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1 43, 1, Web Joint Random Effect Modeling for Repeated Durations and Discrete Choices with Selection Bias Correction: Application to Promotion Policy Planning for Potential Clients Using Web Access-log Data Takahiro Hoshino EM E We point out that the current researches in Big data analytics neglect two important issues, consumer heterogeniety and selection bias, in constructing predivtion modeling. We provide a real example in which we must deal with the two issues properly, joint modeling of repeated duration and pruchase behavior. To be more concrete, we apply the joint modeling to a Web access-log dataset from a very large panel study. To plan a promotion policy for potential clients of a online shopping company, we proposed a propensity score weighted generalized EM algorithm of the proposed model, to adjust for covariate differences between potential clients and current clients. The proposed model incorporates random effects expressing unmeasured heterogeniety, which inevitably requires numerical integration. However in large dataset it is not practical to employ Markov chain Monte Carlo methods in random effect modeling. We applied the fully exponential Laplace approximation to the estimation algorithm of the proposed model, found that the algorithm is less computationally expensive, while it provides accurate estimates. : EM ( hoshino@soec.nagoyau.ac.jp)

5 EM E We point out that the current researches in Big data analytics neglect two important issues, consumer heterogeniety and selection bias, in constructing predivtion modeling.

2 (2012) (2013) ( (2009) Angrist and Pischke (2009))

3 EM Markov Chain Monte Carlo MCMC MCMC MCMC MCMC (Jordan et al. (1999)) ( Wang and Titterington (2004) Braun and McAuliffe (2010)) 18 Tierney and Kadane (1986) 20 MCMC MCMC Rue et al. (2009) latent Gaussian model (integrated nested Laplace approximation: INLA) Fong et al. (2010) INLA Rizopoulos et al. (2009) Bianconcini and Cagnone (2012) (Tierney et al. (1989)) (2012)

4 * EM Web 1 z = 1 1 z = 0 (1) (2) z = 1

5 EM 45 (z = 0) (z = 0) EM 2 3 E EM (recurrent event duration analysis, Seethraman and Chintagunta (2003) Bijwaard et al. (2006)) (recurrent event survival analysis) Web Key Performance Indicator; KPI KPI KPI yij D i j j + 1 j yij B i J i ( j = 1,..., J i ) i x i i j j + 1 w ij i f i f i f i N(0, φ) y D ij yld ij = log y D ij

2.1 (recurrent event duration analysis, Seethraman and Chintagunta (2003) Bijwaard et

6 f ( [ yij LD ) 1 y LD f i, w ij = σ exp ij (β 0 + f i + w t ij β w) σ exp ( y LD ij (β 0 + f i + w t ij β w) σ )] (2.1) (Klein and Moeschberger (2003)) β 0 + f i + w t ij β w σ j + 1 y B ij f i logit [ p ( y B ij = 1 f i, w ij )] = α0 + α f f i + w t ijα w (2.2) J i p(y i1,..., y iji w i1,..., w iji ) = p ( yij LD ) ( ) f i, w ij p y B ij f i, w ij p(f i )df i (2.3) j=1 y ij = (yij LD, yij B)t α f z i = 0 z i i 1 0 z i = 1 y i1,..., y iji z i = 0 x y w (Missing at random) N i=1 z i (1 w(x i )) w(x i ) log p(y i1,..., y iji w i1,..., w iji ) (2.4)

2) J i p(y i1,..., y iji w i1,..., w iji ) = p ( yij LD ) ( ) f i, w ij p y B ij f i, w ij p(f i )df i (2.3) j=1 y ij = (yij LD, yij B)t α f 2.

7 EM 47 z = 0 p(y w, z = 0) ( (2005) Hoshino et al. (2006) Wooldridge (2007) Pan and Schaubel (2009)) w(x i ) x i z i = 1 y (z = 0) w y x y y w x 4 Web 1 z = 0 y w x (z = 1) p(x z = 0) w(x i ) = p(x i z i = 1)p(z i = 1) p(x i z i = 1)p(z i = 1) + p(x i z i = 0)p(z i = 0) (2.5) w(x i ) 2.3 Vaida and Xu (2000) clustered data EM Rizopoulos et al. (2009) 1 Web 2 3. EM α = (α 0, α f, α t w) t β = (σ, β 0, β t w) t θ = (α t, β t, φ) t

y w x (z = 1) p(x z = 0) w(x i ) = p(x i z i = 1)p(z i = 1) p(x i z i = 1)p(z i = 1) + p(x i z i = 0)p(z i

