{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,8,9]],"date-time":"2024-08-09T00:16:20Z","timestamp":1723162580733},"reference-count":62,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,5,8]],"date-time":"2023-05-08T00:00:00Z","timestamp":1683504000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,5,8]],"date-time":"2023-05-08T00:00:00Z","timestamp":1683504000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2024,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider the problem of estimating expectations with respect to a target distribution with an unknown normalising constant, and where even the un-normalised target needs to be approximated at finite resolution. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-index Monte Carlo (MIMC) as well as the efficiency of sequential Monte Carlo (SMC) for inference. In particular, the proposed method provably achieves the canonical complexity of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\hbox {MSE}^{-1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mtext>MSE<\/mml:mtext>\n                    <mml:mrow>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, while single-level methods require <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\hbox {MSE}^{-\\xi }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mtext>MSE<\/mml:mtext>\n                    <mml:mrow>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mi>\u03be<\/mml:mi>\n                    <\/mml:mrow>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\xi &gt;1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03be<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. This is illustrated on examples of Bayesian inverse problems with an elliptic PDE forward model in 1 and 2 spatial dimensions, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\xi =5\/4$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03be<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>5<\/mml:mn>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mn>4<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\xi =3\/2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03be<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, respectively. It is also illustrated on more challenging log-Gaussian process models, where single-level complexity is approximately <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\xi =9\/4$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03be<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>9<\/mml:mn>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mn>4<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and multilevel Monte Carlo (or MIMC with an inappropriate index set) gives <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\xi = 5\/4 + \\omega $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03be<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>5<\/mml:mn>\n                    <mml:mo>\/<\/mml:mo>\n                    <mml:mn>4<\/mml:mn>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mi>\u03c9<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, for any <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\omega &gt; 0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03c9<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, whereas our method is again canonical. We also provide novel theoretical verification of the product-form convergence results which MIMC requires for Gaussian processes built in spaces of mixed regularity defined in the spectral domain, which facilitates acceleration with fast Fourier transform methods via a cumulant embedding strategy, and may be of independent interest in the context of spatial statistics and machine learning.<\/jats:p>","DOI":"10.1007\/s10208-023-09612-z","type":"journal-article","created":{"date-parts":[[2023,5,8]],"date-time":"2023-05-08T16:03:33Z","timestamp":1683561813000},"page":"1249-1304","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Multi-index Sequential Monte Carlo Ratio Estimators for Bayesian Inverse problems"],"prefix":"10.1007","volume":"24","author":[{"given":"Ajay","family":"Jasra","sequence":"first","affiliation":[]},{"given":"Kody J. 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