Causal DAG-Guided Reinforcement Learning for Long-Term Physical Activity Maintenance: Using Prior Data to Inform Future Deployment

Xueqing Liu; Young Hyun Cho; Bhanu Teja Gullapalli; Predrag Klasnja; Susan Murphy

stat.APOct 26, 2026

Causal DAG-Guided Reinforcement Learning for Long-Term Physical Activity Maintenance: Using Prior Data to Inform Future Deployment

Authors: Xueqing Liu, Young Hyun Cho, Bhanu Teja Gullapalli, Predrag Klasnja, Susan Murphy

Organizations: Harvard University · University of Michigan

Abstract

Digital health interventions can provide scalable, personalized support for physical activity (PA), yet sustaining PA over the long-term remains difficult. Because affective experiences with PA are important for activity sustainment, we developed an online reinforcement learning (RL) system for the forthcoming ADAPT factorial trial that personalizes twice-daily activity suggestions aimed at improving affective associations with PA. This setting poses several challenges: participant-specific data are scarce relative to the candidate state space, important states and rewards are partially observed, interventions are frequent while the primary outcome is measured weekly, and activity suggestions can both improve PA-related experiences and increase intervention burden. We address these challenges using an expert-specified causal directed acyclic graph as an integrated RL design tool, guiding state construction, estimation of partially observed states, delayed credit assignment, and representation of beneficial and adverse intervention pathways. We further construct a longitudinal simulator calibrated from a prior micro-randomized trial to compare candidate RL designs under realistic state evolution, missingness, and participant heterogeneity before deployment. Together, this work illustrates how causal domain knowledge and prior trial data can make long-horizon, participant-specific RL more practical for digital health interventions targeting sustained behavior change.

Figures & tables

Panel of Figure 1 — Figure 1: Simplified working causal DAG for the ADAPT RL system. $A$ denotes activity-suggestion actions, $Y$ weekly CAE, $E$ weekly perceived utility, $M^{Y}$ CAE-related mediators, $M^{E}$ perceived-utility-related mediators, and $J$ weekly survey completion. To keep the diagram readable, the 12 within-week decision times and their associated mediators are represented by two illustrative decision points, indexed by $1$ and $2$ ; some daily mediators may depend on both within-day actions. Context variables $C$ , CAE observation variables $O^{Y}$ , and perceived-utility emissions $U$ are omitted.

Process	Coverage
Weekly CAE	92.7%
Next-four-hour step count	89.9%
Anticipated affect	93.8%
Perceived-utility-related mediators	97.7%

Table 1: Participant-averaged coverage of observed MRT measurements by pointwise 95% simulation intervals.

Figure 2: Cumulative 36-week CAE for each policy relative to never send, paired by experiment and participant draw. Panels are the three testbed environments. Dots are means over 200 simulated trials of 75 participants; error bars are 95% confidence intervals.

Appendix figures & tables39 assets

Supplementary material from the paper’s appendix.

Appendix

Factorial Trial Variable	MRT Surrogate
Short weekly survey measuring affective association of physical activity: 1. “For me, being active is …” with response options from unenjoyable (1) to enjoyable (7); 2. “Exercise leaves me feeling …” with response options from exhausted (1) to energized (7); 3. “For me, exercise is …” with response options from stressful (1) to relaxing (7).	Corresponding items from the Core Affective Experiences (CAE) survey: 1. “Exercise feels …” with response options from very terrible (1) to wonderful (7); 2. the CAE item corresponding to the energized–exhausted dimension; 3. the CAE item corresponding to the relaxing–stressful dimension.
Daily morning check-in survey: 1. whether the participant was active yesterday (this can also be reported elsewhere); 2. anticipated affect: “Next time I am active, I expect it will feel …”	The MRT end-of-day survey on day $d$ is used as the surrogate for the factorial trial morning check-in on day $d+1$ . In particular: 1. the MRT end-of-day item “Were you active at least once today?”; 2. the MRT end-of-day anticipated-affect item.
Active minutes: 1. Prior 2 hour active minutes 2. Next 4 hour active minutes	Step count: 1. Prior 2 hour step count 2. Next 4 hour step count

