other than the MSSM fields are heavier than TeV such that they are captured only through the effective higher dimensional operators after integrating out them. Thus the spectrum is the same as the MSSM below TeV. In this framework we propose a possibility of light Higgs which might have been produced at LEP I/II [15] [16] [17]. If CP odd Higgs (A) is lighter than Upsilon (bottomonium), it can alter Higgs decay significantly. In addition if light CP even Higgs (h) is lighter than Z boson such that Z → hA is kinematically allowed, b ¯ b produced from h decay can affect the electroweak precision data measured at LEP. In this case the suppression of Z → Zh is possible with the aid of BMSSM operators. The contents of the Letter is following. Firstly, various Higgs search bounds are briefly reviewed to convince that light Higgs scenario proposed here is compatible with all the existing bounds. Secondly, we take the sample points of the BMSSM which have interesting features and can have interesting implications for the electroweak precision data of LEP. Thirdly, we attempt to soften the discrepancy between the SM predictions and the LEP precision measurements within the scenario. Finally, we summarize and conclude. For the SM Higgs, LEP bound on the Higgs mass is 114 GeV at 95 percent confidence level (C.L.). This equally applies to the Higgs field which has the same coupling to Z boson and decays in the same way as the SM Higgs. Therefore, there are two ways to avoid the bound. If the production is suppressed, the bound becomes weaker. The modification of decay also makes the bound weaker depending on the channel. For the production, Higgsstrahlung is suppressed for light CP even Higgs if ZZh coupling is small. For g 2 ZZh = 0 . 04 g 2 SM ZZh , the Higgs can be as light as 70 to 75 GeV from decay mode independent search [18]. In this case the other CP even Higgs H couples to Z boson with almost the same strength as the SM Higgs since there is a sum rule, g 2 ZZh + g 2 ZZH = g 2 SM ZZh . For H , we can modify its decay if CP odd Higgs decays mostly to AA rather than to b ¯ b . If Br ( H → b ¯ b ) ≤ 0 . 2 and m$_{A}$ < 10 GeV (lighter than 2 m$_{b}$ ), H can be as light as 100 GeV as H → AA → 4 τ does not give a strong constraint. Once m$_{A}$ is very light, we also expect the charged Higgs mass to be close to W boson mass in MSSM-like theories as the tree level mass relation between charged Higgs, CP odd II. LEP HIGGS SEARCH BOUNDS 3