---
title: "When can rust be used?"
author: "Paul Northrop"
date: "`r Sys.Date()`"
output:
  rmarkdown::html_vignette:
    toc: true
vignette: >
  %\VignetteIndexEntry{When can rust be used?}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
bibliography: rust.bib
---

```{r, include = FALSE}
knitr::opts_chunk$set(comment = "#>", collapse = TRUE)

required <- c("bang")

if (!all(unlist(lapply(required, function(pkg) requireNamespace(pkg, quietly = TRUE)))))
  knitr::opts_chunk$set(eval = FALSE)
```

The generalized ratio-of-uniforms can be used to simulate from a wide range of $d$-dimensional multivariate probability densities, provided that $d$ is not so large that its efficiency is prohibitively low (see the vignette [Introducing rust](rust-a-vignette.html)).  However, there are conditions that the target density $f$ must satisfy for this method to be applicable.  This vignette considers instances when these conditions do *not* hold and suggests strategies that may overcome this difficulty.  Although the ratio-of-uniforms method can be used to simulate from multimodal densities, currently **rust** is designed to work effectively with unimodal densities.  This vignette illustrates this using a simple 1-dimensional example.

## Conditions on $f$

The generalized ratio-of-uniforms method is an acceptance-rejection type of algorithm.  It can only be applied to densities for which its acceptance region can be enclosed within a bounding region of finite volume from which it is simple to simulate, usually a cuboidal bounding box.  For a $d$-dimensional density $f(x)$ the bounding box (if it exists) is the $(d+1)$-dimensional set $\{ 0 < u \leq a(r), \, b_i^-(r) \leq v_i \leq b_i^+(r), \, i = 1, \ldots, d \}$,
where
$$
\begin{eqnarray}
a(r) &=& \sup_\chi \, f(x)^{1 / (r d + 1)}, \\
b_i^-(r) &=& \inf_{\chi_i^-} \, x_i \, f(x)^{r / (r d + 1)}, \\  
b_i^+(r) &=& \sup_{\chi_i^+} \, x_i \, f(x)^{r / (r d + 1)},  
\end{eqnarray}
$$
$x =(x_1, \ldots, x_d)$, $\chi \subseteq \mathbb{R}^d$, $\chi_i^- = \{ x \in \chi, x_i \leq 0 \}$ and $\chi_i^+ = \{ x \in \chi, x_i \geq 0 \}$.  See the vignette [Introducing rust](rust-a-vignette.html) for more details.

For a given value of the non-negative tuning parameter $r$ we need $f(x)$ and $x_i ^ {r d + 1} f(x) ^ r, i = 1, \ldots, d$ to be bounded.  If $f(x)$ is unbounded then we could use a transformation of variable to obtain a density that *is* bounded.  For bounded $f(x)$ one or more of $x_i ^ {r d + 1} f(x) ^ r, i = 1, \ldots, d$ can be unbounded if $f(x)$ has heavy-tails.  Again, we could use a transformation of variable to avoid this problem.  For this issue the value of $r$ matters and we may be able to achieve boundedness if a sufficiently large value of $r$ is used.  In **rust** $r = 1/2$ is used by default because this is optimal in the Gaussian case.  For heavy-tailed densities $r$ needs to be larger, perhaps larger than 1. We consider these strategies in the next two sections.

## Unbounded densities

A simple example of an unbounded density is that of a gamma random variable with a shape parameter that is less than 1.  Suppose that $X \sim \mbox{gamma}(\alpha, 1)$ and $\alpha < 1$.  Then the density $f_X(x)$ increases without limit as $x \rightarrow 0$ from above.  Let $Y = (X^\lambda - 1) / \lambda$, that is, a Box-Cox transformation [@BC1964] of $X$.  If we choose $\lambda$ appropriately then the density $f_Y(y)$ of $Y$ is bounded. **rust** has functions for selecting a suitable value of $\lambda$ in a somewhat automatic way (the user needs to specify a range of values (`min_phi`, `max_phi`) over which to perform the calculations).  In the gamma(0.1, 1) case below a value of $\lambda$ that is close to 0 is suggested.  The plot on the right shows that this does the trick.

```{r, fig.show='hold'}
library(rust)
alpha <- 0.1
max_phi <- qgamma(0.999, shape = alpha)
ptr_gam <- create_xptr("logdgamma")
lambda <- find_lambda_one_d_rcpp(logf = ptr_gam, alpha = alpha,
                                 max_phi = max_phi)
# Box-Cox transformation parameter
lambda$lambda
gam <- ru_rcpp(logf = ptr_gam, alpha = alpha, d = 1, n = 1000, trans = "BC",
               lambda = lambda)
plot(gam, xlab = "x")
plot(gam, ru_scale = TRUE, xlab = "y")
```