Appendix

Table 2: Mapping Factorial Trial Target Variables to MRT Surrogates

\widehat{\mathcal{S}}_{w}^{\mathrm{PF}}\leftarrow\textsc{UpdateEBVariancesPF}(H_{w-1}),

Appendix

Algorithm 1 Estimating Belief State

b_{w}(y)

on Monday Night of Week

w

(

w\geq 2

)

Component	Estimated weight
Intercept	$0.915$
$\widehat{e}_{w-1}$	$0.119$
$\overline{M}_{w-1}^{E,PV}$	$0.318$
$\overline{M}_{w-1}^{E,FW}$	$0.232$
$\overline{M}_{w-1}^{E,PJ}$	$0.905$
$J_{w}^{\mathrm{week}}$	$0.000$

Appendix

Table 3: Estimated weights of the perceived-utility approximation

\widehat{e}_{w}

in ( 4 ).

s^{2}=\max\left\{\frac{1}{n}\left\|\mathbf{y}-\mathbf{X}\mu_{0}\right\|_{2}^{2},10^{-12}\right\},\qquad c=\log(s^{2}).

Appendix

Algorithm 5 Generic Empirical-Bayes Variance Update

		$\displaystyle\phi_{1}^{-}\leftarrow\textsc{ConstructFeature}(w,d,t,1;\widehat{e}_{w-1},\widehat{b}_{w-1},\widetilde{b}_{w-1},\mathcal{M}_{w,d,t}^{\mathrm{pre}},C_{w,d,t}),$
		$\displaystyle\phi_{0}^{-}\leftarrow\textsc{ConstructFeature}(w,d,t,0;\widehat{e}_{w-1},\widehat{b}_{w-1},\widetilde{b}_{w-1},\mathcal{M}_{w,d,t}^{\mathrm{pre}},C_{w,d,t}).$

Appendix

Algorithm 6 Online RL Algorithm for the ADAPT Factorial Trial

Variant	Policy class	$\gamma$	Bottleneck TD	Reward design
1	RL	$0.9$	No	None
2	RL	$0.5$	No	None
3	RL	$0.9$	Yes	None
4	RL	$0.9$	No	V1
5	RL	$0.9$	No	V2
6	Never send	–	–	–

Appendix

Table 4: Summary of RL variants and baseline policies. Reward designs: V1 = return-invariant shaping; V2 = return-invariant shaping with redistribution.

Figure 3: Participant-conditional CAE residuals for the 28 fitted participant models. Residuals are expressed on the standardized fitting scale and plotted against study week. Points denote observed assessments, lines connect consecutive observed weeks, and dashed horizontal lines mark zero.

Figure 4: Participant-conditional short CAE residuals for the 28 fitted participant models. Residuals are expressed on the standardized short CAE fitting scale. Points denote observed assessments, lines connect consecutive observed weeks, and dashed horizontal lines mark zero. All panels use common axes and the participant order established for CAE.

Figure 5: Participant-conditional FourSC residuals for the 28 fitted participant models. Points show observed residuals on the standardized log-step-count fitting scale, and dashed lines mark zero. All panels use common axes.

Figure 6: Participant-conditional anticipated-affect residuals for the 28 fitted participant models. Points show observed residuals on the normalized fitting scale, and dashed lines mark zero. All panels use common axes.

Participant	Page-view occurrence $O$				Fitbit wear $M^{E,FW}$				Daily check-in $M^{E,PJ}$
	$n$	$\bar{z}$	$\bar{p}$	BS	$n$	$\bar{z}$	$\bar{p}$	BS	$n$	$\bar{z}$	$\bar{p}$	BS
1	154	0.416	0.418	0.234	76	0.842	0.847	0.129	77	0.714	0.717	0.168
2	154	0.377	0.375	0.224	76	0.895	0.896	0.089	77	0.662	0.662	0.131
3	154	0.026	0.029	0.023	77	0.883	0.879	0.086	77	0.221	0.214	0.146
4	154	0.292	0.288	0.136	76	0.803	0.804	0.151	77	0.169	0.187	0.096
5	154	0.130	0.130	0.111	76	0.868	0.866	0.083	77	0.156	0.157	0.087

Appendix

Table 5: Participant-specific Bernoulli model-fit diagnostics.

