Discovering heterogeneous consumer journeys in online platforms: implications for networking investment

Abstract

We model consumer journeys for user-created programs published in an online programming platform (OPP) and uncover factors that predict their occurrence. We build our model on a theoretical framework where consumer journeys involve three latent stages (Learn, Feel, Do), in which users gather information about, express fondness toward, and try the published items, respectively. Using a dataset from an OPP where users publish multimedia items and follow other users, we find that there is no one dominant consumer journey; instead, the sequences of stages in a journey (e.g., Learn → Feel → Do) vary across published items. Furthermore, we find that the social capital (i.e., social network) of a publisher influences the occurrence of spillover effects between latent stages (the phenomenon that one stage in a period triggers another stage in the next period) for the items posted by the publisher. We also find that a publisher’s social capital has only a transient impact on the consumer journeys for the publisher’s projects, underlining the importance of consistently making new network connections in order to promote the growth of user activities surrounding the publisher’s projects. We apply our findings to the publishers’ networking investment decisions to show that publishers’ networking investment would be severely suboptimal if journey heterogeneity is not considered.

Appendix

MCMC Algorithm

The model is as follows:

$$ {s}_{ijt k}=f\left({y}_{ijt-1,k}\right)\times {\theta}_{it}^{learn}\left( ne{t}_{it}\right)+{v}_{ijt k}\ \mathrm{for}\ k= play, comment $$

(1-1)

$$ {s}_{ijt k}=f\left({y}_{ijt-1,k}\right)\times {\theta}_{it}^{feel}\left( ne{t}_{it}\right)+{v}_{ijt k}\ \mathrm{for}\ k= favorite, love $$

(1-2)

$$ {s}_{ijt k}=f\left({y}_{ijt-1,k}\right)\times {\theta}_{it}^{do}\left( ne{t}_{it}\right)+{v}_{ijt k}\ \mathrm{for}\ k= download $$

(1-3)

$$ f\left({y}_{ijt-1,k}\right)={a}_{ijk}+{b}_{ijk}{y}_{ijt-1,k}+{c}_{ijk}{y}_{ijt-1,k}^2 $$

(2-1)

$$ {a}_{ijk}=\overline{a_k}+{\eta}_{ijk}^a $$

(2-2)

$$ {b}_{ijk}=\overline{b_k}+{\eta}_{ijk}^b $$

(2-3)

$$ {c}_{ijk}=\overline{c_k}+{\eta}_{ijk}^c, $$

(2-4)

(3-1)

$$ {\beta}_i^{learn}=\overline{\beta^{learn}}+{\varepsilon}_i^{learn} $$

(3-2)

$$ {\beta}_i^{feel}=\overline{\beta^{feel}}+{\varepsilon}_i^{feel} $$

(3-3)

$$ {\beta}_i^{do}=\overline{\beta^{do}}+{\varepsilon}_i^{do}, $$

(3-4)

$$ {\gamma}_{ilm}={\overline{\gamma}}_{lm}+{\epsilon}_{ilm}, $$

(3-5)

$$ degre{e}_{it}={\psi}_{it}+{w}_{it}^{degree}, $$

(4-1)

$$ {\psi}_{it}={\upsilon}_i{\psi}_{it-1}+{w}_{it}^{\psi }, $$

(4-2)

$$ {\upsilon}_i=\overline{\upsilon}+{\xi}_i $$

(4-3)

where ψ_it is the dynamic latent instrument. We assume the following distributions for the error terms.

