I have a random walk / Markov chain (is one term more correct than the other?) defined by $$N_{i+1} = N_i + D_i,$$ where $$D_i \sim Poisson(N_i).$$
So, given $N_0$, say, I can compute $P(N_1 = x | N_0 = n_0) = f_{Poisson}(x-n_0; n_0).$
If, instead, I am given $N_1$, can I compute $P(N_0 = z | N_1 = n_1)$? It's certainly not as straightforward as the previous case. I have considered $f_{Poisson}(n_1-z; z)$, but numerically the probabilities do not sum to 1 when considering a range of values for $z$, and I suppose this is suspect anyway given that the random variable is inside its own mean/parameter.
Since the directional dependence is clear ($N_0 \to N_1$), I can naively try to use Bayes's rule: $$P(N_0 | N_1) = \frac{P(N_1 | N_0) P(N_0)}{P(N_1)},$$ but in this case I don't know what it means to say "the probability of $N_i$ (on its own)". For this reason, in fact, I'm having difficulties numerically simulating: I can iterate over values for $N_0$ and simulate the probability of obtaining $N_1$, but this results in a list of probabilities that sum to greater than 1 (in fact it seems like I obtain the same values as in the approach I took above). It's not clear if (or why) I should normalize this whole list of probabilities, or if I should be weighting each value of $N_0$ differently.
It makes much more sense to consider the previous case of computing the probability of $N_1$ if we know $N_0$, but at the same time it feels like we must be able to derive a probability distribution for the range of values of $N_0$ that could give rise to $N_1$ using the above scheme.
Does anyone have any suggestions of how to approach the problem in this way, or any reasons as to why it is illegal?
The problem is that you haven't actually defined a Markov chain yet, because you haven't specified an initial state or distribution. Your application of Bayes' rule is correct, but you know neither $\mathsf P(N_0)$ nor $\mathsf P(N_1)$ because you haven't specified them. Once you do specify an initial distribution (say, $N_0=1$ with probability $1$), you can use it to calculate $\mathsf P(N_1)$. (Alternatively you could specify $\mathsf P(N_k)$ for any $k$ and work forwards and backwards, but the more usual approach to defining a Markov chain is to specify an initial distribution, not an intermediate distribution. Also, in specifying an intermediate distribution, you'd have to make sure that it's compatible with the transition probabilities, whereas for $\mathsf P(N_0)$ you can specify an arbitrary distribution.)