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Learning from Summarized Data: Gaussian Process Regression with Sample Quasi-Likelihood
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Proposes sample quasi-likelihood for Gaussian process regression trained only on summarized data (representative features, sample means, counts), with theoretical error bounds on marginal likelihood and posterior, validated on California housing.
Problem
Gaussian process regression when only summarized spatial data (representative locations, summary statistics, counts) are available due to confidentiality and management costs, so complete inputs and outputs cannot be used.
Approach
Replace the likelihood on complete outputs with a sample quasi-likelihood defined by a variance function on summary statistics, combined with input approximation via representative features; this yields closed-form marginal likelihood and posterior at O(m^3) cost, with approximation error bounds tied to granularity relative to the covariance length scale.
Key results
- Approximation RMSE between summarized and complete GP posterior means is lower for finer grids (e.g., MedValue Laplacian 0.79 at 1.6x1.6 grid vs 0.32 at 0.05x0.05 grid).
- At grid 0.4x0.4 with n=10000, the proposed method achieves comparable or lower RMSE than summarized VI on several outputs (e.g., MedValue Gaussian 0.904 vs 1.008) while running in about 6 s versus about 40 s.
- Errors shrink as summarized data becomes finer relative to the covariance length scale, consistent with the theory.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use summarized-data sample quasi-likelihood instead of complete-data VI | California housing spatial prediction · MedInc, Gaussian | normalized RMSE: 0.881 \pm 0.014 -> 0.970 \pm 0.004 | Hurt |
| Use summarized-data sample quasi-likelihood instead of complete-data VI | California housing spatial prediction · MedInc, Poisson | normalized RMSE: 0.887 \pm 0.024 -> 0.977 \pm 0.008 | Hurt |
| Use summarized-data sample quasi-likelihood instead of complete-data VI | California housing spatial prediction · HouseAge, Gaussian and Poisson | normalized RMSE: 0.880 \pm 0.049 -> 1.000 \pm 0.000; 0.861 \pm 0.049 -> 1.032 \pm 0.060 | Hurt |
| Use summarized-data sample quasi-likelihood instead of summarized VI | California housing spatial prediction · AveRooms, Gaussian and Poisson | normalized RMSE: 0.995 \pm 0.014 -> 0.975 \pm 0.051; 0.995 \pm 0.015 -> 0.949 \pm 0.012 | Helped |
| Use summarized-data sample quasi-likelihood instead of summarized VI | California housing spatial prediction · AveBedrms, Gaussian | normalized RMSE: 0.996 \pm 0.012 -> 0.964 \pm 0.041 | Helped |
| Use summarized-data sample quasi-likelihood instead of summarized VI | California housing spatial prediction · AveBedrms, Poisson | normalized RMSE: 0.941 \pm 0.016 -> 0.962 \pm 0.044 | Hurt |
| Use summarized-data sample quasi-likelihood instead of summarized VI | California housing spatial prediction · Population, Gaussian | normalized RMSE: 1.041 \pm 0.085 -> 1.000 \pm 0.000 | Helped |
| Use summarized-data sample quasi-likelihood instead of complete-data VI | California housing spatial prediction · Population, Poisson | normalized RMSE: 0.990 \pm 0.009 -> 1.004 \pm 0.013 | Hurt |
| Use summarized-data sample quasi-likelihood instead of summarized VI | California housing spatial prediction · AveOccup, Gaussian | normalized RMSE: 1.002 \pm 0.006 -> 1.002 \pm 0.006 | No clear effect |
| Use summarized-data sample quasi-likelihood instead of summarized VI | California housing spatial prediction · MedValue, Gaussian and Poisson | normalized RMSE: 1.008 \pm 0.015 -> 0.904 \pm 0.010; 0.973 \pm 0.038 -> 0.909 \pm 0.011 | Helped |
| Use finer summarized-data grids (ablation) | Input approximation · California housing, MedInc, Laplacian covariance | normalized posterior-mean RMSE: 0.57 \pm 0.04 -> 0.32 \pm 0.05 | Hurt |
| Use finer summarized-data grids (ablation) | Input approximation · California housing, MedValue, Laplacian covariance | normalized posterior-mean RMSE: 0.79 \pm 0.03 -> 0.32 \pm 0.04 | Hurt |
Surprising findings
- For the Gaussian likelihood on the AveRooms output, the Poisson variance function performed better than the Gaussian one in the predictive test.
Limitations
- Input approximation performs poorly when summarized data is coarse.
- Laplace-based approximation degrades for non-Gaussian observations when group sizes are small.
- Theory currently relies on the sample mean; extension to other statistics and hyperparameter dependence remains open.
Experimental setup
Datasets: California housing dataset
Benchmarks: California housing dataset
Metrics: RMSE normalized by test-output standard deviation execution time
Baselines: Complete VI Summarized VI
Builds on: Laplace approximation quasi-likelihood sparse variational Gaussian process
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