v_ijtk~N(0, V_ijk), ${\eta}_{ijk}^a\sim N\left(0,\mathit{\operatorname{var}}\left({\eta}_{ijk}^a\right)\right)$, ${\eta}_{ijk}^b\sim N\left(0,\mathit{\operatorname{var}}\left({\eta}_{ijk}^b\right)\right)$, ${\eta}_{ijk}^c\sim N\left(0,\mathit{\operatorname{var}}\left({\eta}_{ijk}^c\right)\right)$,${\varepsilon}_i^{learn}\sim N\left(0,\mathit{\operatorname{var}}\left({\varepsilon}_i^{learn}\right)\right)$, ${\varepsilon}_i^{feel}\sim N\left(0,\mathit{\operatorname{var}}\left({\varepsilon}_i^{feel}\right)\right)$,${\varepsilon}_i^{do}\sim N\left(0,\mathit{\operatorname{var}}\left({\varepsilon}_i^{do}\right)\right)$, ϵ_ilm~N(0, var(ϵ_ilm)), ${w}_{it}^{\psi}\sim N\left(0,\mathit{\operatorname{var}}\left({w}_{it}^{\psi}\right)\right)$, ξ_i~N(0, var(ξ_i)), and ${\overset{\sim }{\mathbf{w}}}_{it}\equiv \left[{{\mathbf{w}}_{it}^{\boldsymbol{\uptheta}}}^{\prime}\kern0.5em {w}_{it}^{degree}\right]\sim MVN\left(0,{\boldsymbol{\Omega}}_i\right)$, where

$$ {\boldsymbol{\Omega}}_i=\left[\begin{array}{cccc}{W}_i^{ll}& {W}_i^{lf}& {W}_i^{ld}& {W}_i^{l\mathrm{n}}\\ {}{W}_i^{fl}& {W}_i^{ff}& {W}_i^{fd}& {W}_i^{fn}\\ {}{W}_i^{dl}& {W}_i^{df}& {W}_i^{dd}& {W}_i^{dn}\\ {}{W}_i^{nl}& {W}_i^{nf}& {W}_i^{nd}& {W}_i^{nn}\end{array}\right]=\left[\begin{array}{cc}{\mathbf{W}}_i^{\boldsymbol{\uptheta}}& {\boldsymbol{\Omega}}_i^{\theta n}\\ {}{{\boldsymbol{\Omega}}_i^{\theta n}}^{\prime }& {W}_i^{nn}\end{array}\right]. $$

The MCMC algorithm consists of two parts. In Part 1, we draw parameters for individual publishers and projects using Equations (1–1)-(1–3), (2–1), (3–1), (4–1), and (4–2). In Part 2, we shrink the individual-level parameters using Equations (2-2)-(2–4), (3–2)-(3–5), and (4–3). We iterate the two parts sufficiently to collect a representative sample from the posterior distribution. In this appendix, we illustrate the algorithm assuming J2: Feel → Learn → Do. Adaptation to other adoption paths is straightforward.

Part 1: Individual User-Level Parameters

Sample θ_it

We represent the model in the state-space framework. For user i, Equation (A.1), which is the reiteration of Equation (5), is the observation equation. Equation (A.2) is the state equation.

(A.1)

(A.2)

Let V_i be the diagonal matrix whose diagonal terms consist of V_ij(k) and let W_i be the covariance matrix of the composite error vector ${\mathbf{w}}_{it}^{\boldsymbol{\uptheta}}$ conditional on ${w}_{it}^{degree}$—i.e., ${\mathbf{W}}_i={\mathbf{W}}_i^{\boldsymbol{\uptheta}}-{\boldsymbol{\Omega}}_i^{\theta n}{\left({W}_i^{nn}\right)}^{-1}{{\boldsymbol{\Omega}}_i^{\theta n}}^{\prime }$. We apply the forward-filtering/backward-sampling (FF/BS) algorithm (West and Harrison 1997) to draw θ_it. Let D_it denote the information set at t for user i.

- Forward Filtering.

(a) Posterior at t − 1: θ_it − 1 ∣ D_it − 1~N(m_{i, t − 1}, C_{i, t − 1}).

(b) Prior at t: θ_it ∣ D_it − 1~N(a_it, R_it) where ${\mathbf{a}}_{it}={\mathbf{G}}_i{\mathbf{m}}_{i,t-1}+{\mathbf{u}}_{it}+{\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}{\left({W}_i^{nn}\right)}^{-1}\left[ degre{e}_{it}-{\psi}_{it}\right],\kern0.5em {\mathbf{R}}_{it}={\mathbf{G}}_i{\mathbf{C}}_{i,t-1}{\mathbf{G}}_i^{\prime }+{\mathbf{W}}_i^{\boldsymbol{\uptheta}}-{\boldsymbol{\Omega}}_i^{\theta n}{\left({W}_i^{nn}\right)}^{-1}{{\boldsymbol{\Omega}}_i^{\theta n}}^{\prime }$.

