# Modeling count data with the Bell distribution

    library(bellreg)

    data(faults)

    # ML approach:
    mle <- bellreg(nf ~ lroll, data = faults, approach = "mle")
    summary(mle)
    #> Call:
    #> bellreg(formula = nf ~ lroll, data = faults, approach = "mle")
    #>
    #> Coefficients:
    #>               Estimate     StdErr z.value   p.value
    #> (Intercept) 0.98524996 0.33219431  2.9659  0.003018 **
    #> lroll       0.00190934 0.00049003  3.8963 9.766e-05 ***
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> logLik = -88.96139   AIC = 181.9228

    # Bayesian approach:
    bayes <- bellreg(nf ~ lroll, data = faults, approach = "bayes", refresh = FALSE)
    summary(bayes)
    #>
    #> bellreg(formula = nf ~ lroll, data = faults, approach = "bayes",
    #>     refresh = FALSE)
    #>
    #>              mean se_mean    sd  2.5%   25%   50%   75% 97.5%    n_eff  Rhat
    #> (Intercept) 0.977   0.007 0.331 0.304 0.757 0.980 1.201 1.620 2326.404 1.002
    #> lroll       0.002   0.000 0.000 0.001 0.002 0.002 0.002 0.003 2648.453 1.001
    #>
    #> Inference for Stan model: bellreg.
    #> 4 chains, each with iter=2000; warmup=1000; thin=1;
    #> post-warmup draws per chain=1000, total post-warmup draws=4000.

    log_lik <- loo::extract_log_lik(bayes$fit)
    loo::loo(log_lik)
    #> Warning: Relative effective sample sizes ('r_eff' argument) not specified.
    #> For models fit with MCMC, the reported PSIS effective sample sizes and
    #> MCSE estimates will be over-optimistic.
    #>
    #> Computed from 4000 by 32 log-likelihood matrix
    #>
    #>          Estimate  SE
    #> elpd_loo    -91.0 3.9
    #> p_loo         2.0 0.6
    #> looic       182.1 7.9
    #> ------
    #> Monte Carlo SE of elpd_loo is 0.0.
    #>
    #> All Pareto k estimates are good (k < 0.5).
    #> See help('pareto-k-diagnostic') for details.
    loo::waic(log_lik)
    #> Warning:
    #> 1 (3.1%) p_waic estimates greater than 0.4. We recommend trying loo instead.
    #>
    #> Computed from 4000 by 32 log-likelihood matrix
    #>
    #>           Estimate  SE
    #> elpd_waic    -91.0 3.9
    #> p_waic         1.9 0.6
    #> waic         182.0 7.9
    #>
    #> 1 (3.1%) p_waic estimates greater than 0.4. We recommend trying loo instead.