8 S(θ) = N i=1 N i=1 z i (1 w(x i )) S i (θ) w(x i ) z i (1 w(x i )) w(x i ) log g(fi, y i, w i θ) p(f i y θ i, w i, θ)df i (3.1) S i (θ) = θ log p(y i1,..., y iji w i1,..., w iji ) (3.2) J i g (f i, y i, w i θ) = p ( yij LD ) ( ) f i, w ij p y B ij f i, w ij p(f i φ) (3.3) j=1 p(f i y i, w i, θ) = g(f i, y i, w i θ) g(fi, y i, w i θ)df i θ log g(f i, y i, w i θ) = α + β J i j=1 J i j=1 log p ( y B ij f i, w ij ) log p ( yij LD ) f i, w ij + φ log p(f i φ) (Tierney et al. (1989)) S i (θ) r ( ) log g ˆfi, y i, w i θ Ŝ ir (θ) = 1 θ r 2 γ ir (3.4) O(J 2 i ) (Rizopoulos et al. (2009)) θ r θ r { γ ir = Σ 1 Σi i Σ 1 i f i Σ i = 2 f 2 i 2 Σ i} θ r log g(f i, y f i θ i, w i θ) (3.5) r f i= ˆf i log g(f i, y i, w i, z i θ) (3.6) fi = ˆf i EM (1)

3) j=1 p(f i y i, w i, θ) = g(f i, y i, w i θ) g(fi, y i, w i θ)df i θ log g(f i, y i, w i θ) = α + β J i j=1 J i j=1 log p ( y B ij f i, w ij ) log p ( yij LD ) f i, w ij + φ log p(f i φ)

9 EM 49 (2) 1 (3.3) f i ˆf i log g(f i, y i, w i θ) (3) (3.1) (4) θ ˆθ (5) (2) (4) EM MCMC S(θ) w(x i ) 1 w ij N = 5, ,000 2 J i J i (1) (2) MCMC MCMC iteration Burn-in phase Geweke 1 MCMC

1) (4) θ ˆθ (5) (2) (4) EM MCMC S(θ) w(x i ) 1 w ij 2 1 0.

10 J i MCMC MCMC SAS/IML Window7 64bit Intel Core i7-3930k (6 12way/3.20GHz/3.80GHz/12MB) 32GB N = 100,000 J i MCMC iteration ( ) MCMC MCMC 3000 iteration MCMC URL URL URL web Random Digit Dialing; RDD A URL

11 EM 51 URL A 3 2 (z = 1) B 2 A 3 2 (z = 0) x (5 ) (9 ) (6 ) ( 3 ) (13 ) w ij 24 4 Web 3 (yahoo google ) blog SNS (twitter facebook mixi) ( ) 8 URL A URL A (z = 1) B )

12 Web Web (z = 1) (z = 0) A (z = 1) 4483 (z = 0) (1) (2.4) EM (2) EM (3)

13 EM 53 (4) EM 4 2 (4) (4) (3) α f φ 2 (1) (2) (3) (1) (4) 5% (1) (3) (4) (2) (3) α w Yahoo! (4) A A (1) (4) google (2) (3) Yahoo! Yahoo! A Yahoo! google Web 10% ROI (Return on investment) (4) 2 3

14 Web 4 (1) (2) (4) (3) σ β Yahoo! google βw blog SNS α α f Yahoo! google αw blog SNS φ (4) %

06824 google 0.299 0.07304 0.093 0.06985 0.101 0.06208 0.367 0.08607 βw 0.109 0.10028 0.034 0.07709 0.060 0.08094 0.130 0.11276 blog 0.038 0.06904 0.110 0.07090 0.381 0.05033 0.093 0.06687 SNS 0.