(c) One-step ahead forecast of s_it at t: s_it ∣ D_{i, t − 1}~N(f_it, B_it) where ${\mathbf{f}}_{it}={\mathbf{F}}_{it}^{\prime }{\mathbf{a}}_{it}$ and ${\mathbf{B}}_{it}={\mathbf{F}}_{it}{\mathbf{R}}_{it}{\mathbf{F}}_{it}^{\prime }+{\mathbf{V}}_i$.

(d) Posterior at t: θ_it ∣ D_it~N(m_it, C_it), where ${\mathbf{m}}_{it}={\mathbf{a}}_{it}+{\mathbf{R}}_{it}{\mathbf{F}}_{it}{\mathbf{B}}_{it}^{-1}\left({\mathbf{s}}_{it}-{\mathbf{f}}_{it}\right)$ and ${\mathbf{C}}_{it}={\mathbf{R}}_{it}-{\mathbf{R}}_{it}{\mathbf{F}}_{it}{\mathbf{B}}_{it}^{-1}{\mathbf{F}}_{it}^{\prime }{\mathbf{R}}_{it}$.

- Backward Sampling.

at t = T: θ_iT ∣ D_iT~N(m_iT, C_iT).

at t = T − 1, …, 0: θ_it ∣ θ_it − 1,D_it~N(g_it, K_it), where ${\mathbf{g}}_{it}={\mathbf{m}}_{it}+{\mathbf{C}}_{it}{\mathbf{G}}_i^{\prime }{\mathbf{R}}_{i,t+1}^{-1}\left({\boldsymbol{\uptheta}}_{i,t+1}-{\mathbf{a}}_{i,t+1}\right)$ and ${\mathbf{K}}_{it}={\mathbf{C}}_{it}-{\mathbf{C}}_{it}{\mathbf{G}}_i^{\prime }{\mathbf{R}}_{i,t+1}^{-1}{\mathbf{G}}_i{\mathbf{C}}_{it}$.

• Step 1-1) Sample γ_i11 and ${\beta}_i^{learn}$

Let ${\overset{\sim }{\boldsymbol{\upbeta}}}_i={\left[{\gamma}_{i11}\kern0.5em {\beta}_i^{learn}\ \right]}^{\prime }$, ${\overset{\sim }{\mathbf{x}}}_{it}=\left[{\theta}_{i,t-1}^{learn}\kern0.75em degre{e}_{it}\ \right],{\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}\left[p\right]$ be the p^th element of the column vector ${\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}=\left[{W}_i^{ln}\kern0.5em {W}_i^{fn}\kern0.5em {W}_i^{dn}\right]$, ${\overset{\sim }{\mathbf{y}}}_{it}={\theta}_{it}^{learn}-{\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}\left[1\right]{\left({W}_i^{nn}\right)}^{-1}\left( degre{e}_{it}-{\psi}_{it}\right)$, and ${\overset{\sim }{\mathbf{W}}}_i^{\boldsymbol{\uptheta}}=\boldsymbol{\Omega} \left[1,1\right]-{\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}\left[1\right]{\left({W}_i^{nn}\right)}^{-1}{{\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}}^{\prime}\left[1\right]$. Then, ${\overset{\sim }{\boldsymbol{\upbeta}}}_i\sim MVN\left({\overset{\sim }{\mathbf{b}}}_i,\kern0.5em {\overset{\sim }{\mathbf{S}}}_i\right)$, where ${\overset{\sim }{\mathbf{S}}}_i={\left[{\left({\overset{\sim }{\mathbf{W}}}_i^{\boldsymbol{\uptheta}}\right)}^{-1}{{\overset{\sim }{\mathbf{X}}}_i}^{\prime }{\overset{\sim }{\mathbf{X}}}_i+{\Sigma}_{{\overset{\sim }{\boldsymbol{\upbeta}}}_i}^{-1}\right]}^{-1}$ and ${\overset{\sim }{\mathbf{b}}}_i={\overset{\sim }{\mathbf{S}}}_i\left[{\left({\overset{\sim }{\mathbf{W}}}_i^{\boldsymbol{\uptheta}}\right)}^{-1}{{\overset{\sim }{\mathbf{X}}}_i}^{\prime }{\overset{\sim }{\mathbf{y}}}_i+{\Sigma}_{{\overset{\sim }{\boldsymbol{\upbeta}}}_i}^{-1}{\overline{\overset{\sim }{\boldsymbol{\upbeta}}}}_i\right]$. ${\Sigma}_{{\overset{\sim }{\boldsymbol{\upbeta}}}_i}^{-1}$ is the prior covariance matrix and ${\overline{\overset{\sim }{\boldsymbol{\upbeta}}}}_i$ are the prior mean. That is, ${\overline{\overset{\sim }{\boldsymbol{\upbeta}}}}_i={\left[{\overline{\gamma}}_{11}\kern0.75em \overline{\beta^{learn}}\kern0.5em \right]}^{\prime }$ and ${\Sigma}_{{\overset{\sim }{\boldsymbol{\upbeta}}}_i}$ is a diagonal matrix whose diagonal elements are var(ϵ_i11) and $\mathit{\operatorname{var}}\left({\varepsilon}_i^{learn}\right)$.

