# Estimating credit transition probabilities from additional information

Let say $$P_{i,j}, j = 1,2,3, DEF$$ are the probabilities of transitions from an initial rating $$i$$ to rating $$j$$, where $$P_{i, DEF}$$ represents the default probability from that initial rating.

Now let say, based on some other information I require to modify above default probability to $$R_{i, DEF}$$ - which is considered to be more accurate estimate of probability.

Now, I need to adjust non-default probabilities, as I have now better information. I consider following 3 approached.

First approach is to use the formula $$\frac{1-P_{i,j}}{1-R_{i,j}} = \frac{1-P_{i, DEF}}{1-R_{i, DEF}}, j = 1,2,3$$

Second approach is to use the formula $$\frac{P_{i,j}}{R_{i,j}} = \frac{P_{i, DEF}}{R_{i, DEF}}, j = 1,2,3$$

And, third approach is to use the formula $$\frac{P_{i,j}}{R_{i,j}} = \frac{1-P_{i, DEF}}{1-R_{i, DEF}}, j = 1,2,3$$

Among above three approaches, which approach can be considered as best?