Dynamic Belief Networks (DBNs) have become a popular method for monitoring dynamical processes in real-time. However DBN evaluation has the same problems of computational intractability as ordinary belief networks, with additional exponential complexity as the number of time-slices increases. Several approximate methods for fast DBN evaluation have been devised [1,3,11]. We present a new method which simplifies evaluation by selectively “forgetting” past events and their relationships to the present. This is done by pruning, from past time-slices, arcs and nodes which are deemed less relevant to the current time-slice, as determined by the arc weight measure introduced in [15]. This approach is more flexible than a fixed-size window and can be combined with other approximate evaluation techniques.