For information about the model underlying the inferences performed in this vignette, including the (adjusted) likelihood for \((p_u, \sigma_u, \xi, \theta)\) and the Cheeseboro wind gusts dataset used below, see the vignette Frequentist Likelihood-Based Inference for Time Series Extremes.

We perform Bayesian inference for \((p_u, \sigma_u, \xi, \theta)\), combining a likelihood with a prior distribution for these parameters. For \(\theta\) the likelihood is based on the \(K\)-gaps model. For \((p_u, \sigma_u, \xi)\) the likelihood is based on vertically-adjusted log-likelihoods for \(p_u\) and \((\sigma_u, \xi)\). This latter aspect is an example of Bayesian inference using a composite likelihood (Ribatet, Cooley, and Davison (2012)).

Currently, **lite** allows only priors where \(p_u\), \((\sigma_u, \xi\)) and \(\theta\) are independent a priori. In the
example below, we use the `blite`

function’s default priors
for the exceedance probability \(p_u\)
(the Jeffreys’ prior), the generalised Pareto parameters \((\sigma_u, \xi)\) (and maximal data
information prior) and \(\theta\) (a
uniform prior on [0,1]). Different priors can be set using the
respective arguments `gp_prior`

, `b_prior`

and
`theta_prior_pars`

.

The `blite`

function samples from the posterior
distribution for \((p_u, \sigma_u,
\xi)\) based on vertically-adjusted log-likelihoods for \(p_u\) and \((\sigma_u, \xi)\). To use unadjusted
log-likelihoods set the argument `type = "none"`

.

```
library(lite)
<- exdex::cheeseboro
cdata # Each column of the matrix cdata corresponds to data from a different year
# blite() sets cluster automatically to correspond to column (year)
<- blite(cdata, u = 45, k = 3, ny = 31 * 24, n = 10000) cpost
```

The `summary`

and `plot`

methods produce
numerical and graphical marginal summaries of the posterior samples.

```
summary(cpost)
#>
#> Call:
#> blite(data = cdata, u = 45, k = 3, ny = 31 * 24, n = 10000)
#>
#> Posterior mean Posterior SD
#> p[u] 0.02838 0.006032
#> sigma[u] 9.95900 2.376000
#> xi -0.07280 0.094370
#> theta 0.24270 0.023430
plot(cpost)
```

We perform posterior predictive inference for the largest value \(M_N\) to be observed over a future time
period of length \(N\) years. Objects
returned from the `blite`

function have a
`predict`

method based on the `predict.evpost`

method in the **revdbayes** package (Northrop 2020). See the Posterior Predictive
Extreme Value Inference using the revdbayes Package vignette for
information. The effect of adjusting for the values of the extremal
index in the posterior sample is to change the effective time horizon
from \(N\) to \(\theta N\).

The function `predict.blite`

can estimate predictive
intervals, quantiles, distribution and density functions for \(M_N\) and simulate from the predictive
distribution for \(M_N\). For example,
the following code estimates a 95% highest predictive density (HPD)
interval for \(M_{100}\) and plots the
predictive densities of \(M_{100}\) and
\(M_{1000}\).

```
# Interval estimation
predict(cpost, hpd = TRUE)$short
#> lower upper n_years level
#> [1,] 73.01946 140.6364 100 95
```

```
# Density function
plot(predict(cpost, type = "d", n_years = c(100, 1000)))
```

Northrop, Paul J. 2020. *revdbayes:
Ratio-of-Uniforms Sampling for Bayesian Extreme Value Analysis*. https://CRAN.R-project.org/package=revdbayes.

Ribatet, Mathieu, Daniel Cooley, and Anthony C. Davison. 2012.
“Bayesian Inference from Composite Likelihoods, with an
Application to Spatial Extremes.” *Statistica Sinica* 22
(2): 813–845. https://www.jstor.org/stable/24310036.