We propose a retrodictive forecasting paradigm for time series: instead of predicting the future from the past, we identify the future that best explains the observed present via inverse MAP optimization over a Conditional Variational Autoencoder (CVAE). This conditioning is a statistical modeling choice for Bayesian inversion; it does not assert that future events cause past observations. The approach is theoretically grounded in an information-theoretic arrow-of-time measure: the symmetrized Kullback-Leibler divergence between forward and time-reversed trajectory ensembles provides both the conceptual rationale and an operational GO/NO-GO diagnostic for applicability. We implement the paradigm as MAP inference over an inverse CVAE with a learned RealNVP normalizing-flow prior and evaluate it on six time series cases: four synthetic processes with controlled temporal asymmetry and two ERA5 reanalysis datasets (wind speed and solar irradiance). The work makes four contributions: (i) a formal retrodictive inference formulation; (ii) an inverse CVAE architecture; (iii) a model-free irreversibility diagnostic; and (iv) a falsifiable validation protocol with four pre-specified predictions. All pre-specified predictions are empirically supported: the diagnostic correctly classifies all six cases; the learned flow prior improves over an isotropic Gaussian baseline on GO cases; the inverse MAP yields no spurious advantage on time-reversible dynamics; and on irreversible GO cases, it achieves competitive or superior RMSE relative to forward baselines, with a statistically significant 17.7% reduction over a forward MLP on ERA5 solar irradiance. These results provide a structured proof-of-concept that retrodictive forecasting can constitute a viable alternative to conventional forward prediction when statistical time-irreversibility is present and exploitable.
Flow prior improves inverse inference vs N(0,I) on GO cases
A, C
RMSE
RMSE flow< RMSE N(0,I)
A: 1.038<1.074 ; C: 0.782<0.872
P3
No meaningful directional advantage on NO-GO cases
B, D
Ratio Inv/MLP
≥0.95 ( ±5% )
B: 1.870; D: 0.984
P4
Inverse MAP competitive with or beats MLP on GO cases
A, C, ERA5, ERA_ssrd
Ratio Inv/MLP
≤1.05 ( ±5% )
A: 0.897; C: 1.014; ERA5: 0.991; ERA_ssrd: 0.823
Table 3: Falsifiable predictions. Four pre-specified predictions with pass criteria and observed outcomes (Section 9 ).
Figure 1: Training curves (ELBO loss) for all six cases. Solid: training loss; dashed: validation loss. Stable convergence without overfitting across all cases confirms correct CVAE training.
Figure 2: Cross-case RMSE comparison (all six cases, five methods). The inverse MAP (flow, green) is competitive or best on all GO cases, while the forward MLP (orange) clearly dominates on Case B ( NO-GO ). Case D shows near-perfect equivalence across methods.
Figure 3: RMSE per forecast horizon h∈{1,…,16} for all six cases and five methods. GO cases (A, C, ERA5, ERA_ssrd): the inverse MAP with flow prior (dark green) is competitive with or below the forward MLP (orange) across most horizons, with the advantage widening at longer horizons on ERA_ssrd. NO-GO cases (B, D): the forward MLP dominates on Case B; Case D shows near-equivalence consistent with time-reversibility. The N(0,I) prior (pink) uniformly underperforms the flow prior on GO cases. Compare with Fig. 10 ( GO cases only, overlay).
Figure 4: Falsifiable predictions scorecard. All 4/4 predictions PASS. Summary panel produced by the automated verification module.
Case
Verdict
Naïve
MLP
CVAE
Inv(flow)
Ratio
DM stat
DM p -value
A
GO
1.030
1.156
1.048
1.038
0.897
−22.9
<0.001 ✓
B
NO-GO
1.728
0.066
0.087
0.124
1.870
+19.3
<0.001
C
GO
0.883
0.772
0.749
0.782
1.014
+1.7
0.092 (ns)
D
NO-GO
1.001
0.622
0.616
0.612
0.984
−8.1
<0.001
ERA5
GO
1.384
1.034
0.931
1.025
0.991
−0.5
0.642 (ns)
ERA_ssrd
GO
0.963
0.195
0.168
0.160
0.823
−10.4
<0.001 ✓
Table 4: Main results. RMSE in standardized space for all six cases and five methods. Ratio = RMSE Inv(flow) / RMSE MLP . Green ratios satisfy the paradigm prediction; DM stat and p-value from two-sided Diebold–Mariano test vs. MLP. ✓ = paradigm-confirming significant result.
