A search for exotic decays of the Higgs boson to a pair of light pseudoscalar particles $\mathrm{a_1}$ is performed under the hypothesis that one of the pseudoscalars decays to a pair of opposite sign muons and the other decays to a $\mathrm{b}\mathrm{\overline{b}}$ pair. Such signatures are predicted in a number of extensions of the standard model (SM), including next-to-minimal supersymmetry and two-Higgs-doublet models with an additional scalar singlet. A data sample corresponding to an integrated luminosity of 35.9\,\mathrm{fb}^{-1} recorded with the CMS detector in 2016 is studied. No statistically significant excess is observed with respect to the SM backgrounds in the search region for pseudoscalar masses from 20\,\mathrm{GeV} to half of the Higgs boson mass. Upper limits at $95\%$ confidence level are set on the production cross section times branching ratio, $\sigma_{\mathrm{h}}\times{\mathcal B}(\mathrm{h}\to \mathrm{ a_1}\mathrm{ a_1}\to\mu^+\mu^{-}\mathrm{b}\mathrm{\overline{b}})$, ranging from 5 to 36\,\mathrm{fb}, depending on the pseudoscalar mass. Corresponding limits on the branching ratio, assuming the SM prediction for $\sigma_{\mathrm{h}}$, are $(1$--$6)\times 10^{-4}$.

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