• Step 1-2) Sample γ_i21, γ_i22, and ${\beta}_i^{feel}$

Let ${\overset{\sim }{\boldsymbol{\upbeta}}}_i={\left[{\gamma}_{i21}\kern0.5em {\gamma}_{i22}\ {\beta}_i^{feel}\ \right]}^{\prime }$, ${\overset{\sim }{\mathbf{x}}}_{it}=\left[{\theta}_{i,t-1}^{learn}\kern0.5em {\theta}_{i,t-1}^{feel}\kern0.5em degre{e}_{it}\ \right]$, ${\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}\left[p\right]$ be the p^th element of the column vector ${\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}=\left[{W}_i^{ln}\kern0.5em {W}_i^{fn}\kern0.5em {W}_i^{dn}\right]$, ${\overset{\sim }{\mathbf{y}}}_{it}={\theta}_{it}^{feel}-{\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}\left[2\right]{\left({W}_i^{nn}\right)}^{-1}\left( degre{e}_{it}-{\psi}_{it}\right)$, and ${\overset{\sim }{\mathbf{W}}}_i^{\boldsymbol{\uptheta}}=\boldsymbol{\Omega} \left[2,2\right]-{\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}\left[2\right]{\left({W}_i^{nn}\right)}^{-1}{{\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}}^{\prime}\left[2\right]$. Then, ${\overset{\sim }{\boldsymbol{\upbeta}}}_i\sim MVN\left({\overset{\sim }{\mathbf{b}}}_i,\kern0.5em {\overset{\sim }{\mathbf{S}}}_i\right)$, where ${\overset{\sim }{\mathbf{S}}}_i={\left[{\left({\overset{\sim }{\mathbf{W}}}_i^{\boldsymbol{\uptheta}}\right)}^{-1}{{\overset{\sim }{\mathbf{X}}}_i}^{\prime }{\overset{\sim }{\mathbf{X}}}_i+{\Sigma}_{{\overset{\sim }{\boldsymbol{\upbeta}}}_i}^{-1}\right]}^{-1}$ and ${\overset{\sim }{\mathbf{b}}}_i={\overset{\sim }{\mathbf{S}}}_i\left[{\left({\overset{\sim }{\mathbf{W}}}_i^{\boldsymbol{\uptheta}}\right)}^{-1}{{\overset{\sim }{\mathbf{X}}}_i}^{\prime }{\overset{\sim }{\mathbf{y}}}_i+{\Sigma}_{{\overset{\sim }{\boldsymbol{\upbeta}}}_i}^{-1}{\overline{\overset{\sim }{\boldsymbol{\upbeta}}}}_i\right]$. ${\Sigma}_{{\overset{\sim }{\boldsymbol{\upbeta}}}_i}^{-1}$ is the prior covariance matrix and ${\overline{\overset{\sim }{\boldsymbol{\upbeta}}}}_i$ are the prior mean. That is, ${\overline{\overset{\sim }{\boldsymbol{\upbeta}}}}_i={\left[{\overline{\gamma}}_{21}\ {\overline{\gamma}}_{22}\kern0.5em \overline{\beta^{feel}}\kern0.5em \right]}^{\prime }$ and ${\Sigma}_{{\overset{\sim }{\boldsymbol{\upbeta}}}_i}$ is a diagonal matrix whose diagonal elements are var(ϵ_i21), var(ϵ_i22),and $\mathit{\operatorname{var}}\left({\varepsilon}_i^{feel}\right)$.

