The task of model selection targets the question: If there are several competing models, how do I choose the most appropriate one? This vignette1 outlines the model selection tools implemented in RprobitB.
For illustration, we revisit the probit model of travelers deciding between two fictional train route alternatives from the vignette on model fitting:
form <- choice ~ price + time + change + comfort | 0
data <- prepare_data(form = form, choice_data = train_choice, id = "deciderID", idc = "occasionID")
model_train <- fit_model(
data = data,
scale = "price := -1",
R = 1000, B = 500, Q = 10
)As a competing model, we consider explaining the choices only by the
alternative’s price, i.e. the probit model with the formula
choice ~ price | 0:
model_train_sparse <- update(model_train, form = choice ~ price | 0)The update() function helps to estimate a new version of
model_train with new specifications. Here, only
form has been changed.
The model_selection() function
RprobitB provides the convenience function
model_selection(), which takes an arbitrary number of
RprobitB_fit objects and returns a matrix of model
selection criteria:
model_selection(model_train, model_train_sparse)
#> model_train model_train_sparse
#> npar 4 1
#> LL -1727.73 -1865.86
#> AIC 3463.46 3733.72
#> BIC 3487.39 3739.71Specifying criteria is optional. Per default,
criteria = c("npar", "LL", "AIC", "BIC").2 The available model
selection criteria are explained in the following.
npar
"npar" yields the number of model parameters, which is
computed by the npar() method:
npar(model_train, model_train_sparse)
#> [1] 4 1Here, model_train has 4 parameters (a coefficient for
price, time, change, and comfort, respectively), while
model_train_sparse has only a single price coefficient.
LL
If "LL" is included in criteria,
model_selection() returns the model’s log-likelihood
values. They can also be directly accessed via the logLik()
method:3
AIC
Including "AIC" yields the Akaike’s Information
Criterion (Akaike 1974), which is computed as \[-2 \cdot \text{LL} + k \cdot
\text{npar},\] where \(\text{LL}\) is the model’s
log-likelihood value, \(k\) is the
penalty per parameter with \(k = 2\)
per default for the classical AIC, and \(\text{npar}\) is the number of parameters
in the fitted model.
Alternatively, the AIC() method also returns the AIC
values:
AIC(model_train, model_train_sparse, k = 2)
#> df AIC
#> model_train 4 3463.458
#> model_train_sparse 1 3733.723The AIC quantifies the trade-off between over- and under-fitting,
where smaller values are preferred. Here, the increase in goodness of
fit justifies the additional 3 parameters of
model_train.
BIC
Similar to the AIC, "BIC" yields the Bayesian
Information Criterion (Schwarz 1978), which is defined as \[-2 \cdot \text{LL} + \log{(\text{nobs})} \cdot
\text{npar},\] where \(\text{LL}\) is the model’s
log-likelihood value, \(\text{nobs}\) is the number of data points
(which can be accessed via the nobs() method), and \(\text{npar}\) is the number
of parameters in the fitted model. The interpretation is analogue to
AIC.
RprobitB also provided a method for the
BIC value:
BIC(model_train, model_train_sparse)
#> df BIC
#> model_train 4 3487.387
#> model_train_sparse 1 3739.706
WAIC (with se(WAIC) and
pWAIC)
WAIC is short for Widely Applicable (or Watanabe-Akaike) Information
Criterion (Watanabe and Opper 2010). As for AIC and
BIC, the smaller the WAIC value the better the model. Including
"WAIC" in criteria yields the WAIC value, its
standard error se(WAIC), and the effective number of
parameters pWAIC, see below.
The WAIC is defined as
\[-2 \cdot \text{lppd} + 2\cdot p_\text{WAIC},\]
where \(\text{lppd}\) stands for log pointwise predictive density and \(p_\text{WAIC}\) is a penalty term proportional to the variance in the posterior distribution that is sometimes called effective number of parameters, see McElreath (2020) p. 220 for a reference.
The \(\text{lppd}\) is approximated as follows. Let \[p_{si} = \Pr(y_i\mid \theta_s)\] be the probability of observation \(y_i\) (here the single choices) given the \(s\)-th set \(\theta_s\) of parameter samples from the posterior. Then
\[\text{lppd} = \sum_i \log \left( S^{-1} \sum_s p_{si} \right).\] The penalty term is computed as the sum over the variances in log-probability for each observation: \[p_\text{WAIC} = \sum_i \mathbb{V}_{\theta} \log (p_{si}) . \] The \(\text{WAIC}\) has a standard error of \[\sqrt{n \cdot \mathbb{V}_i \left[-2 \left(\text{lppd} - \mathbb{V}_{\theta} \log (p_{si}) \right)\right]},\] where \(n\) is the number of choices.
