Statistical Detection of Errors/Fraud in Announced Election Results: Model and Its Verification. The Case of the 2025 Elections in Poland.

The potential use of statistical models to detect election fraud has long been explored, with numerous effective models documented in the literature. These models apply to various types of elections - both majoritarian/personal and proportional (party-list) systems. However, due to the inherent nature and politically charged context of the issue, these models rarely undergo empirical verification, rendering discussions of their properties and value largely theoretical. The 2025 Polish presidential election stands out as a unique case. Doubts about the accuracy of vote counting in Poland's 2025 presidential election prompted the Attorney General of the Republic of Poland to commission the author of this presentation to develop a model capable of identifying polling stations where improper vote tallying likely occurred with high probability. Subsequently, prosecutors manually recounted votes in the stations flagged by the model. Of the 247 stations identified by my model, 83 showed discrepancies between the announced results and the prosecutors' recounts, while 164 revealed no differences. Although the state's actions were not intended to validate the statistical model's accuracy, they inadvertently created conditions for such verification. A relatively straightforward model -employing multiple linear regression of second-round results against first-round outcomes -proved effective at capturing tallying errors (without distinguishing unintentional mistakes from deliberate fraud). Opportunities for refinement are clear, however. The recount focused solely on model-identified stations, providing insight into false positives among "suspicious" cases but no direct measure of false negatives - stations with irregularities overlooked by the model. A refined model, using an adjusted criterion for flagging suspicious stations, allows comparison with prosecutorial findings: it boasts a high true positive rate, few missed irregularities (that the original model had flagged and prosecutors confirmed), and relatively few novel predictions (unconfirmed by recounts). This demonstrates that, while not optimal, the original model was sufficiently accurate and efficient to estimate errors in announced results with enough precision to reliably confirm they did not alter the ultimate election winner. The presentation will cover the original model developed for the Prosecutor's Office, a comparison of its predictions against recount outcomes in flagged stations, the refined model, and a discussion of how the refined model aligns with verified recount results.