• Step 1-3) Sample γ_i32, γ_i33, and ${\beta}_i^{do}$

Let ${\overset{\sim }{\boldsymbol{\upbeta}}}_i={\left[{\gamma}_{i32}\kern0.5em {\gamma}_{i33}\ {\beta}_i^{do}\ \right]}^{\prime }$, ${\overset{\sim }{\mathbf{x}}}_{it}=\left[{\theta}_{i,t-1}^{feel}\kern0.5em {\theta}_{i,t-1}^{do}\kern0.5em degre{e}_{it}\ \right]$, ${\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}\left[p\right]$ be the p^th element of the column vector ${\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}=\left[{W}_i^{ln}\kern0.5em {W}_i^{fn}\kern0.5em {W}_i^{dn}\right]$, ${\overset{\sim }{\mathbf{y}}}_{it}={\theta}_{it}^{feel}-{\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}\left[3\right]{\left({W}_i^{nn}\right)}^{-1}\left( degre{e}_{it}-{\psi}_{it}\right)$, and ${\overset{\sim }{\mathbf{W}}}_i^{\boldsymbol{\uptheta}}=\boldsymbol{\Omega} \left[3,3\right]-{\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}\left[3\right]{\left({W}_i^{nn}\right)}^{-1}{{\boldsymbol{\Omega}}_i^{\boldsymbol{\uptheta} n}}^{\prime}\left[3\right]$. Then, ${\overset{\sim }{\boldsymbol{\upbeta}}}_i\sim MVN\left({\overset{\sim }{\mathbf{b}}}_i,\kern0.5em {\overset{\sim }{\mathbf{S}}}_i\right)$, where ${\overset{\sim }{\mathbf{S}}}_i={\left[{\left({\overset{\sim }{\mathbf{W}}}_i^{\boldsymbol{\uptheta}}\right)}^{-1}{{\overset{\sim }{\mathbf{X}}}_i}^{\prime }{\overset{\sim }{\mathbf{X}}}_i+{\Sigma}_{{\overset{\sim }{\boldsymbol{\upbeta}}}_i}^{-1}\right]}^{-1}$ and ${\overset{\sim }{\mathbf{b}}}_i={\overset{\sim }{\mathbf{S}}}_i\left[{\left({\overset{\sim }{\mathbf{W}}}_i^{\boldsymbol{\uptheta}}\right)}^{-1}{{\overset{\sim }{\mathbf{X}}}_i}^{\prime }{\overset{\sim }{\mathbf{y}}}_i+{\Sigma}_{{\overset{\sim }{\boldsymbol{\upbeta}}}_i}^{-1}{\overline{\overset{\sim }{\boldsymbol{\upbeta}}}}_i\right]$. ${\Sigma}_{{\overset{\sim }{\boldsymbol{\upbeta}}}_i}^{-1}$ is the prior covariance matrix and ${\overline{\overset{\sim }{\boldsymbol{\upbeta}}}}_i$ are the prior mean. That is, ${\overline{\overset{\sim }{\boldsymbol{\upbeta}}}}_i={\left[{\overline{\gamma}}_{32}\ {\overline{\gamma}}_{33}\kern0.5em \overline{\beta^{do}}\kern0.5em \right]}^{\prime }$ and ${\Sigma}_{{\overset{\sim }{\boldsymbol{\upbeta}}}_i}$ is a diagonal matrix whose diagonal elements are var(ϵ_i32), var(ϵ_i33),and $\mathit{\operatorname{var}}\left({\varepsilon}_i^{do}\right)$.

