How to find time-dependent probabilities that are satisfied by a given time series?

Question

I'm not good in statistics but have a rough understanding of frequentist vs. propensity probability.

This is the problem I stumbled upon:

Given a smooth time-dependent (propensity) probability of an event to occur, i.e. a function $p: \mathbb{R} \rightarrow [0,1]$ (without further restrictions than being "smooth"). The task is to produce a time-series that satisfies $p$.

The solution seems straight-forward:

1. Choose an arbitrary time unit $\Delta t$. Divide time into steps of size $\Delta t$ (ignoring a phase).

2. For time step $t \in \mathbb{Z}$ create a random number $\rho(t) \in [0,1]$.

3. If $\rho(t) < p(t\Delta t)$, let the event occur at time step $t$.

What you get is a time series $e:\mathbb{Z} \rightarrow \{0,1\}$ ($e$ for "event"), which by construction satisfies $p$ with a (meta-)probability* that depends somehow on $\Delta t$ and the volatility of $p$.

This is the easy part (or direction): from a given propensity probability to one of a many time series that satisfy it (with rather high (meta-)probability).

But what about the other direction? Given a time series $e:\mathbb{Z} \rightarrow \{0,1\}$ and finding

• an arbitrary or
• one of a kind or
• one distinguished

probability function $p:\mathbb{R} \rightarrow [0,1]$ which is satisfied by $e$. Like above the choice of a time unit $\Delta t$ seems to be essential. Assume $\Delta t$ is choosen:

How do I find at least one single $p$ – given as a smooth function $p: \mathbb{R} \rightarrow [0,1]$ – that is satisfied by $e$?

And what about the (meta-)probability that this $p$ is "correct"?

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* Maybe it's not about the probability of an arbitrary time series $e$ to satisfy $p$ (yes/no?), but about the average degree by which specific time series $e$ satisfy probability $p$.

Answer 1

The technical term for this is density estimation:

Density estimation is the construction of an estimate, based on observed data, of an unobservable underlying probability density function.

One straight forward approach would be kernel density estimation with the standard normal density function $\phi$.

The function $p_0(t) = \frac{1}{N}\sum_{i=1}^N \delta(t-t_i)$ does perfectly reproduce the distribution of events $e:\mathbb{Z} \rightarrow \{0,1\}$ (assuming we have $N$ events at times $t_i$). Note, that $\delta(x) = \lim_{\sigma \rightarrow 0}\phi_\sigma(x) $ with

$$\phi_\sigma(x) = \frac{1}{\sqrt{2\pi}\sigma}e^{-x^2/2\sigma^2}$$

The functions $p_\sigma(t) = \frac{1}{N}\sum_{i=1}^N \phi_\sigma(t-t_i)$ are approximations of the "perfect" distribution.

Intuitively one wants to choose $\sigma$ as large as possible (to get a somehow smooth distribution) but as small as the data will allow (see Wikipedia).