Recent growing adoption of experimentation in practice has led to a surge of attention to multiarmed bandits as a technique to reduce the opportunity cost of online experiments. In this setting, a decision-maker sequentially chooses among a set of given actions, observes their noisy rewards, and aims to maximize her cumulative expected reward (or minimize regret) over a horizon of length $T$. In this paper, we introduce a general analysis framework and a family of algorithms for the stochastic linear bandit problem that includes well-known algorithms such as the optimism-in-the-face-of-uncertainty-linear-bandit (OFUL) and Thompson sampling (TS) as special cases. Our analysis technique bridges several streams of prior literature and yields a number of new results. First, our new notion of optimism in expectation gives rise to a new algorithm, called sieved greedy (SG) that reduces the overexploration problem in OFUL. SG utilizes the data to discard actions with relatively low uncertainty and then choosing one among the remaining actions greedily. In addition to proving that SG is theoretically rate optimal, our empirical simulations show that SG outperforms existing benchmarks such as greedy, OFUL, and TS. The second application of our general framework is (to the best of our knowledge) the first polylogarithmic (in $T$) regret bounds for OFUL and TS, under similar conditions as the ones by Goldenshluger and Zeevi (2013). Finally, we obtain sharper regret bounds for the $k$-armed contextual MABs by a factor of $\sqrt{k}$.