We develop a statistically explicit sentiment index for Google Play user reviews and establish the mathematical results supporting its construction. Normalized star ratings and text-sentiment scores are treated as noisy measures of latent review valence and fused by covariance-aware inverse-variance weighting. Review-level estimates are aggregated with bounded helpfulness and recency weights, then shrunk toward a population mean using estimated precision rather than an arbitrary review-count threshold. App-level rating histograms provide a distributional diagnostic for samples returned under different API sort orders; because star ratings are discrete, classical continuous Kolmogorov-Smirnov critical values are not used. A local-level state-space model and the Kalman filter provide a denoised temporal trend. Full proofs cover the BLUE and Gaussian maximum-likelihood result, Gaussian-conjugate shrinkage, the Glivenko-Cantelli and Donsker theorems, count transformations via the delta method, and exact Gaussian Kalman filtering. A worked three-review example shows how textual complaints can materially reduce an apparently perfect star-only score.
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