---
title: "Distributions"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Distributions}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  eval = identical(Sys.getenv("TORCH_TEST", unset = "0"), "1"),
  purl = FALSE
)
```

```{r setup}
library(torch)
torch_manual_seed(1) # setting seed for reproducibility
```

This vignette showcases the basic functionality of distributions in torch.
Currently the distributions modules are considered 'work in progress'
and are still experimental features in the torch package. You can see
the progress in this [link](https://github.com/mlverse/torch/issues/479).

The distributions modules in torch are modelled after PyTorch's [distributions module](https://pytorch.org/docs/stable/distributions.html#) which in turn is based on the TensorFlow [Distributions package](https://arxiv.org/abs/1711.10604).

This vignette is based in the TensorFlow's distributions [tutorial](https://www.tensorflow.org/probability/examples/TensorFlow_Distributions_Tutorial#basic_univariate_distributions).

## Basic univariate distributions

Let's start and create a new instance of a normal distribution:

```{r}
n <- distr_normal(loc = 0, scale = 1)
n
```
We can draw samples from it with:

```{r}
n$sample()
```

or, draw multiple samples:

```{r}
n$sample(3)
```
We can evaluate the log probability of values:

```{r}
n$log_prob(0)
log(dnorm(0)) # equivalent R code
```
or, evaluate multiple log probabilities:

```{r}
n$log_prob(c(0, 2, 4))
```

## Multiple distributions

A distribution can take a tensor as it's parameters:

```{r}
b <- distr_bernoulli(probs = torch_tensor(c(0.25, 0.5, 0.75)))
b
```
This object represents 3 independent Bernoulli distributions, one for each element
of the tensor.

We can sample a single observation:

```{r}
b$sample()
```
or, a batch of `n` observations:

```{r}
b$sample(6)
```
## Using distributions within models

The `log_prob` method of distributions can be differentiated, thus, distributions
can be used to train models in torch.

Let's implement a Gaussian linear model, but first let's simulate some data

```{r}
x <- torch_randn(100, 1)
y <- 2*x + 1 + torch_randn(100, 1)
```

and plot:

```{r}
plot(as.numeric(x), as.numeric(y))
```

We can now define our model:

```{r}
GaussianLinear <- nn_module(
  initialize = function() {
    # this linear predictor will estimate the mean of the normal distribution
    self$linear <- nn_linear(1, 1)
    # this parameter will hold the estimate of the variability
    self$scale <- nn_parameter(torch_ones(1))
  },
  forward = function(x) {
    # we estimate the mean
    loc <- self$linear(x)
    # return a normal distribution
    distr_normal(loc, self$scale)
  }
)

model <- GaussianLinear()
```

We can now train our model with:

```{r}
opt <- optim_sgd(model$parameters, lr = 0.1)

for (i in 1:100) {
  opt$zero_grad()
  d <- model(x)
  loss <- torch_mean(-d$log_prob(y))
  loss$backward()
  opt$step()
  if (i %% 10 == 0)
    cat("iter: ", i, " loss: ", loss$item(), "\n")
}
```
We can see the parameter estimates with:

```{r}
model$parameters
```

and quickly compare with the `glm()` function:

```{r}
summary(glm(as.numeric(y) ~ as.numeric(x)))
```