---
title: "Model fitting"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Model fitting}
  %\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)
```

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.] is a documentation of the estimation procedure `fit_model()` in `{RprobitB}`.

## Bayes estimation of the probit model

Bayes estimation of the probit model builds upon the work of @McCulloch1994, @Nobile1998, @Allenby1998, and @Imai2005. A key ingredient is the concept of data augmentation, see @Albert1993: The idea is to treat the latent utilities $U$ in the model equation $U = X\beta + \epsilon$ as additional parameters. Then, conditional on $U$, the probit model constitutes a standard Bayesian linear regression set-up. Its posterior distribution can be approximated by iteratively drawing and updating each model parameter conditional on the other parameters (the so-called Gibbs sampling approach).

A priori, we assume the following (conjugate) parameter distributions:

- $(s_1,\dots,s_C)\sim D_C(\delta)$, where $D_C(\delta)$ denotes the $C$-dimensional Dirichlet distribution with concentration parameter vector $\delta = (\delta_1,\dots,\delta_C)$,

- $\alpha\sim \text{MVN}_{P_f}(\psi,\Psi)$, where $\text{MVN}_{P_f}$ denotes the $P_f$-dimensional normal distribution with mean $\psi$ and covariance $\Psi$,

- $b_c \sim \text{MVN}_{P_r}(\xi,\Xi)$, independent for all $c$,

- $\Omega_c \sim W^{-1}_{P_r}(\nu,\Theta)$, independent for all $c$, where $W^{-1}_{P_r}(\nu,\Theta)$ denotes the $P_r$-dimensional inverse Wishart distribution with $\nu$ degrees of freedom and scale matrix $\Theta$,

- and $\Sigma \sim W^{-1}_{J-1}(\kappa,\Lambda)$.

These prior distributions imply the following conditional posterior distributions:

- The class weights are drawn from the Dirichlet distribution
\begin{equation}
(s_1,\dots,s_C)\mid \delta,z \sim D_C(\delta_1+m_1,\dots,\delta_C+m_C),
\end{equation}
where for $c=1,\dots,C$, $m_c=\#\{n:z_n=c\}$ denotes the current absolute class size.^[Mind that the model is invariant to permutations of the class labels $1,\dots,C$. For that reason, we accept an update only if the ordering $s_1>\dots>s_C$ holds, thereby ensuring a unique labeling of the classes.]

- Independently for all $n$, we update the allocation variables $(z_n)_n$ from their conditional distribution
\begin{equation}
\text{Prob}(z_n=c\mid s,\beta,b,\Omega )=\frac{s_c\phi_{P_r}(\beta_n\mid b_c,\Omega_c)}{\sum_c s_c\phi_{P_r}(\beta_n\mid b_c,\Omega_c)}.
\end{equation}

- The class means $(b_c)_c$ are updated independently for all $c$ via
\begin{equation}
b_c\mid \Xi,\Omega,\xi,z,\beta \sim\text{MVN}_{P_r}\left( \mu_{b_c}, \Sigma_{b_c}  \right),
\end{equation}
where $\mu_{b_c}=(\Xi^{-1}+m_c\Omega_c^{-1})^{-1}(\Xi^{-1}\xi +m_c\Omega_c^{-1}\bar{b}_c)$, $\Sigma_{b_c}=(\Xi^{-1}+m_c\Omega_c^{-1})^{-1}$, $\bar{b}_c=m_c^{-1}\sum_{n:z_n=c} \beta_n$.

- The class covariance matrices $(\Omega_c)_c$ are updated independently for all $c$ via
\begin{equation}
\Omega_c \mid \nu,\Theta,z,\beta,b \sim W^{-1}_{P_r}(\mu_{\Omega_c},\Sigma_{\Omega_c}),
\end{equation}
where $\mu_{\Omega_c}=\nu+m_c$ and $\Sigma_{\Omega_c}=\Theta^{-1} + \sum_{n:z_n=c} (\beta_n-b_c)(\beta_n-b_c)'$.

- Independently for all $n$ and $t$ and conditionally on the other components, the utility vectors $(U_{nt:})$ follow a $J-1$-dimensional truncated multivariate normal distribution, where the truncation points are determined by the choices $y_{nt}$. To sample from a truncated multivariate normal distribution, we apply a sub-Gibbs sampler, following the approach of @Geweke1998:
\begin{equation}
U_{ntj} \mid U_{nt(-j)},y_{nt},\Sigma,W,\alpha,X,\beta 
\sim \mathcal{N}(\mu_{U_{ntj}},\Sigma_{U_{ntj}}) \cdot \begin{cases}
1(U_{ntj}>\max(U_{nt(-j)},0) ) & \text{if}~ y_{nt}=j\\
1(U_{ntj}<\max(U_{nt(-j)},0) ) & \text{if}~ y_{nt}\neq j
\end{cases},
\end{equation}
