---
title: "Model selection"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Model selection}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
bibliography: ../inst/REFERENCES.bib
link-citations: yes
---

```{r, label = setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.path = "img/",
  fig.align = "center",
  fig.dim = c(8, 6),
  out.width = "75%"
)
library("RprobitB")
options("RprobitB_progress" = FALSE)
```

The task of model selection targets the question: If there are several competing models, how do I choose the most appropriate one? This vignette^[This vignette is built using R `r paste(R.Version()$major, R.Version()$minor, sep = ".")` with the `{RprobitB}` `r utils::packageVersion("RprobitB")` package.] 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](https://loelschlaeger.de/RprobitB/articles/v03_model_fitting.html):

```{r, message = FALSE}
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`:

```{r, message = FALSE}
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:

```{r, model-selection-example}
model_selection(model_train, model_train_sparse)
```

Specifying `criteria` is optional. Per default, `criteria = c("npar", "LL", "AIC", "BIC")`.^[To show the model formulas in the output of `model_selection()`, add the argument `add_form = TRUE`.] 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:

```{r, npar-example}
npar(model_train, model_train_sparse)
```

Here, `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:^[The log-likelihood values are per default computed at the point estimates derived from the Gibbs sample means. `logLik()` has the argument `par_set`, where alternative statistics for the point estimate can be specified.]

```{r, loglik-example}
logLik(model_train)
logLik(model_train_sparse)
```

### `AIC`

Including `"AIC"` yields the Akaike's Information Criterion [@Akaike1974], which is computed as $$-2 \cdot \text{LL} + k \cdot \text{npar},$$
where $\text{LL}$ is the [model's log-likelihood value](#ll), $k$ 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:

```{r, aic-example}
AIC(model_train, model_train_sparse, k = 2)
```

The 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 [@Schwarz1978], which is defined as 
$$-2 \cdot \text{LL} + \log{(\text{nobs})} \cdot \text{npar},$$
where $\text{LL}$ is the [model's log-likelihood value](#ll), $\text{nobs}$ 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](#npar). The interpretation is analogue to AIC.

`{RprobitB}` also provided a method for the `BIC` value:

```{r, bic-example}
BIC(model_train, model_train_sparse)
```

### `WAIC` (with `se(WAIC)` and `pWAIC`)

WAIC is short for Widely Applicable (or Watanabe-Akaike) Information Criterion [@Watanabe2010]. 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 @McElreath2020 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 \code{RprobitB_fit} object, the probabilities $p_{si}$ must be computed via the `compute_p_si()` function: ^[This computation has been outsourced because it is very time consuming. For example, the computation for `model_train` took about 5 minutes. To decrease the computation time, the function offers parallelization via specifying the number `ncores` of parallel CPU cores.]

```{r, eval = FALSE}
model_train <- compute_p_si(model_train)
```

Afterwards, the WAIC can be accessed as follows:

```{r, waic-example, eval = FALSE}
WAIC(model_train)
WAIC(model_train_sparse)
```

### `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 [@Newton1994]: 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](#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"`:^[Note that the marginal likelihood value is very small. The given representation is required so that the value is not rounded to 0 by the computer.]

```{r, eval = FALSE}
model_train <- mml(model_train)
model_train$mml
attr(model_train$mml, "mmll")
```

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 [@Hammersley1964]: 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`:^[The `mml()` function offers parallelization via specifying the number `ncores` of parallel CPU cores.]

```{r, eval = FALSE}
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 @Gronau2017 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 [@Marin2014]. 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](#mmll). 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 [@Jeffreys1998] in favor of `model_train`.

```{r, bayes-factor-example, eval = FALSE}
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`:^[See the vignette on choice prediction for more nuanced performance comparison in terms of sensitivity and specificity.]

```{r, pred-acc-example}
pred_acc(model_train, model_train_sparse)
```

## References