Threshold-censored normal likelihood ($L(\mu ,\sigma )={\prod }_i\Phi (s_i(\mu -c_i)/\sigma )$)

Suppose conditionally independent latent values have normal distribution $N(\mu ,{\sigma }^2)$, with $\sigma >0$, but the observation only says which side of a known threshold $c_i$ contains each value. With $s_i=1$ for a value above the threshold and $s_i=-1$ for a value below, the likelihood is the displayed product, where $\Phi$ is the standard normal distribution function. Repeated thresholds give corresponding powers. These factors are tail probabilities, not probability densities evaluated at the thresholds. Multiplying normal densities at $c_i$ would describe a different observation process. The thresholds must be treated as fixed or otherwise modelled appropriately; informative selection of thresholds can change the likelihood.