Let say we have a vector indicating how many times an event occurred in the past 10 years as [2,0,0,1,0,3,1,0,1,0].
How can I build a (Bayesian) stochastic model that tells me what is the probability of the same event to happen this year based on the observed past events?
You could do this utilizing no temporal dependency by modeling it as: \begin{equation} \begin{split} Y_i & \overset{iid}{\sim} \textrm{Poi}(\lambda)\\ \lambda & \sim \textrm{Gamma}(\alpha_0,\beta_0) \end{split} \end{equation}
Where the prior parameters represent $\alpha_0$ prior expected occurrences in $\beta_0$ observations.
(Relevant point: This ignores the zero-inflated nature of the data. Often zero-inflatedness is incorporated into a model to account for the overdispersion of the data. The Poisson distribution has only one parameter and the mean and variance are equivalent, so if you have an excess of 0's you could have difficulty dealing with the mean and variance at the same time. This doesn't seem to be a huge issue with your data because of the scale of the non-zero values and the fact that the mean and variance of the observations are fairly close (.8 and ~1.07))
Then the posterior distribution of $\lambda$ is given by: $\lambda | Y \sim \textrm{Gamma}(\alpha_0 + \sum_i Y_i,\beta_0 + n)$.
Then the posterior predicted value for a new $X$ is given by: $\int_{\lambda}\frac{\lambda^{X} e^{-\lambda}}{X!} d\lambda|Y$. Observe that this is a Poisson-Gamma mixture, which has a Negative Binomial distribution; you can read about this here and here.
This gives us that the posterior predicted distribution of a new observation is:
\begin{equation} X|Y \sim \textrm{NB}(r = \beta_0 + n, p = \frac{1}{\alpha_0+\sum_i Y_i+1}) \end{equation}
If you specify $\beta_0 = .1$ and $\alpha_0 =.1$, which equates to a relatively high variance prior with expectation 1. You get $X|Y \sim \textrm{NB}(r = 10.1, p = \frac{1}{9.1}) = \textrm{NB}(r = 10.1, p = .109)$. Which has expected value 1.25. This doesn't seem unreasonable.
Temporal dependence is more complicated, I'm working on a write up of that now.