where $U_{nt(-j)}$ denotes the vector $(U_{nt:})$ without the element $U_{ntj}$, $\mathcal{N}$ denotes the univariate normal distribution, $\Sigma_{U_{ntj}} = 1/(\Sigma^{-1})_{jj}$ and
\begin{equation}
\mu_{U_{ntj}} = W_{ntj}'\alpha + X_{ntj}'\beta_n - \Sigma_{U_{ntj}} (\Sigma^{-1})_{j(-j)}   (U_{nt(-j)} - W_{nt(-j)}'\alpha - X_{nt(-j)}' \beta_n ),
\end{equation}
where $(\Sigma^{-1})_{jj}$ denotes the $(j,j)$th element of $\Sigma^{-1}$, $(\Sigma^{-1})_{j(-j)}$ the $j$th row without the $j$th entry, $W_{nt(-j)}$ and $X_{nt(-j)}$ the coefficient matrices $W_{nt}$ and $X_{nt}$, respectively, without the $j$th column.

- Updating the fixed coefficient vector $\alpha$ is achieved by applying the formula for Bayesian linear regression of the regressors $W_{nt}$ on the regressands $(U_{nt:})-X_{nt}'\beta_n$, i.e.
\begin{equation}
\alpha \mid \Psi,\psi,W,\Sigma,U,X,\beta \sim \text{MVN}_{P_f}(\mu_\alpha,\Sigma_\alpha),
\end{equation}
where $\mu_\alpha = \Sigma_\alpha (\Psi^{-1}\psi + \sum_{n=1,t=1}^{N,T} W_{nt} \Sigma^{-1} ((U_{nt:})-X_{nt}'\beta_n) )$ and $\Sigma_\alpha = (\Psi^{-1} + \sum_{n=1,t=1}^{N,T} W_{nt}\Sigma^{-1} W_{nt}^{'} )^{-1}$.

- Analogously to $\alpha$, the random coefficients $(\beta_n)_n$ are updated independently via 
\begin{equation}
\beta_n \mid \Omega,b,X,\Sigma,U,W,\alpha \sim \text{MVN}_{P_r}(\mu_{\beta_n},\Sigma_{\beta_n}),
\end{equation}
where $\mu_{\beta_n} = \Sigma_{\beta_n} (\Omega_{z_n}^{-1}b_{z_n} + \sum_{t=1}^{T} X_{nt} \Sigma^{-1} (U_{nt:}-W_{nt}'\alpha) )$ and $\Sigma_{\beta_n} = (\Omega_{z_n}^{-1} + \sum_{t=1}^{T} X_{nt}\Sigma^{-1} X_{nt}^{'} )^{-1}$ .

- The error term covariance matrix $\Sigma$ is updated by means of 
\begin{equation}
\Sigma \mid \kappa,\Lambda,U,W,\alpha,X,\beta \sim W^{-1}_{J-1}(\kappa+NT,\Lambda+S), \\
\end{equation}
where $S = \sum_{n=1,t=1}^{N,T} \varepsilon_{nt} \varepsilon_{nt}'$ and $\varepsilon_{nt} = (U_{nt:}) - W_{nt}'\alpha - X_{nt}'\beta_n$.

### Parameter normalization

Samples obtained from the updating scheme described above lack identification (except for $s$ and $z$ draws), compare to the vignette on the model definition. Therefore, subsequent to the sampling, the following normalizations are required for the $i$th updates in each iterations $i$:

- $\alpha^{(i)} \cdot \omega^{(i)}$,

- $b_c^{(i)} \cdot \omega^{(i)}$, $c=1,\dots,C$,

- $U_{nt}^{(i)} \cdot \omega^{(i)}$, $n = 1,\dots,N$, $t = 1,\dots,T$,

- $\beta_n^{(i)} \cdot \omega^{(i)}$, $n = 1,\dots,N$,

- $\Omega_c^{(i)} \cdot (\omega^{(i)})^2$, $c=1,\dots,C$, and

- $\Sigma^{(i)} \cdot (\omega^{(i)})^2$, 

where either $\omega^{(i)} = \sqrt{\text{const} / (\Sigma^{(i)})_{jj}}$ with $(\Sigma^{(i)})_{jj}$ the $j$th diagonal element of $\Sigma^{(i)}$, $1\leq j \leq J-1$, or alternatively $\omega^{(i)} = \text{const} / \alpha^{(i)}_p$ for some coordinate $1\leq p \leq P_f$ of the $i$th draw for the coefficient vector $\alpha$. Here, $\text{const}$ is any positive constant (typically 1). The preferences will be flipped if $\omega^{(i)} < 0$, which only is the case if $\alpha^{(i)}_p < 0$.

### Burn-in and thinning

The theory behind Gibbs sampling constitutes that the sequence of samples produced by the updating scheme is a Markov chain with stationary distribution equal to the desired joint posterior distribution. It takes a certain number of iterations for that stationary distribution to be approximated reasonably well. Therefore, it is common practice to discard the first $B$ out of $R$ samples (the so-called burn-in period). Furthermore, correlation between nearby samples should be expected. In order to obtain independent samples, we consider only every $Q$th sample when computing Gibbs sample statistics like expectation and standard deviation. The independence of the samples can be verified by computing the serial correlation and the convergence of the Gibbs sampler can be checked by considering trace plots, see below.

## The `fit_model()` function

The Gibbs sampling scheme described above can be executed by applying the function

