Explain the bias-variance tradeoff for an estimator

In robotics state estimation and learning, we often evaluate an estimator $\hat{\theta}$ of an unknown quantity $\theta$ using its mean squared error. Explain what the bias and variance of an estimator are, how they combine to determine the expected squared error, and what the resulting tradeoff means in practice. Give a concrete example from robotics (e.g. tuning a filter or fitting a sensor-calibration model) where deliberately accepting some bias reduces overall error.