Appropriate Distribution for Survival Probability Parameters I have a model that assumes a probability of survival over discrete time (example: decades)
P  = [P1, P2, P3 .., P10]
Pi = [0,1]. These are survival probabilities after the ith decade.
P does not need to sum to 1.
P10 = 0. We assume that everyone is dead after the 10th decade.

If an individual is dead after 50, then
Observation-Likelihood = P1 * P2 * P3 * P4 * (1-P5)

I would like to infer the P parameters, given some observed death data.
Does this map to any known model?
 A: It looks like your parameters are discrete survival rates for those decades.  Suppose you observe data vector $\mathbf{x} = (x_1,...,x_n)$ which are the ages at death, measured in decades for your sample.  (Since you have not specified to the contrary, I will assume that you observe all the times of death, so there is no censored data.)  Let $\mathbf{p} = (p_1,...,p_{10})$ be the survival rates each decade, which are defined by $p_k = \mathbb{P}(X > k | X \geqslant k)$.  For a single observation you have sampling density:
$$p(X=x|\mathbf{p}) = (1-p_x) \prod_{k=1}^{x-1} p_k.$$
Hence, your log-likelihood for the whole data vector $\mathbf{x}$ is:
$$\begin{equation} \begin{aligned}
\ell_\mathbf{x}(\mathbf{p}) 
&= \sum_{i=1}^n \Big[ \ln (1-p_{x_i}) +  \sum_{k=1}^{x_i-1} \ln (p_k) \Big] \\[6pt]
&= \sum_{k=1}^{10} n_k \Big[ \ln (1-p_k) +  \sum_{j=1}^{k-1} \ln (p_j) \Big] \\[6pt]
&= \sum_{k=1}^{10} \Big[ n_k \cdot \ln (1-p_k) + (n_k^+ - n_k) \cdot \ln(p_k) \Big], \\[6pt]
\end{aligned} \end{equation}$$
where $n_k \equiv \sum_{i=1}^n \mathbb{I}(X_i = k)$ and $n_k^+ \equiv \sum_{i=1}^n \mathbb{I}(X_i \geqslant k)$ are counts of the number of people that died in decade $k$ and survived up to decade $k$ respectively.

Maximum likelihood estimation:  The partial derivatives of the log-likelihood for the model are:
$$\begin{equation} \begin{aligned}
\frac{d \ell_\mathbf{x}}{d p_k}(\mathbf{p}) 
&= \frac{n_k}{1-p_k} - \frac{n_k^+-n_k}{p_k}. \\[6pt]
\end{aligned} \end{equation}$$
Setting $\nabla \ell_\mathbf{x}(\hat{\mathbf{p}}) = \boldsymbol{0}$ you obtain the MLEs:
$$\hat{p}_k = \frac{n_k^+ - n_k}{n_k^+}.$$
As you can see, these are just sample proportions  --- specifically, the estimator for the survival rate is the proportion of people that survived until decade $k$ that also survived past that decade.  This is an intuitively reasonable estimator that emerges from this simple non-parametric model.  It can easily be shown that the estimator is unbiased and consistent.  This should give you reasonable estimates for the unknown survival rates for the decades.
