I do not fully grasp the second correct option from the 2019 quiz. Could you please help me understand why both options are equivalent?
I think this is because we can write P(T1=t1, ... Tn=tn | W1=w1, ... Wn = wn) in the following way:
P(T1=t1, ... Tn=tn, W1=w1, ... Wn = wn) / P(W1=w1, ... Wn = wn)
For a fixed word sequence, P(W1=w1, ... Wn = wn) will be fixed. Thus, maximizing P(T1=t1, ... Tn=tn | W1=w1, ... Wn = wn) will be the same as maximizing P(T1=t1, ... Tn=tn,W1=w1, ... Wn = wn).
P(T1=t1, ... Tn=tn, W1=w1, ... Wn = wn) / P(W1=w1, ... Wn = wn)
For a fixed word sequence, P(W1=w1, ... Wn = wn) will be fixed. Thus, maximizing P(T1=t1, ... Tn=tn | W1=w1, ... Wn = wn) will be the same as maximizing P(T1=t1, ... Tn=tn,W1=w1, ... Wn = wn).