Figure 5: Arrow-of-time diagnostic (block permutation test) for all six cases. Top panels: Jobs(w) at w=2,4,8 for LEVEL (left) and DIFF (right) representations. Bottom panels: −log10(p) with the α=0.05 threshold (red dashed). GO cases (A, C, ERA5, ERA_ssrd) exceed the threshold; NO-GO cases (B, D) remain below. Case C dominates with Jobs up to 22.9 in LEVEL; ERA_ssrd leads in DIFF with Jobs=5.3 .
Figure 6: Arrow-of-time strength per case ( Jobs median across all windows and scales). GO cases (green) are clearly separated from NO-GO cases (orange). Case C dominates ( Jobs=12.84 ); ERA_ssrd is the strongest real-world case (5.30). Cases B and D score ≈0 .
Figure 7: Prior ablation study (P2). RMSE comparison between Inverse MAP with flow prior (green) and N(0,I) prior (pink) for all six cases. The flow prior is systematically better on GO cases; improvements range from 3.3% (Case A) to 37% (ERA_ssrd).
Figure 8: MAP loss distributions (flow prior vs. N(0,I) prior) for all six cases. Lower MAP loss indicates better objective convergence. On GO cases (A, C, ERA5, ERA_ssrd), the flow prior distribution (green) is shifted toward lower loss values. The separation is largest for ERA_ssrd (flow median −121.6 vs. N(0,I) median −80.6 ). NO-GO cases (B, D) show overlapping distributions.
Figure 9: Forward-Inverse Chaining (FIC) landscape diagnostic for GO cases (A, C, ERA5, ERA_ssrd). Left panel: scatter of multi-start standard deviation (optimization roughness) vs. best-of-K MAP loss; right panel: cumulative distribution of MAP loss (flow prior). Low dispersion and a concentrated CDF indicate a tractable landscape. ERA_ssrd (P10–P90 range: 14.7 units, σ≈0.05 ) and Case A ( σ<0.15 ) exhibit well-conditioned landscapes; Case C ( σ=0.35 ) and ERA5 ( σ=0.23 ) show greater roughness.
Figure 10: RMSE per forecast horizon h for GO cases only (forward CVAE vs. inverse MAP; shaded = inverse advantage). ERA_ssrd shows expanding advantage at h>8 ; ERA5 shows crossover at h≈6 ; Cases A and C show moderate inverse advantage at mid-range horizons.
Figure 11: Example reconstructions for Case ERA_ssrd (representative). Left: best sample (RMSE =0.029 ); centre: median (0.130); right: worst (0.357). Inverse MAP (green) tracks the true future (dashed black) closely in the best and median cases; forward MLP (orange) degrades at longer horizons.
Figure 12: Multi-start dispersion (mean std of best- K MAP solutions) per case. Higher dispersion indicates a rugged optimization landscape. Case C (0.35) and ERA5 (0.23) are most challenging; ERA_ssrd (0.05) and Case A (0.08) are best-conditioned—consistent with their superior inverse MAP performance.
Figure 13: RetroNLL ( −logpθ(xobs∣y^,z^) ) vs. per-sample RMSE for GO cases. Theoretically expected: a negative correlation (low RetroNLL ⇔ accurate prediction). Case A: r=−0.037 , p=0.100 (ns)—no significant correlation, consistent with correct decoder coupling. Case C: r=−0.070 , p=0.002 ✓—expected negative correlation confirmed. Cases ERA5 ( r=+0.172 , p<0.001 ) and ERA_ssrd ( r=+0.088 , p=0.012 ) show positive correlations, indicating partial decoder likelihood–RMSE decoupling attributable to the homogeneous Gaussian reconstruction assumption.
Time series forecasting predicts future values from ordered observations using statistical, neural, and probabilistic models to support decisions under uncertainty.
Graduate School of China Academy of Engineering Physics, Beijing 100193, China · Institute of Applied Physics and Computational Mathematics, Beijing 100094, China