Participant	Prior two-hour steps			Positive page-view intensity
	$n$	Mean residual	RMSE	$n$	Mean residual	RMSE
1	134	0.005	0.808	64	0.002	0.814
2	141	0.019	1.266	58	0.008	0.848
3	139	-0.012	0.813	4	0.000	0.002
4	132	0.008	0.670	45	0.017	0.574
5	132	-0.004	0.927	20	-0.020	0.694

Appendix

Table 6: Participant-specific fit diagnostics for prior two-hour steps and positive page-view intensity.

Participant	Helpfulness $U_{w,1}$			Pleasantness $U_{w,2}$
	$n$	Mean residual	RMSE	$n$	Mean residual	RMSE
1	11	-0.009	0.078	11	0.001	0.007
2	10	-0.006	0.167	10	0.032	0.216
3	6	0.000	0.009	6	0.008	0.135
4	5	-0.004	0.216	5	0.019	0.208
5	3	0.000	0.000	3	0.000	0.002

Appendix

Table 7: Participant-specific model-fit diagnostics for the two perceived-utility items.

Participant	CAE		Short CAE		$U_{w,1}$		$U_{w,2}$		$J_{w}$
	$N$	$\widehat{\rho}_{1}$	$N$	$\widehat{\rho}_{1}$	$N$	$\widehat{\rho}_{1}$	$N$	$\widehat{\rho}_{1}$	$N$	$\widehat{\rho}_{1}$
1	10	-0.111	10	0.113	10	-0.549	10	-0.009	10	0.167
2	9	-0.723	9	—	9	-0.305	9	0.195	10	0.250
3	4	-0.152	4	-0.789	4	-0.284	4	0.197	10	0.370
4	1	—	1	—	1	—	1	—	10	-0.306
5	1	—	1	—	1	—	1	—	10	0.002

Appendix

Table 8: Participant-specific one-week lagged residual correlations.

Participant	Anticipated affect				Fitbit wear
	$N_{1}$	$\widehat{\rho}_{1}$	$N_{7}$	$\widehat{\rho}_{7}$	$N_{1}$	$\widehat{\rho}_{1}$	$N_{7}$	$\widehat{\rho}_{7}$
1	39	-0.111	39	0.026	75	0.059	69	0.020
2	40	0.533	37	0.173	75	0.041	69	-0.096
3	4	0.995	5	0.612	76	0.025	70	0.040
4	4	0.941	1	—	75	-0.017	69	-0.159
5	4	-0.997	3	0.243	75	0.312	69	-0.021

Appendix

Table 9: Participant-specific lagged residual correlations for daily components.

AM $\rightarrow$ same-day PM
Participant	FourSC		Prior 2h steps		PV occurrence		PV intensity		Interaction
	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$
1	61	0.110	60	-0.014	77	-0.052	15	0.085	20	0.241
2	34	-0.237	67	-0.059	77	0.020	13	0.440	25	0.092
3	0	—	67	-0.050	77	-0.331	0	—	16	1.000
4	64	0.107	59	0.127	77	-0.098	9	-0.335	26	-0.109

Appendix

Table 10: Participant-specific residual correlations between adjacent AM/PM occasions.

Participant	$Y,\widetilde{Y}$		$Y,U_{1}$		$Y,U_{2}$		$\widetilde{Y},U_{1}$		$\widetilde{Y},U_{2}$		$U_{1},U_{2}$
	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$
1	11	0.392	11	0.599	11	-0.144	11	0.334	11	0.107	11	-0.717
2	10	—	10	-0.600	10	0.035	10	—	10	—	10	0.343
3	6	-0.633	6	0.193	6	-0.493	6	0.435	6	-0.022	6	-0.794
4	5	0.566	5	0.103	5	-0.781	5	0.609	5	-0.613	5	0.206
5	3	-0.395	3	—	3	-0.974	3	—	3	0.174	3	—

Appendix

Table 11: Participant-specific residual correlations among continuous end-of-week assessments.

