Are likelihood functions unique?

Here is my problem:

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A life office uses the three-state healthy-sick-dead model in the pricing of its long term sickness policies. The transition rates are assumed to be constant.

Denote the state space R={H,S,D} and transition rates q(hs), q(hd), q(sh) and q(sd).

For a group of policy holders, over a one year period the following data were recorded:

Transition from: Number that made this transition

H to S: 15

H to D: 6

S to H: 5

S to D: 1

The total times spent in states H and S were 625 years and 35 years, respectfully.

Write down the likelihood function for this model and show that it is maximised when q(hs)=0.024.

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There are two ways to set this up as far as I can see:

I can treat it as poisson processes, and obtain a likelihood function from that.

I can treat it as n independent lifetimes and then factorise to remove the n.

However, I seem to obtain two different likelihood functions depending on the way I do it. I think this is because the first one is a discrete method, the other is a continuous method. Luckily, they are multiples of each other with respect to the parameters I want to find, and since the only point of a likelihood function is to maximise it, it doesn't matter which one I use.

But I would like to know whether they are actually both likelihood functions, or if one is correct and one is false. The book says to do it the second way, but I find the first method more intuitive.