Otherwise your games with the “definition” of initial and final states and with the sign of t are completely immaterial. “Initial” and “final” states are, according to logic, qualitatively different things, and the usual convention for the sign of t is that t_{initial} < t_{final}. But I have never even used this convention.

Even if I had, it wouldn’t matter. One can easily rewrite all proofs to the opposite convention by replacing t with −t; all those things are physically vacuous. The non-vacuous claim is that the future and past don’t play symmetric roles in logic.

— Physics Stack Exchange

— Jan 25 ’12 at 9:49

— Lubos Motl

2013.09.26 Thursday ACHK

That’s why the retrodicted probabilities of initial states pi=P(Hi) always depend on some subjective choices. What we think about the past inevitably depends on other things we have learned about the past. This is a totally new property of retrodictions that doesn’t exist for predictions. Predictions may be probabilistic (and in quantum mechanics and statistical physics, they are inevitably “just” probabilistic) but the predicted probabilities are objectively calculable for certain input data. The formulae that objectively determine these probabilities are known as the laws of physics. But the retrodicted probabilities of the past are not only probabilistic; their values inevitably depend on the subjective knowledge, too!

— Prediction isn’t the right method to learn about the past

— Lubos Motl

The future is not fixed, in the sense that the present chooses among the potential futures to evolve to. Since each higher entropy macrostate by definition is corresponding to more microstates, it has a higher probability to occur. 

However, the past is fixed. The probability is subjective probability. The present cannot “choose” among the “potential” pasts from which the present is evolved. The fact that there are more one possible pasts is due to your subjective ignorance about the past. If someone else has more data about the past, his number of possible pasts will be much smaller. If that person has a record of the past, there will be only one possible past.

The probability for predicting the future is objective, because by the definition of the word “future”, no one can have any data about the future. No one can have a record of the future now.

— Me@2013-07-26 6:01 PM

The difference between the future and the past is that logically, no one can have any data of future, but someone may have some data of the past. Also, different people can have different sets of data about the past.

— Me@2013-08-08 8:43 AM

2013.08.08 Thursday (c) All rights reserved by ACHK