Estimating credit transition probabilities from additional information

Question

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?

Your pointer will be very helpful

Answer 1

I found this thread on the math stack exchange: https://math.stackexchange.com/questions/3988333/how-to-redistribute-probabilities-when-one-outcomes-probabilities-changes

It suggests keeping the ratio of the remaining probabilities constant which in your case might be a good idea since you might want to avoid making the other states more or less probable in relation to one another as this would implicitly mean you have been given more information.