dataset. The ori K, m1 m1_ K, and m2 K represent the estimated number of latent factors K for the original model, our proposed Model 1, and Model 2, respectively. Given that the optimal K value is influenced by various factors, such the selection of the factorization model and the characteristics of the dataset, it is challenging to identify the optimal K value as a benchmark. Based on different matrix factorization experiments, reasonable K value is typically selected belween 50 and 200. Our additional results demonstrate that our Model generates the most comparable m1_ K values to the most commonly chosen value. Conversely, the exceptionally large m2 K values suggest that our Model may not be adequately suited to these MovieLens datasets. In summary, our proposed Model provides more appropriate estimations of K compared to the original model, particularly for these MovieLens datasets with varying sizes and sparsities.
We measured the execution time of three different models on both our synthetic and the real-world datasets mentioned above.
Table 6: Execution time (in seconds) of the three different models on our synthetic an the MovieLens datasets
dataset
ori (s)
m1 (s)
m2
(s)
sampling (s)
synthetic
8.11 10
7.15 10
4.05
10-6
0.01
ml-100k
9.06 * 10
6.91 * 1
4.77 10-6
0.56
ml-1m
1.29 * 10
6.19 * 10
3.81
10-6
1.10
ml-10m
5.61 * 10-5
2.79 * 10-5
6.91 10-6
211.19
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In Table 6 the ori, mi, and m2 represent single estimation time for the original model, our proposed Model 1, and Model 2, respectively. It shows that the estimation times for our synthetic datasets and real-world datasets are within a minuscule range and exhibit no significant difference.
In fact, the most computationally expensive aspect is the sampling process of the correlation coefficienss shared by all models. The sampling column represents the sampling time of datasets with varying scales. The sampling time increases exponentially the dataset size increases, accounting for the majority of the total execution time.
23


Reviews:
[{"review_id": "Mtz6gLmAwe", "reviewer": "Reviewer_yKEJ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n\n- The paper addresses a practical and well-motivated problem in Bayesian recommender systems, focusing on improving prior specification under sparse data.\n- It introduces an exposure-based modeling approach that aligns with how missing data typically arises in real-world recommendation settings.\n- The proposed method is supported by detailed theoretical derivations and shows consistent improvements in estimation accuracy in controlled synthetic experiments.\n\n### Weaknesses\n\n- The evaluation is limited to synthetic data.\n- The approach assumes knowledge of exposure probabilities, which are generally unobserved.\n- The method is primarily evaluated on estimating the number of latent factors, without broader analysis of downstream performance.\n- The added modeling complexity is not accompanied by discussion of computational overhead.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "- This paper investigates how Prior Predictive Matching struggles to set good priors in Bayesian matrix factorization when data is sparse.\n- To fix that, the authors propose a new model that adds an \"exposure\" variable, capturing whether a user even had the chance to interact with an item.\n- They use this exposure-aware model to build better estimators for key hyperparameters like the number of latent factors.\n- In experiments with synthetic data, their method consistently beats the original PPM approach, especially when lots of data is missing.\n- Overall, they show that modeling exposure helps make prior specification more accurate and reliable in realistic recommender system settings.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- It is difficult to analyze Figures 1, 2, and 3 and Table 3; please elaborate on these.\n- I’m not sure why real-world datasets weren’t included, assuming I understood the setup correctly.\n- The approach assumes exposure probabilities are known, but these are usually unobserved and tricky to estimate in practice.\n- The model adds complexity, but there’s no discussion about how much computation it requires.\n- The tables use vertical bars, which isn’t typical for academic formatting.\n- Figure 4 has a rounded box around it.\n- The figures don’t have grid lines, which would make the plots easier to read and compare visually.\n- In Equation (15), the word \"if\" should be in regular text, not italic, since it’s not a variable.\n- What are M1 and M2 in Model 1 and Model 2, respectively?\n- A space is missing after (1), (2), and (3) in Section 4.2.\n- Parentheses are missing in the references to equations throughout the manuscript.\n- Figure 1 appears after Figures 2 and 3, which should be reordered.", "broader_impact_concerns": "There are no broader impact concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "o5L7HzzOsz", "reviewer": "Reviewer_Ck3Y", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The main strength of this paper is that it clearly articulates the challenge of prior specification in Bayesian Matrix Factorization under data sparsity, grounding it in real-world issues common to recommender systems. It effectively motivates the need for an exposure-based refinement of existing methods.\nAnother strength is that the synthetic experiments systematically vary sparsity, latent dimensions, and prior configurations to stress-test the estimators. The inclusion of baseline comparisons and quantile-based error plots contributes to a solid and reproducible evaluation framework.\n\nThe main weakness is that, although the paper is framed around practical recommender system applications, all experiments are conducted on synthetic datasets. The lack of experiments on real-world data limits insights into practical deployment and generalization performance.