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\begin{document}

\title{Homotopy Sampling and Data Assimilation}



\author[1]{Juan Restrepo}%
\author[2]{Jorge Ramirez}%
\author[1]{Robert Miller}%
\affil[1]{Oregon State University}%
\affil[2]{Universidad Nacional de Colombia}%


\vspace{-1em}



  
  \date{May 28, 2024}


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\begin{abstract}
Importance sampling is modified via \{\textbackslash{}it homotopy
continuation\} to improve the efficiency and success of the sampler. The
homotopy will use a known distribution as a starting empirical
importance sampling distribution and generate a continuous schedule
which culminates with the target distribution. The focus is the
estimation of the normalization constant of the target distribution. The
homotopy method is extended to a Bayesian setting, for stationary and
time dependent posterior distributions. The numerical implementation
uses a combination of sample averages, with sampling parameter N, and
homotopy stages M, where M is typically a small number. The algorithm
replaces homotopy stages for sampling steps, potentially resulting in a
better or more efficient importance sampler. Numerical experiments
suggest this is the case. The results also suggest that the method may
improve the efficiency of the sampler by concentrating the samples in
regions of greater impact.%
\end{abstract}%



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