Sample Ω_i

The relevant equations are

$$ {\theta}_{it}^{learn}-{\beta}_i^{learn} degre{e}_{it}-{\gamma}_{i11}{\theta}_{i,t-1}^{learn}={w}_{it}^{learn}, $$

$$ {\theta}_{it}^{feel}-{\beta}_i^{feel} degre{e}_{it}-{\gamma}_{i21}{\theta}_{i,t-1}^{learn}-{\gamma}_{i22}{\theta}_{i,t-1}^{feel}={w}_{it}^{feel}, $$

$$ {\theta}_{it}^{do}-{\beta}_i^{do} degre{e}_{it}-{\gamma}_{i32}{\theta}_{i,t-1}^{feel}-{\gamma}_{i33}{\theta}_{i,t-1}^{do}={w}_{it}^{do}, $$

$$ degre{e}_{it}-{\psi}_{it}={w}_{it}^{degree}. $$

The posterior distribution of Ω_i is given by ${\boldsymbol{\Omega}}_i\sim IW\left({T}_i+{q}_i,\left({V}_i+{\sum}_{t=1}^{T_i}\left({\boldsymbol{\upphi}}_{it}\right){\left({\boldsymbol{\upphi}}_{it}\right)}^{\prime}\right)\right),$

where q_i = 6, V_i = 10⁻⁶I₄, and ${\boldsymbol{\upphi}}_{it}=\left[\begin{array}{c}{\theta}_{it}^{learn}-{\beta}_i^{learn} degre{e}_{it}-{\gamma}_{i11}{\theta}_{i,t-1}^{learn}\\ {}{\theta}_{it}^{feel}-{\beta}_i^{feel} degre{e}_{it}-{\gamma}_{i21}{\theta}_{i,t-1}^{learn}-{\gamma}_{i22}{\theta}_{i,t-1}^{feel}\\ {}{\theta}_{it}^{do}-{\beta}_i^{do} degre{e}_{it}-{\gamma}_{i32}{\theta}_{i,t-1}^{feel}-{\gamma}_{i33}{\theta}_{i,t-1}^{do}\\ {} degre{e}_{it}-{\psi}_{it}\end{array}\right]$ .

Sample ψ_it

We transform into the reduced form of the model. Equations (A.6) is the observation equation. Equation (A.7) is the state equation.

(A.6)

$$ {\psi}_{it}={\upsilon}_i{\psi}_{it-1}+{w}_{it}^{\psi }, $$

(A.7)

Let ${\overset{\sim }{\mathbf{V}}}_i$ be the covariance matrix of the composite error ${\overset{\sim }{\mathbf{v}}}_{it}$: ${\overset{\sim }{\mathbf{V}}}_i={\mathbf{L}}_i{\boldsymbol{\Omega}}_i{\mathbf{L}}_i^{\prime }$, where

$$ {\mathbf{L}}_i=\left[\begin{array}{cccc}1& 0& 0& {\beta}_i^{learn}\\ {}0& 1& 0& {\beta}_i^{feel}\\ {}0& 0& 1& {\beta}_i^{do}\\ {}0& 0& 0& 1\end{array}\right]. $$

We apply the FFBS algorithm (West and Harrison 1997) to draw ψ_it.

• Step 1-4) Sample υ_i and $\mathit{\operatorname{var}}\left({w}_{it}^{\psi}\right)$

The relevant equation is ${\psi}_{it}={\upsilon}_i{\psi}_{it-1}+{w}_{it}^{\psi }$. We apply a normal-inverse Gamma prior. The prior distribution of υ_i comes from Equation (4-3): ${\upsilon}_i\sim N\left(\overline{\upsilon},\kern0.5em \mathit{\operatorname{var}}\left({\xi}_i\right)\right)$. We use a diffuse inverse-Gamma prior for $\mathit{\operatorname{var}}\left({w}_{it}^{\psi}\right)$: $\mathit{\operatorname{var}}\left({w}_{it}^{\psi}\right)\sim IG\left(1,0.001\right)$.