```{r, eval = FALSE}
fit_model(data = data)
```

where `data` must be an `RprobitB_data` object (see the vignette about choice data). The function has the following optional arguments:

- `scale`: A character which determines the [utility scale](#parameter-normalization). It is of the form `"<parameter> := <value>"`, where `<parameter>` is either the name of a fixed effect or `Sigma_<j>,<j>` for the `<j>`th diagonal element of `Sigma`, and `<value>` is the value of the fixed parameter (i.e. $\text{const}$ introduced [above](#parameter-normalization)). Per default `scale = "Sigma\_1,1 := 1"`, i.e. the first error-term variance is fixed to 1.

- `R`: The number of iterations of the Gibbs sampler. The default is `R = 10000`.

- `B`: The length of the burn-in period, i.e. a non-negative number of samples to be discarded. The default is `B = R/2`.

- `Q`: The thinning factor for the Gibbs samples, i.e. only every `Q`th sample is kept. The default is `Q = 1`.

- `print_progress`: A boolean, determining whether to print the Gibbs sampler progress.

- `prior`: A named list of parameters for the prior distributions (their default values are documented in the `check_prior()` function):

  - `eta`: The mean vector of length `P_f` of the normal prior for `alpha`.
  
  - `Psi`: The covariance matrix of dimension `P_f` x `P_f` of the normal prior for `alpha`.
   
  - `delta`: The concentration parameter of length 1 of the Dirichlet prior for `s`.
  
  - `xi`: The mean vector of length `P_r` of the normal prior for each `b_c`.
  
  - `D`: The covariance matrix of dimension `P_r` x `P_r` of the normal prior for each `b_c`.
  
  - `nu`: The degrees of freedom (a natural number greater than `P_r`) of the Inverse Wishart prior for each `Omega_c`.
  
  - `Theta`: The scale matrix of dimension `P_r` x `P_r` of the Inverse Wishart prior for each `Omega_c`.
   
  - `kappa`: The degrees of freedom (a natural number greater than `J-1`) of the Inverse Wishart prior for `Sigma`.
  
  - `E`: The scale matrix of dimension `J-1` x `J-1` of the Inverse Wishart prior for `Sigma`.

- `latent_classes`: A list of parameters specifying the number and the updating scheme of latent classes, see the vignette [on modeling heterogeneity fitting](https://loelschlaeger.de/RprobitB/articles/v04_modeling_heterogeneity.html).

## Example

In [the previous vignette on choice data](https://loelschlaeger.de/RprobitB/articles/v02_choice_data.html), we introduced the `train_choice` data set that contains 2922 choices between two fictional train route alternatives. The following lines fit a probit model that explains the chosen trip alternatives (`choice`) by their `price`, `time`, number of `change`s, and level of `comfort` (the lower this value the higher the comfort). For normalization, the first linear coefficient, the `price`, was fixed to `-1`, which allows to interpret the other coefficients as monetary values:

