I thought I should illustrate the following story with two pictures.
Columbo (C) collects documents (*) about the murder case (the black vertical lines). He can easily predict the future of those documents, because they are stable (otherwise they would not be good documents). However, he can postdict the past of those documents only up to a certain point, when they have been created.
Different documents tell a coherent story, therefore he can assume that they were created by the same event E. But notice that he cannot postdict the state of those documents before E. The future of those documents is known, but their past is uncertain beyond a particular point and this is what makes them memories of the past event E.
This picture of the 'arrow of time' is somewhat different from the usual image of entropy being low in the past and uncertainty increasing in the future.
(*) 'document' is used in a general sense - a hot cup of coffee in an empty apartment 'documents' that a person was in that apartment not too long ago. We know this because we cannot postdict the temperature of that coffee beyond 100C - so we know somebody had to be there to make it.