Participant	$Y,J^{\mathrm{week}}$		$\widetilde{Y},J^{\mathrm{week}}$		$U_{1},J^{\mathrm{week}}$		$U_{2},J^{\mathrm{week}}$
	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$
1	11	-0.627	11	-0.475	11	-0.498	11	-0.116
2	10	0.631	10	—	10	-0.130	10	0.401
3	6	0.224	6	-0.388	6	-0.571	6	0.160
4	5	-0.654	5	-0.365	5	0.327	5	0.882
5	3	0.450	3	0.643	3	—	3	-0.642

Appendix

Table 12: Participant-specific residual associations between check-in completion and continuous assessments.

Participant	AA–FW		AA–PJ		AA–AD		FW–PJ		FW–AD		PJ–AD
	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$
1	55	-0.085	55	-0.038	12	0.143	77	0.017	12	-0.275	12	-0.370
2	51	-0.042	51	-0.444	13	0.646	77	0.054	13	-0.219	13	-0.224
3	17	-0.309	17	-0.023	13	0.184	77	-0.024	13	0.013	13	0.178
4	13	0.060	13	-0.200	7	—	77	-0.100	7	—	7	—
5	12	-0.039	12	0.108	10	-0.358	77	0.112	10	0.541	10	0.489

Appendix

Table 13: Participant-specific joint residual correlations for daily components.

Participant	AM FourSC–AM P2h		PM FourSC–PM P2h		AM FourSC–PM P2h
	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$
1	62	-0.039	63	-0.004	66	0.530
2	67	0.155	34	-0.321	72	0.293
3	67	0.053	0	—	69	0.212
4	61	-0.108	64	0.046	70	0.488
5	64	-0.032	59	0.114	63	0.300

Appendix

Table 14: Participant-specific joint residual correlations between step-count components.

Participant	FourSC– $O$		FourSC–PV +		FourSC–AS		P2h– $O$		P2h–PV +		P2h–AS
	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$
1	68	0.007	34	-0.108	32	0.104	65	0.343	32	0.407	31	0.236
2	72	-0.006	28	0.337	44	0.090	69	0.245	28	0.114	41	0.038
3	70	-0.186	4	-0.264	32	-0.759	69	0.090	4	-0.293	33	-0.035
4	72	-0.088	21	0.130	38	0.026	62	-0.009	19	0.040	34	0.047
5	65	-0.204	9	-0.065	30	-0.251	67	0.043	8	-0.708	31	-0.008

Appendix

Table 15: Participant-specific residual correlations between step-count and engagement components at AM occasions.

Participant	FourSC– $O$		FourSC–PV +		FourSC–AS		P2h– $O$		P2h–PV +		P2h–AS
	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$
1	65	-0.018	26	-0.110	35	-0.240	69	-0.158	27	-0.202	36	0.020
2	35	-0.076	11	0.585	19	0.144	72	0.156	25	-0.229	37	-0.022
3	0	—	0	—	0	—	70	0.081	0	—	32	0.344
4	64	0.041	20	-0.432	32	-0.172	70	-0.175	21	0.112	36	-0.017
5	60	-0.010	7	-0.434	25	-0.114	65	-0.088	8	-0.191	28	-0.462

Appendix

Table 16: Participant-specific residual correlations between step-count and engagement components at PM occasions.

AM
Participant	$O$ –PV +		$O$ –AS		PV + –AS
	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$	$N$	$\widehat{\rho}$
1	34	0.084	35	0.478	15	0.497
2	30	-0.011	48	0.447	16	0.116
3	4	0.154	35	0.532	3	0.277
4	21	0.141	41	0.448	14	-0.167

Appendix

Table 17: Participant-specific joint residual correlations among page-view and interaction components.