15 EM 55 3 * z ( (2009)) c z 5. Web MCMC (transfer learning) ( (2010)) (Shimodaira (2000) Sugiyama et al. (2007)) (Rosenbaum and Rubin (1983))

16 z z f (Follman and Wu (1995) (2009)) MCMC ( Hjort et al. (2010)) (Hoshino, in press) MCMC URL A A (4) 2 URL

17 EM 57 ( ) (A) A. θ [ [ ] log g(f i, y f i, w i, z i θ) = f Ji i i φ 1 σ + 1 y LD σ exp ij β t w ij σ j=1 ( ) ] α f y B ij p ij [ [ Σ i = 1 J i φ 1 y LD σ 2 exp ij J i Σ i = f i j=1 [ j=1 [ 1 y LD σ 3 exp ij β t w ij σ β t w ij σ ] ] + α 2 f p ij (1 p ij ) + α 3 f p ij (1 p ij )(1 2p ij ) ] ] (A.1) (A.2) (A.3) w ij = (1, f i, w t ij )t p ij = p(y B ij = 1 f i, x i, w ij ) Σ/ θ β β Σ i = J i j=1 [ 1 y LD ij σ 3 w ij exp β t w ij σ ] (A.4) α p ij αp ij (1 p ij ) Angrist, J. and Pischke, J. S. (2009). Mostly Harmless Econometrics: An Empiricist s Companion, Princeton University Press, London. Bianconcini, S. and Cagnone, S. (2012). Estimation of generalized linear latent variable models via fully exponential Laplace approximation, J. Multivar. Anal., 112, Bijwaard, G. E., Franses, P. H. and Paap, R. (2006). Modeling purchases as repeated events, J. Bus. Econ. Stat., 24, Braun, M. and McAuliffe, J. (2010). Variational inference for large-scale models of discrete choice, J. Am. Stat. Assoc., 105,

w ij σ β t w ij σ ] ] + α 2 f p ij (1 p ij ) + α 3 f p ij (1 p ij )(1 2p ij ) ] ] (A.1) (A.2) (A.

18 Follman, D. and Wu, M. (1995). An approximate generalized linear model with random effects for informative missing data, Biometrics, 51, Fong, Y., Rue, H. and Wakefield, J. (2010). Bayesian inference for generalized linear mixed model, Biostatistics, 11, Hjort, N. L., Holmes, C., Müller, O. and Walker, S. G. (2010). Bayesian Nonparametrics, Cambridge University Press, Cambridge. (2005). M 32 2, (2009).. Hoshino, T. (in press). Semiparametric Bayesian estimation for marginal parametric potential outcome modeling: Application to causal inference, J. Am. Stat. Assoc. Hoshino, T., Kurata, H. and Shigemasu, K. (2006). A propensity score adjustment for multiple group structural equation modeling, Psychometrika, 71, (2013). AI Jordan, M. I., Ghahramani, Z., Jaakkola, T. S. and Saul, L. (1999). An introduction to variational methods for graphical models, Mach. Learn., 37, (2010) Klein, J. P. and Moeschberger, M. L. (2003) Survival Analysis: Techniques for Censored and Truncated Data, 2nd ed., Springer, New York. (2012). 60 1, Pan, Q. and Schaubel, D. E. (2009). Evaluating bias correction in weighted proportional hazard regression, Lifetime Data Analysis, 15, Rizopoulos, D., Verbeke, G. and Lesaffre, E. (2009). Fully exponential laplace approximations for the joint modeling of survival and longitudinal data, J. R. Stat. Soc., Ser. B, 71, Rosenbaum, P. R. and Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects, Biometrika, 70, Rue, H., Martino, S. and Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models by using integrated nested laplace approximations (with discussion), J. R. Stat. Soc., Ser. B, 71, Seethraman, P. B. and Chintagunta, P. K. (2003). The proportional hazard model for purchase timing: A comparison of alternative specifications, J. Bus. Econ. Stat., 21, Shimodaira, H. (2000). Improving predictive inference under covariate shift by weighting the log-likelihood function, J. Stat. Plan. Inf., 90, (2012).. Sugiyama, M., Krauledat, M. and Müller, K.-R. (2007). Covariate shift adaptation by importance weighted cross validation, J. Mach. Learn. Res., 8, Tierney, L. and Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densities, J. Am. Stat. Assoc., 81, Tierney, L., Kass, R. and Kadanae, J. B. (1989). Fully exponential Laplace approximations to expectations and variances of nonpositive functions, J. Am. Stat. Assoc., 84, Vaida, F. and Xu, R. (2000). Proportional hazards models with random effects, Stat. Med., 19, Wang, B. and Titterington, D. M. (2004). Lack of consistency of mean field and variational Bayes approximations for state space models, Neural Process. Lett., 20, Wooldridge, J. M. (2007) Inverse probability weighted M-estimation for general missing data problems, J. Econom., 141,

Bayesian Nonparametrics, Cambridge University Press, Cambridge. (2005). M 32 2, 121 132. (2009).. Hoshino, T. (in press).

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