Here we have used `find_lambda_one_d_rcpp` and `ru_rcpp` (see the vignette [Rusting faster: Speedy Simulation using Rcpp](rust-c-using-rcpp-vignette.html)) but we could equally have used `ru` and `find_lambda_one_d`: this example appears in the help file for `find_lambda_one_d`.
 
## Heavy-tailed densities

Now we consider densities that are bounded but have sufficiently heavy tails that one or more of $x_i ^ {r d + 1} f(x) ^ r, i = 1, \ldots, d$ are unbounded if $r = 1/2$ is used.

### Cauchy density

Suppose that $X$ has a standard Cauchy density, that is $f(x) \propto (1 + x^2)^{-1}, -\infty < x < \infty$.  The bounding box cannot be constructed if $r < 1$.  For $r = 1$ the bounding box parameters $b_1^-(r) = -1$ and $b_1^+(r) = 1$ are attained in the limits as $x$ decreases/increases to infinity respectively.  This is fine in theory but using $r > 1$ avoids this problem and the largest probability of acceptance is obtained for $r \approx 1.26$. Below are the estimated bounding boxes and probabilities of acceptance for $r = 1$ (left) and $r = 1.26$ (right).  

```{r}
ptr_c <- create_xptr("logcauchy")
cauchy1 <- ru_rcpp(logf = ptr_c, log = TRUE, init = 0, r = 1, n = 1000)
cauchy126 <- ru_rcpp(logf = ptr_c, log = TRUE, init = 0, r = 1.26, n = 1000)
cbind(cauchy1$box, cauchy126$box)
c(cauchy1$pa, cauchy126$pa)
```

For $r = 1$ the algorithm finds essentially the correct values of $b_1^-$ and $b_1^+$ even though these do not occur at a turning point of the objective functions.  The estimated probability of acceptance is indeed larger for $r = 1.26$.

### Hierarchical 1-way ANOVA with few groups

Consider the hierarchical 1-way ANOVA model
\begin{equation*}
Y_{ij} = \mu + \alpha_i + \epsilon_{ij}, \quad\mbox{for} \,\, i = 1, \ldots, I, \, 
j = 1, \ldots, n_i,  \label{eqn:1way}
\end{equation*}
where  $\alpha_i \sim N(0, \sigma_\alpha^2)$ and $\epsilon_{ij} \sim N(0, \sigma^2)$
and all random variables are independent.

If the number $I$ of groups is small (in the example we consider below $I = 4$) then there is little information about the parameter $\sigma_\alpha$ that summarizes the variability in the mean of the response variable $Y$ across groups.  Unless strong prior information about $\sigma_\alpha$ is provided the posterior distribution for $\sigma_\alpha$ will tend to have a heavy right tail [@Gelman2006].  Indeed, for a commonly-used `noninformative' prior $\pi(\mu, \sigma_\alpha, \log\sigma) \propto 1$ [@BDA2014] a proper posterior density is not obtained if $I < 3$ [@Gelman2006].

To illustrate the problem that a heavy-tailed density can cause a naive implementation of the generalized ratio-of-uniforms algorithm we use the blood coagulation time data presented in Table 11.2 of @BDA2014.  These data are available as the dataframe `coagulation` in the **bang** package [@bang]: the first column holds the response values (the coagulation times of blood drawn from 24 animals), the second column a factor indicating to which of 4 diets the animal is allocated.  

