\[\mathcal{L}_{Y_{1:n}}(y_1, \dots, y_N) = \prod_{i\in R} f_X(y_i) \prod_{i\not\in R} (1-F_X(y_i))\]
Finally, we show how we can use these results to forecast future records.

Notation

Suppose we would like to model some time series for the running record for some task, where the record is the maximum or minimum over some sequence of attempts. We'll first consider the case where the record is of a minimum - the derivation for the maximum requires only a slight modification. We assume that the record is a continuous quantity, rather than discrete. Let \(\ \left\{X_t\right\}_{t\ \in \mathbb{N}}\) denote a discrete-time stochastic process representing the results from some sequence of attempts at a task. We assume that the \(X_i\) are i.i.d. according to some random variable \(X\) with a common CDF given by \(F_X\) and PDF given by \(f_X\). When we make inferences about \(X\) we will assume that \(X\) lies in some parametric family parameterized by \(\theta \in \Theta\).
Our observed data of the record is some time-series \(\left\{r_1, r_2,\ldots, r_n \right\}\) where \(n\) is the number of time periods for which the record has been observed. In the case the record is of a minimum, note that we must have \(r_{i} \ge r_j\) whenever \(i \le j\). To match our observed data, we define a sequence \(\ \left\{Y_t\right\}_{t\ \in \mathbb{N}}\) where\[Y_i\ :=\max\left\{X_1,\ldots,X_i\ \right\}.\]We treat \(\left\{r_1, r_2,\ldots, r_n \right\}\) as noiseless, truncated observations along some sample path \(\omega = \{ r_1, r_2, \ldots, r_n, \ldots \}\).  

The Likelihood Function

To perform Bayesian inference on the model parameters \(\theta\), we need to be able to compute the likelihood function. The marginal distribution of each record is easy enough to derive:
Lemma 1: Marginal distribution of a historical record
Let \(X_1, \dots, X_n\) be a collection of i.i.d. i.id random continuous variables with PDF \(f_X\) and CDF \(F_X\), and define \(Y_n := \max_{i\le n} X_i\). Then the marginal likelihood of \(Y_n\) is equal to:
\[\mathcal{L}_{Y_{n}}(o_n) = n [F_X(o_n)]^{n-1}f_X(o_n)\]
Proof: The CDF of \(Y_n\) is:\[\begin{align} F_{Y_n}(o_n) &= P(Y_n \le o_n) = P(\max\{X_1, \dots, X_n\} \le o_n)\\ &= \prod_{i\le n} P(X_i \le o_n) \\ &= \prod_{i\le n} F_X(o_n) \\ &= [F_X(o_n)]^n \end{align}\]
Differentiation of the CDF gives us the desired result. \(\square\)