Before computing the WAIC of an object, the probabilities \(p_{si}\) must be computed via the
compute_p_si() function: 4
model_train <- compute_p_si(model_train)Afterwards, the WAIC can be accessed as follows:
MMLL
"MMLL" in criteria stands for marginal
model log-likelihood. The model’s marginal likelihood \(\Pr(y\mid M)\) for a model \(M\) and data \(y\) is required for the computation of
Bayes factors, see below. In general, the term has no closed form and
must be approximated numerically.
RprobitB uses the posterior Gibbs samples derived from
the mcmc() function to approximate the model’s marginal
likelihood via the posterior harmonic mean estimator (Newton
and Raftery 1994): Let \(S\)
denote the number of available posterior samples \(\theta_1,\dots,\theta_S\). Then, \[\Pr(y\mid M) = \left(\mathbb{E}_\text{posterior}
1/\Pr(y\mid \theta,M) \right)^{-1} \approx \left( \frac{1}{S} \sum_s
1/\Pr(y\mid \theta_s,M) \right) ^{-1} = \tilde{\Pr}(y\mid
M).\]
By the law of large numbers, \(\tilde{\Pr}(y\mid M) \to \Pr(y\mid M)\) almost surely as \(S \to \infty\).
As for the WAIC, computing the MMLL relies on the
probabilities \(p_{si} = \Pr(y_i\mid
\theta_s)\), which must first be computed via the
compute_p_si() function. Afterwards, the mml()
function can be called with an RprobitB_fit object as
input. The function returns the RprobitB_fit object, where
the marginal likelihood value is saved as the entry "mml"
and the marginal log-likelihood value as the attribute
"mmll":5
There are two options for improving the approximation: As for the WAIC, you can use more posterior samples. Alternatively, you can combine the posterior harmonic mean estimate with the prior arithmetic mean estimator (Hammersley and Handscomb 1964): For this approach, \(S\) samples \(\theta_1,\dots,\theta_S\) are drawn from the model’s prior distribution. Then,
\[\Pr(y\mid M) = \mathbb{E}_\text{prior} \Pr(y\mid \theta,M) \approx \frac{1}{S} \sum_s \Pr(y\mid \theta_s,M) = \tilde{\Pr}(y\mid M).\]
Again, it hols by the law of large numbers, that \(\tilde{\Pr}(y\mid M) \to \Pr(y\mid M)\) almost surely as \(S \to \infty\). The final approximation of the model’s marginal likelihood than is a weighted sum of the posterior harmonic mean estimate and the prior arithmetic mean estimate, where the weights are determined by the sample sizes.
To use the prior arithmetic mean estimator, call the
mml() function with a specification of the number of prior
draws S and set recompute = TRUE:6
model_train <- mml(model_train, S = 1000, recompute = TRUE)Note that the prior arithmetic mean estimator works well if the prior
and posterior distribution have a similar shape and strong overlap, as
Gronau et al. (2017) points out. Otherwise, most of the
sampled prior values result in a likelihood value close to zero, thereby
contributing only marginally to the approximation. In this case, a very
large number S of prior samples is required.
BF
The Bayes factor is an index of relative posterior model plausibility of one model over another (Marin and Robert 2014). Given data \(\texttt{y}\) and two models \(\texttt{mod0}\) and \(\texttt{mod1}\), it is defined as
\[ BF(\texttt{mod0},\texttt{mod1}) = \frac{\Pr(\texttt{mod0} \mid \texttt{y})}{\Pr(\texttt{mod1} \mid \texttt{y})} = \frac{\Pr(\texttt{y} \mid \texttt{mod0} )}{\Pr(\texttt{y} \mid \texttt{mod1})} / \frac{\Pr(\texttt{mod0})}{\Pr(\texttt{mod1})}. \]
The ratio \(\Pr(\texttt{mod0}) / \Pr(\texttt{mod1})\) expresses the factor by which \(\texttt{mod0}\) a priori is assumed to be the correct model. Per default, RprobitB sets \(\Pr(\texttt{mod0}) = \Pr(\texttt{mod1}) = 0.5\). The front part \(\Pr(\texttt{y} \mid \texttt{mod0} ) / \Pr(\texttt{y} \mid \texttt{mod1})\) is the ratio of the marginal model likelihoods. A value of \(BF(\texttt{mod0},\texttt{mod1}) > 1\) means that the model \(\texttt{mod0}\) is more strongly supported by the data under consideration than \(\texttt{mod1}\).
Adding "BF" to the criteria argument of
model_selection yields the Bayes factors. We find a
decisive evidence (Jeffreys 1998) in favor of
model_train.
model_selection(model_train, model_train_sparse, criteria = c("BF"))
pred_acc
Finally, adding "pred_acc" to the criteria
argument for the model_selection() function returns the
share of correctly predicted choices. From the output of
model_selection() above (or alternatively the one in the
following) we deduce that model_train correctly predicts
about 6% of the choices more than model_train_sparse:7
pred_acc(model_train, model_train_sparse)
#> [1] 0.6940935 0.6340048