\nFrom a presentation perspective, some parts of the paper—especially in the theoretical and experimental sections—suffer from verbose text and dense notations. This may hinder readability and accessibility for a broader audience. For instance, while the paper refers to $\\rho$ and $\\rho*$ in Equation (5), Equations (17) and (19) refer to $\\rho*$ but no clarification is provided and only further reading, including the appendix, clarifies this. Another important weakness is that the paper underemphasizes potential limitations of the proposed approach (e.g., sensitivity to the choice of exposure model, assumptions of independence or correlation), and does not compare against alternative methods beyond da Silva et al. (2023).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes an enhanced prior specification method for Bayesian Matrix Factorization (MF) that integrates exposure-based modeling to improve hyperparameter estimation in the presence of increased data sparsity. The paper builds upon the Prior Predictive Matching (PPM) framework, which aligns prior distributions with observed data to estimate hyperparameters without costly posterior inference. While PPM performs well in dense datasets, its performance deteriorates significantly as sparsity increases—a common characteristic in real-world recommender systems. To address this, the paper introduces an exposure-based Bayesian MF model that accounts for missing-not-at-random (MNAR) data mechanisms by incorporating a Bernoulli exposure variable into the probabilistic framework. This allows the model to distinguish between true relevance (latent interaction) and observed data, better reflecting implicit feedback behavior. Two estimators for the number of latent factors are derived under different assumptions of exposure variable correlation: one assumes all exposure variables share the same distribution, and the other assumes they are i.i.d. These estimators utilize expectations, variances, and correlation statistics of the sparse observation matrix to infer model hyperparameters. The method is empirically evaluated on synthetic datasets across different sparsity levels and prior specifications, comparing the original PPM-based estimator with the new exposure-based versions. This approach is tested empirically using two models and compared with various baseline models.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I would suggest working on the weaknesses mentioned above. Specifically, it would be interesting to see how the proposed exposure-based estimator perform on real-world recommendation datasets (e.g., MovieLens, Amazon, or Netflix), particularly those with high sparsity and implicit feedback. I know that the synthetic experiments aim to have a controlled setting for known conditions of the priors. However, it will be good to verify the applicability to a couple of real scenarios. \n\nA controlled analysis that could also clarify the utility of the model is to test how sensitive the proposed estimators are to dataset characteristics such as the user-item interaction distribution or popularity bias.\n\nFinally, a minor comment regarding the readability issue mentioned above is to ensure that variables are introduced and discussed when they are used. \n\nI also have a few questions. \n\nHow robust are the estimators to misspecification of the exposure model (e.g., if the true missingness mechanism is not Bernoulli)?\n\nUnder what conditions do the proposed estimators break down or yield unstable estimates?", "broader_impact_concerns": "There is no broader impact statement, but the paper can be used in areas beyond recommender systems. A paragraph could be added to discuss potential impacts.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "wvVFHHZ6h1", "reviewer": "Reviewer_xU9C", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n- The write up is well done and clear. The problem considered seems intersting even though I can't say how important missing values are in recommender systems as I'm not really familiar with them.\n\n- The paper discuessed theory and there expermients on synthetic data and it mentions both adavantages but also limitations of their methods.\n\nWeaknesses:\n\n- their model only perfoms better for sparse data.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Authors consider recommender systems (RSs).They analyze how Prior Predictive Matching (PPM) which has been used before performs as as data sparsity increases. They present an enhanced method for specifying priors in Bayesian matrix factorization models and improve the estimators by implementing an exposure-based model to better simulate data scarcity.\nTheir method demonstrates accuracy improvements in hyperparameter estimation during synthetic experiments. \nThey also explore the feasibilityof applying this method to real-world datasets and provide insights into how the model’s behavior adapts to varying levels of data sparsity.\n\nThey are making three assumptions on the model that uses exposure variable to simulate data sprsity/missing values:\n \n1) the exposure variable $O_{ij}$ is independent of the truerelevance $R_{ij} $  \n\n2) the observation model $F_R$ is Poisson distribution\n\n3a)  the random variable $O_{ij}$ and $O_{tl}$ follow exactly the same distribution $O_{ij}, O_{tl} ∼ Bern(\\theta)$ and $Cov(O_{ij}, O_{tl})=Var(\\theta)=\\theta(1-\\theta)$. (Model 1) or\n\n3b) the random variable $O_{ij}$ and $O_{tl}$ are independent and identically distributed,$O_{ij}, O_{tl} ∼ Bern(\\theta)$ (Model 2)\n\nAuthors justify their models by stating theoretical propositions and model equations with proofs in the appendix.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Is there also a difference in running times between the different models?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)