If we imagine we have a bunch of polynomials $P_n(t)$, each describe probability densities that an event happening at some time $t$ and we know that exactly one of the $N$ events will happen during this time.
$$\int_{0}^{1}\sum_n P_n(t)dt = 1, \hspace{1cm}t\in[0,1]$$ And we have the following events happening at time points $t_k$ at polynomial $n_k$.
Which mathematical techniques can we use to find the most likely polynomials $\{P_n\}$ given we know timepoints and polynomial for each event that has occurred?
It seems I will need to try to explain better what I mean.
Once an event occurs at one of the $n$ polynomials, probability density rakes up to 100% at this place, ok?
$$P_{m_k}(t_k) = \delta_k(t-t_k)$$
But which polynomial $m_k$ of the $n$ it happens for can vary for each new sample: $k$, in this sense it is like a mixture as @nathan asked about. Every new sample could be from any of the $n$ polynomials and together they make up to 100 percent. But for every new sample we restart to count from $t=0$. Assume we get $N$ samples at $\left\{P_{m_k}(t_k)\right\}$ So we may want to minimize in some sense
$$\sum_{k=1}^{N}\|P_{l}(t) - \delta(t-t_k)\cdot \delta(m_k-l)\|$$
The probability that the first event occurs at time $t$ using polynomial $P_n$ is given by the product $P_n(t)P_{n_1}(t_1-t)$ where $n_1$ is the polynomial used for the event immediately following, $t_1$ is the time of that event, and we assume that all events are independent. The actual probability is then given by integrating that expression, $$\int_{\max\{1,t_1\}-1}^{\min\{1,t_1\}}P_n(t)P_{n_1}(t_1-t)\,\mathrm{d}t$$ and so we need only check each of the $n$ possible integrals to see which is largest, and hence likeliest. That integral should be very easy to evaluate for polynomials, of course.