The **bang** function `hanova1` samples from the marginal posterior distribution of $(\mu, \sigma_\alpha, \sigma)$ given data based on a user-supplied prior distribution.  The default prior is $\pi(\mu, \sigma_\alpha, \log\sigma) \propto 1$.  Under this prior, or indeed any prior in which $\mu$ is normally distributed and independent of $(\sigma_\alpha, \sigma)$ *a priori*, the generalized ratio-of-uniforms method can be used to sample from the marginal posterior distribution of $(\sigma_\alpha, \sigma)$.  By default (argument `param = "trans"`) then `hanova1` parameterizes this marginal posterior in terms of $(\log \sigma_\alpha, \log \sigma)$.  If instead we use `param = "original"`, so that this posterior is parameterized in terms of $(\mu, \sigma_\alpha, \sigma)$, then, with the default $r = 1/2$, we find that the bounding box cannot be found because the right tail of the posterior for $\sigma_\alpha$ is heavy enough to prevent this.  However, if we use $r = 1$ then the bounding box can be found.  The two successful approaches (reparameterization or use of $r=1$) are illustrated below.

```{r, fig.show='hold'}
library(bang)
coag1 <- hanova1(resp = coagulation[, 1], fac = coagulation[, 2], n = 1000)
coag2 <- hanova1(resp = coagulation[, 1], fac = coagulation[, 2], n = 1000,
                 param = "original", r = 1)
probs <- c(2.5, 25, 50, 75, 97.5) / 100
all1 <- cbind(coag1$theta_sim_vals, coag1$sim_vals)
all2 <- cbind(coag2$theta_sim_vals, coag2$sim_vals)
round(t(apply(all1, 2, quantile, probs = probs)), 1)
round(t(apply(all2, 2, quantile, probs = probs)), 1)
```

These posterior summaries are similar to those presented in Table 11.3 of @BDA2014 (where $\sigma_\alpha$ is denoted $\tau$), which were obtained using Gibbs sampling. 

The reparameterization strategy has the higher estimated probability of acceptance.

```{r}
coag1$pa
coag2$pa
```

## Multimodal densities

Consider the simple bimodal univariate density produced by a mixture of N(0,1) and N($m$, 1) densities, with probability $p$ that a value comes from the first component.  In principle the generalized ratio-of-uniforms can be used to sample from this density but this relies on the a bounding box being found that includes the entire acceptance region.  Currently, `ru` (and `ru_rcpp`) search for bounding box parameters in a way that is designed to work well when the density is unimodal.  The following examples demonstrate that currently `ru` isn't guaranteed to find a suitable bounding box for multimodal densities.

```{r, fig.show='hold'}
normal_mixture <- function(x, mu, p) {
  return(log(p * dnorm(x) + (1 - p) * dnorm(x, mean = mu)))
}

res1 <- ru(logf = normal_mixture, mu = 10, p = 0.25, init = -1, n = 1000)
plot(res1, main = "(a)")
res2 <- ru(logf = normal_mixture, mu = 10, p = 0.25, init = 11, n = 1000)
plot(res2, main = "(b)")
res3 <- ru(logf = normal_mixture, mu = 4, p = 0.25, init = 5, n = 1000)
plot(res3, main = "(c)")
res3$pa
res4 <- ru(logf = normal_mixture, mu = 4, p = 0.25, init = -1, n = 1000)
plot(res4, main = "(d)")
```

In (a), using the initial value `init = -1` means that the smaller of the two modes is found in the search for $a(r)$.  As a consequence most of the acceptance region for the other component of the mixture is not contained in the bounding box and this component is effectively missing from the sample produced. Case (b) is similar but the larger of the two modes is found.  In (c) and (d) the two components are closer, in the sense that the component distributions overlap to a greater degree.  In (c) the larger mode is found, the bounding box contains the entire acceptance region and a valid sample is produced.  In (d) the smaller mode is found and much of the acceptance region corresponding to the other component is not included in the acceptance region.

A future release of **rust** will include an option to employ a more extensive search for the bounding box parameters so that some simple multimodal densities can be accommodated.  However, multimodality will tend to reduce the probability of acceptance.  In example (c) above it is reasonable (approximately 0.55) but as the number of modes and/or dimensions of the density increase the probability of acceptance will decrease.

## References

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