We deal with the situation covered by the univariate general linear model, that is, it is intended thatnobservations be generated in accordance with the usual model y =Xβ + ε however, it is feared thatkof the observations are spurious, that is, not generated in the manner intended, so that for an unknown set ofkdistinct integers, say (i1, …ik), a subset of the firstnintegers, we have, specifically, thatytj, =xtj′, β +aj, + ∊tj, where in generalxt′ denotes thet-th row ofX, and where (a1, …,ak), so called shift parameters, are such that – ∞ <aj, < ∞. In this paper, we discuss the posterior distribution of β, when indeed it is assumed a priori that any given set ofkobservations has uniform probability I/(nk) of being spurious. The properties of the posterior of β are discussed, and the results used in an example using data generated from a response surface design. Ad hoc procedures are discussed for gaining information onk, whenkis unknown. These ad hoc procedures are illustrated using a famous set of data from Darwin