A detailed prescription for postdictions and an explanation why it differs from time-reverted predictions is given at the end of the article Myths about the arrow of timeOne topic on the boundary between physics and philosophy that is never discussed too explicitly are the rules how to "predict" the past, i.e. how to use the information about the present together the physical laws to reconstruct the conditions in the past.
Arrows of time
Whenever you study macroscopic systems that interact with the environment, you seem to end up with some time-asymmetric phenomena. Friction always slows objects down: it never accelerates them. Heat is transferred in such a way that the temperature becomes more uniform, not less, and so forth. Differential equations for heat conductivity, friction etc. have a clear time-reversal asymmetry.
A related arrow of time is the logical arrow of time. You should always assume that you know the initial conditions in the past - or the present - and use the physical laws to predict the future. At least The Reference Frame tells you that you should never do it in the opposite way. Decoherence is a phenomenon whose origin is analogous to friction and it has thus an arrow of time, too. See
Exchanging the cause and the effect
Many physicists seem to think that the laws of physics can be used to predict the past in the very same way as if you predict the future. You take the matrix element of the evolution operator between the two states and square its absolute value to obtain a probability that one of these two states evolves into the second state.
When you study microscopic systems with a full knowledge of all degrees of freedom and no friction or decoherence, there is clearly a full time-reversal symmetry, or at least a CPT-symmetry if you are a quantum field theory purist, and no one can protest, not even your humble correspondent.
However, if someone uses the same rule even for macroscopic objects where you have to trace over the environment in such a way that friction or decoherence become important, I happen to think that the prescription is simply wrong.
Start with a piece of ice immersed in water at t=0 - everything embedded into a perfectly isolated box - and try to predict the probabilities of different states at t=-1 hour. Many people think that the right prediction is that the ice is gonna melt during the negative hour exactly as it would melt in a positive hour. Experiments combined with memory show that this conclusion is clearly wrong: at t=-1 hour, the ice was bigger than at t=0, not smaller.
If you took all the degrees of freedom, including the environmental ones, into account and if you could moreover insert extremely accurate information about the state of the atoms of ice and water, surely you could evolve this state backwards to find that the ice was bigger at t=-1 hour. That's true both in classical physics as well as quantum mechanics. However, the required accuracy would be exponential because the non-uniformities in the diffusion equation decrease exponentially with time - or, in other words, non-uniformities increase exponentially if you go backwards in time.
It's clearly unrealistic and one must work with approximations of the states described by density matrices that don't provide you with the exponentially accurate knowledge about the atoms of ice. Is it still true that you should simply evolve this density matrix into the past in the same way as you would evolve it into the future and use the evolved density matrix to reconstruct the probabilities in the past?
No, no, no
I think that the answer is clearly No. If this procedure were correct, you would indeed deny the existence of the arrow of time. The entropy would be minimized at t=0 and it would grow in both directions. That clearly doesn't occur. Entropy is an increasing function of time both for t positive as well as t negative. Not only we remember it was the case: if it were not the case, our laws of physics would be incompatible with the time-translational symmetry because they would predict something special about an arbitrary moment t=0, namely the minimization of the entropy.
Another way to see a problem with the time-symmetric prescription is to look at the environment - and the environment is the real source of the asymmetry (both for friction and decoherence). A piece of ice emits things like thermal photons into the environment. These photons always travel from a warm piece of ice to infinity along a light-like trajectory. The number of these photons leaving a piece of ice is clearly greater than the number of photons that are going from infinity directly to the ice. That's why the piece of ice is able to transfer some heat to the cosmic microwave background, among other recipients of energy.
So the assumption that the evolution before t=0 should work just like the evolution to positive t is flawed. It's not surprising that the mathematical framework to predict probabilities sees a difference between the past and the future. If you recall some formulae for the Bayesian inference etc., you will realize that the probabilities of assumptions or initial conditions are treated rather differently from the probabilities of your predictions. The logical framework to deal with probabilities is not invariant under the interchange of the past and the future or the interchange of the causes and effects, or assumptions and predictions.
If you naively performed the time-reversal symmetric calculation, you would predict that the microstates corresponding to macroscopic configurations with a high entropy would be overwhelmingly preferred. That's not the case. You can say that the entropy in the past was lower than today because of a universal fact - the low entropy right after the Big Bang. That's what Brian Greene decided to say in The Fabric of the Cosmos.
Rules of postdiction are local
However, every sane person feels that the method how we use the physical laws to reconstruct (or try to reconstruct) the state of your lab - or the box with water and ice - one hour ago shouldn't depend on some details of cosmology. And indeed, it doesn't. I believe that even without assuming that there has ever been anything like the Big Bang, you may define meaningful rules how to try to reconstruct the past. Such rules will never be practical - exactly because of the exponentially increasing non-uniformities etc. if you go to the past that make all postdictions practically impossible - but one of the features of the rules should be, I think, that the low-entropy states that you find in the evolution backwards should be subjects to some heavy positive discrimination. Whenever you have some positive discrimination, it becomes an ordinary discrimination for others: in this case, the victims are high-entropy states in the past.
I think that you should assign an extra factor of exp(-S) to a candidate microstate in the past where S is the entropy of this microstate viewed as a representative of a family of macroscopically indistinguishable states of your physical system (without the environment). Of course, you should normalize the probabilities at the end so that they add up to one.
Something like that is necessary to avoid ludicrous conclusions that would tell you that eggs were unbreaking and dead people were becoming alive during the last hour. Many physicists are making a demonstrably wrong assumption that the predictions of the future and the past follows the same logical rules even if you consider subsystems and approximately defined states where dissipation and decoherence play a role. That's why they derive various wrong conclusions, e.g. that the young Universe should "naturally" have a high entropy.
There is no rational reason for a young Universe to have a high entropy and indeed, I think that even children are correctly taught laws that can be used to conclude that the young Universe had a much lower entropy than today.
Also, we should never think about the early cosmology as if we were evolving the present state backwards in time. We should always start to think about the past and try to figure out whether the present state of the Cosmos could have evolved from the past. The complex matrix elements of the evolution operator are identical, up to a complex conjugation, but the full rules to calculate the probabilities of subsystems interacting with the environment are not identical.
Those who pay more attention to reality know that the second law of thermodynamics will work not only in the future but it has also worked in the past: there is nothing special about the present. Those who carefully avoid theoretical errors and unjustified incorrect assumptions also know that the causes and predictions enter the rules of logical inference asymmetrically. When this asymmetry is appreciated, there is nothing inconsistent about the thermodynamical arrow of time in the real world and there is nothing unnatural about the low-entropy initial conditions of the Universe.
And that's the memo.