```{r, echo = FALSE}
set.seed(1)
```

```{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"
)
```

The estimated coefficients (using the mean of the Gibbs samples as a point estimate) can be printed via

```{r, coef-model-train}
coef(model_train)
```

and visualized via

```{r, plot-coef-model-train}
plot(coef(model_train), sd = 3)
```

The results indicate that the deciders value one hour travel time by about 25€, an additional change by 5€, and a more comfortable class by 14€.^[These results are consistent with the ones that are presented [in a vignette of the mlogit package](https://cran.r-project.org/package=mlogit/vignettes/c5.mxl.html#train) on the same data set but using the logit model.]

## Checking the Gibbs samples

The Gibbs samples are saved in list form in the `RprobitB_fit` object at the entry `"gibbs_samples"`, i.e.

```{r, str-gibbs-samples}
str(model_train$gibbs_samples, max.level = 2, give.attr = FALSE)
```

This object contains 2 elements:

- `gibbs_samples_raw` is a list of the raw samples from the Gibbs sampler,

- and `gibbs_samples_nbt` are the Gibbs samples used for parameter estimates, i.e. the normalized and thinned Gibbs samples after the burn-in.

Calling the summary function on the estimated `RprobitB_fit` object yields additional information about the Gibbs samples `gibbs_samples_nbt`. You can specify a list `FUN` of functions that compute any point estimate of the Gibbs samples^[Use the function `point_estimates()` to access the Gibbs sample statistics as an `RprobitB_parameter` object.], for example

- `mean` for the arithmetic mean,

- `stats::sd` for the standard deviation,

- `R_hat` for the Gelman-Rubin statistic [@Gelman1992] ^[A Gelman-Rubin statistic close to 1 indicates that the chain of Gibbs samples converged to the stationary distribution.],
 
- or custom statistics like the absolute difference between the median and the mean.

```{r, summary-model-train}
summary(model_train,
  FUN = c(
    "mean" = mean,
    "sd" = stats::sd,
    "R^" = R_hat,
    "custom_stat" = function(x) abs(mean(x) - median(x))
  )
)
```

Calling the `plot` method with the additional argument `type = "trace"` plots the trace of the Gibbs samples `gibbs_samples_nbt`: 

```{r, plot-trace-model-train}
par(mfrow = c(2, 1))
plot(model_train, type = "trace")
```

Additionally, we can visualize the serial correlation of the Gibbs samples via the argument `type = "acf"`. The boxes in the top-right corner state the total sample size TSS (here `R` - `B` = 10000 - 5000 = 5000), the effective sample size ESS, and the factor by which TSS is larger than ESS.

```{r, plot-acf-model-train}
par(mfrow = c(2, 3))
plot(model_train, type = "acf")
```

Here, the effective sample size is the value $\text{TSS} / (1 + \sum_{k\geq 1} \rho_k)$, where $\rho_k$ is the auto correlation between the chain offset by $k$ positions. The auto correlations are estimated via the `stats::acf()` function.

## Model transformation after estimation

The `transform` method can be used to transform an `RprobitB_fit` object in three ways:

1. change the length `B` of the burn-in period, for example 
```{r, transform-model-train}
model_train <- transform(model_train, B = 1)
```

2. change the thinning factor `Q` of the Gibbs samples, for example 
```{r, eval = FALSE} 
model_train <- transform(model_train, Q = 100)
```

3. or change the model normalization `scale`, for example
```{r, eval = FALSE} 
model_train <- transform(model_train, scale = "Sigma_1 := 1")
```

## References
