---
title: "Assessing the sampling efficiency"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Assessing the sampling efficiency}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

```{r setup}
library(malt)
```

Goal: generate samples $X_1,\cdots,X_n\in\mathbb{R}^d$ approximately distributed from
$$
\Pi(x)\propto e^{-\Phi(x)}
$$
by evaluating a potential function $\Phi$ and its gradient.

Example 1: Gaussian
$$
\begin{aligned}
\Phi(x)&=\frac{1}{2}(x-\mu)^\top\Sigma^{-1}(x-\mu)\\
\nabla\Phi(x)&=\Sigma^{-1}(x-\mu)
\end{aligned}
$$
Suppose we want to sample from a Gaussian distribution with heterogeneous scales, such that
$$
\mu=0_d,\qquad \Sigma=\underset{1\le i\le d}{\rm diag}(\sigma_i^2),\qquad \sigma_i^2=i/d,\qquad d=50.
$$
We specify the corresponding potential function and its gradient, as well as the starting values.
```{r}
d=50
sigma=((d:1)/d)^(1/2)
U=function(x){sum(0.5*x^2/sigma^2)}
grad=function(x){x/sigma^2}
init=rep(5,d)
```

We choose a friction, time step and integration time. We run malt from a deterministic initialisation and draw $n=10000$ samples. We also run hmc (friction: g=0) to compare. Both outputs are stored.
```{r}
n=10^4
g=1.5
h=0.2
L=10
output_malt=malt(init,U,grad,n,g,h,L)
output_hmc=malt(init,U,grad,n,0,h,L)
```
We obtain the acceptance rates of malt and hmc.
```{r}
output_malt$acceptance
output_hmc$acceptance
```