Figure 7: Observed and simulated CAE for the 28 participants under their recorded MRT action histories. Gray bands show pointwise 95% simulation intervals, red points and lines show observed CAE, and crosses mark missing MRT assessments. All panels use common axes.

Figure 8: Observed and simulated short CAE for the 28 participants under their recorded MRT action histories. Gray bands show pointwise 95% simulation intervals, and red points and lines show observed assessments. Crosses below each panel mark missing assessments, which are not joined by lines. All panels use the original 1–7 scale and a common study-week axis.

Figure 9: Observed and simulated FourSC for the 28 participants under their recorded MRT action histories. Gray bands show pointwise 95% simulation intervals, and red points and lines show observed step counts. All panels use a common

\log(1+\mathrm{steps})

axis labeled in step counts. Missing observations are not joined.

Figure 10: Observed and simulated anticipated affect for the 28 participants under their recorded MRT action histories. Gray bands show pointwise 95% simulation intervals, and red points and lines show recorded responses. All panels use the original 1–7 scale and a common study-day axis. Missing responses are not joined.

Figure 11: Observed and simulated weekly page-view occurrence for the 28 participants under their recorded MRT action histories. Red points and lines show the percentage of decision occasions with at least one page view. Gray bands show pointwise 95% simulation intervals calculated after weekly aggregation within each trajectory. All panels share the same axes.

Figure 12: Observed and simulated mean page-view counts among positive occasions in each study week. Gray bands show pointwise 95% simulation intervals and red points show observed means. Crosses mark MRT weeks with no positive count, rather than missing logging data. All panels share a logarithmic count axis.

Figure 13: Observed and simulated Fitbit-wear proportions by study week. Gray bands show pointwise 95% simulation intervals and red points and lines show MRT proportions. Outcomes are aligned to their actual morning dates. All panels share the same percentage axis.

Figure 14: Observed and simulated daily check-in completion proportions by study week. Each proportion uses seven daily opportunities. Gray bands show pointwise 95% simulation intervals and red points and lines show observed proportions. Noncompletion remains an observed zero.

Figure 15: Observed and simulated assessment-completion proportions over the most recent four study weeks. The initial points use the shorter available history. Gray bands show pointwise 95% simulation intervals and red points and lines show the corresponding MRT summaries. Adjacent plotted points share weeks.

Figure 16: Observed and simulated usefulness responses. Gray bands show pointwise 95% intervals from all generated responses and red points show MRT responses. Missing assessments remain gaps and are marked by crosses below the panels. All panels share the original 1–7 scale.

Figure 17: Observed and simulated pleasantness responses. Gray bands show pointwise 95% intervals from all generated responses and red points show MRT responses. Missing assessments remain gaps and are marked by crosses below the panels. All panels share the original 1–7 scale.

Figure 18: Pointwise 95% simulation intervals for CAE across 36 weeks under the never-send policy. Each panel represents one participant and uses 100 trajectories. Sunday suggestions are fixed at zero. No observed MRT trajectory is superimposed on this target-schedule assessment.

Figure 19: Pointwise 95% simulation intervals for CAE across 36 weeks under the always-send policy. Each panel represents one participant and uses 100 trajectories. Sunday suggestions are fixed at zero. No observed MRT trajectory is superimposed on this target-schedule assessment.

Figure 20: Pointwise 95% simulation intervals for CAE across 36 weeks under the random-send policy. Each panel represents one participant and uses 100 trajectories. Sunday suggestions are fixed at zero. No observed MRT trajectory is superimposed on this target-schedule assessment.

Figure 21: Population-average send probability by decision slot and week, with 95% confidence intervals across 200 simulation replications. Rows correspond to morning and afternoon decision slots; columns correspond to STE $\approx 0.2$ , vanilla (STE $\approx 0.5$ ), and STE $\approx 0.8$ environments. The gray dashed line denotes random send ( $p=0.5$ ).

Figure 22: Participant-specific send probability by decision slot and week for an illustrative participant (user 13), based on 186 replications in which this participant model was sampled. The fitted environment for this participant favors morning activity suggestions.

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