A search for the standard model Higgs boson (H) decaying to $\mathrm{b\bar b}$ when produced in association with a weak vector boson (V) is reported for the following modes: \mathrm{W}(\mu\nu)\mathrm{H},~ \mathrm{W}(e\nu)\mathrm{H},~ \mathrm{Z}(\mu\mu)\mathrm{H},~ \mathrm{Z}(ee)\mathrm{H} and $\mathrm{Z}(\nu\nu)\mathrm{H}$. The search is performed in data samples corresponding to integrated luminosities of $5.0$ fb$^{-1}$ at $\sqrt{s}=$7 TeV and 12.1 fb$^{-1}$ at $\sqrt{s}=$8 TeV, recorded by the CMS experiment at the LHC. Upper limits, at the 95$\%$ confidence level, on the VH production cross section times the $\mathrm{H} \to \mathrm{b\bar b}$ branching ratio, with respect to the expectations for a standard model Higgs boson, are derived for a Higgs boson in the mass range 110-135 GeV. In this range, the observed upper limits vary from 1.0 to 4.2 times the standard model prediction; the corresponding expected limits vary from 0.9 to 1.9. At a Higgs boson mass of 125 GeV the observed limit is 2.5 and the expected limit is 1.2. An excess of events is observed above the expected background with a local significance of 2.2 standard deviations, which is consistent with the expectation from the production of the standard model Higgs boson.

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