Reversals of signal-posterior monotonicity for any bounded prior

Paul Milgrom [Milgrom, P.R., 1981. Goods news and bad news: representation theorems and applications. The Bell Journal of Economics 12, 380-391] showed that if the strict monotone likelihood ratio property (MLRP) does not hold for a conditional distribution then there exists some non-degenerate prior and pair of signals where the higher-signal posterior does not stochastically dominate the lower-signal posterior. We show that for any non-degenerate prior with bounded support there exists a conditional distribution (satisfying several natural properties) and pair of signals such that the lower signal's posterior stochastically dominates that of the higher signal. Thus, for every bounded prior, higher signals may represent strictly "worse" news.