Here is the formula for the probability of taking any of the possible paths in our claim transition model for one claim (the “i-th” claim).
is the number of observed periods after the last period with a payment,
is the number of periods up to and including the last period with a payment,
is the number of periods with payments in them
As our claims are independent, the likelihood given all of the observed claims is the product of this formula over all of the claims: