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In this paper, the <jats:italic>maximum a posteriori estimate<\/jats:italic> is studied under the class of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\nu $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03bd<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> times differentiable linear time-invariant Gauss\u2013Markov priors, which can be computed with an iterated extended Kalman smoother. The maximum a posteriori estimate corresponds to an optimal interpolant in the reproducing kernel Hilbert space associated with the prior, which in the present case is equivalent to a Sobolev space of smoothness <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\nu +1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03bd<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Subject to mild conditions on the vector field, convergence rates of the maximum a posteriori estimate are then obtained via methods from nonlinear analysis and scattered data approximation. These results closely resemble classical convergence results in the sense that a <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\nu $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03bd<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> times differentiable prior process obtains a global order of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\nu $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03bd<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, which is demonstrated in numerical examples.<\/jats:p>","DOI":"10.1007\/s11222-021-09993-7","type":"journal-article","created":{"date-parts":[[2021,3,3]],"date-time":"2021-03-03T08:09:43Z","timestamp":1614758983000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["Bayesian ODE solvers: the maximum a posteriori estimate"],"prefix":"10.1007","volume":"31","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1102-7706","authenticated-orcid":false,"given":"Filip","family":"Tronarp","sequence":"first","affiliation":[]},{"given":"Simo","family":"S\u00e4rkk\u00e4","sequence":"additional","affiliation":[]},{"given":"Philipp","family":"Hennig","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,3,3]]},"reference":[{"key":"9993_CR1","doi-asserted-by":"crossref","unstructured":"Abdulle, A., Garegnani, G.: Random time step probabilistic methods for uncertainty quantification in chaotic and geometric numerical integration. 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