Parametric Estimation Under Diffuse Observations: An Application On Election Polls

28 Pages Posted: 18 Oct 2022 Last revised: 4 Nov 2022

See all articles by Charles Thraves

Charles Thraves

University of Chile - Industrial Engineering; Instituto Sistemas Complejos de Ingenieria (ISCI)

Sebastián Morales

affiliation not provided to SSRN

Date Written: October 12, 2022


Most techniques to estimate distributions consider observations as a single-point. However, there are several applications in which observations have some degree of uncertainty which is known by the researcher. We propose a method in which Maximum Likelihood Estimation (MLE) methods can incorporate this noise dimension of the observed data. In this context, classical MLE can be seen as a particular case. In this more general setting, we show how the likelihood can be written as a product integral, while the log-likelihood can be expressed as a tractable expression, i.e., a function with sums and integrals. We show how the introduced technique can be used as a novel methodology to aggregate polls for election forecasting. In addition, this can also account for state biases and house effects, while also consider a time-decay weight factor for polls depending on how afar are these with respect to the prediction date. We apply the presented method in the US President Election of 2020, in which polls at each state are used to estimate the probability distribution of votes during the election period. In particular, each poll’ distribution, due to its sampling error, is considered as a noisy observation represented by a known probability density function. States biases and house effects are estimated from the US presidential election of 2016. The results obtained by the proposed method outperforms the ones obtained with a classical MLE in terms of achieving a higher log-likelihood value. In addition, the distributions estimated by the former method have more variability than those obtained with the latter one. The proposed framework can be applied to several other applications in which observations can be considered as known probability distribution.

Keywords: Maximum Likelihood Estimation, Election Forecasting, Polls, Diffuse Observations, Product Integral

Suggested Citation

Thraves, Charles and Morales, Sebastián, Parametric Estimation Under Diffuse Observations: An Application On Election Polls (October 12, 2022). Available at SSRN: or

Charles Thraves (Contact Author)

University of Chile - Industrial Engineering ( email )

República 701, Santiago

Instituto Sistemas Complejos de Ingenieria (ISCI) ( email )

Republica 695

Sebastián Morales

affiliation not provided to SSRN

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