• Step 1-5) Sample a_ij(k), b_ij(k), c_ij(k), and V_ij(k)

The relevant regression equation comes from (WA.1): ${s}_{ijt k}={a}_{ijk}{\theta}_{it}^{stag{e}_k}+{b}_{ijk}{y}_{ijt-1,k}{\theta}_{it}^{stag{e}_k}+{c}_{ijk}{y}_{ijt-1,k}^2{\theta}_{it}^{stag{e}_k}+{v}_{ijt k}$, where stage_k is the journey stage corresponding to activity k (e.g., stage_play = learn, stage_download = do). We use a normal-inverse Gamma prior. The prior distribution of a_ijk, b_ijk, and c_ijk come from Equations (2-2)-(2–4)—for example, the prior distribution of a_ijk is ${a}_{ijk}\sim N\left(\overline{a_k},\kern0.5em \mathit{\operatorname{var}}\left({\eta}_{ijk}^a\right)\right)$. We use a diffuse inverse-Gamma prior for V_ijk: V_ijk~IG(1, 0.0001).

Part 2: Shrinkage (The Population Parameters)

• Step 2-1) Sample $\overline{a_k}$ and ${\Xi}_k^a\equiv \mathit{\operatorname{var}}\left({\eta}_{ijk}^a\right)$

The relevant equation is Equation (2-2): ${a}_{ijk}=\overline{a_k}+{\eta}_{ijk}^a$ where a_ijk is drawn in Step 1–8). We use a diffuse normal-inverse Gamma prior to sample $\overline{a_k}$ and ${\Xi}_k^a$.

• Step 2-2) Sample $\overline{b_k}$ and ${\Xi}_k^b\equiv \mathit{\operatorname{var}}\left({\eta}_{ijk}^b\right)$

The relevant equation is Equation (2-3): ${b}_{ijk}=\overline{b_k}+{\eta}_{ijk}^b$ where b_ijk has been drawn in Step 1–8). The sampling procedure is identical to that in Step 2–1).

• Step 2-3) Sample $\overline{c_k}$, and ${\Xi}_k^c\equiv \mathit{\operatorname{var}}\left({\eta}_{ijk}^c\right)$

The relevant equation is Equation (2-4): ${c}_{ijk}=\overline{c_k}+{\eta}_{ijk}^c$ where c_ijk has been drawn in Step 1–8). The sampling procedure is identical to that in Step 2–1).

• Step 2-4) Sample $\overline{\beta^{stage}}$ and ${\varOmega}^{stage}\equiv \mathit{\operatorname{var}}\left({\varepsilon}_i^{stage}\right)$

The relevant equation is Equations (3-2)-(3–4). For example, the relevant equation for Learn stage is Equation (3-2): ${\beta}_i^{learn}=\overline{\beta^{learn}}+{\varepsilon}_i^{learn}$. We use a diffuse normal-inverse Gamma prior to sample $\overline{\beta^{stage}}$ and Ω^stage.

• Step 2-5) Sample ${\overline{\gamma}}_{lm}$ and Σ_lm ≡  var (ϵ_ilm)

The relevant equation is Equation (3-5): ${\gamma}_{ilm}={\overline{\gamma}}_{lm}+{\epsilon}_{ilm}$. We use a diffuse normal-inverse Gamma prior to sample ${\overline{\gamma}}_{lm}$ and Σ_lm.

• Step 2-6) Sample $\overline{\upsilon}$ and var(ξ_i).

The relevant equation is Equation (4-3): ${\upsilon}_i=\overline{\upsilon}+{\xi}_i$. We use a diffuse normal-inverse Gamma prior to sample $\overline{\upsilon}$ and var(ξ_i).