ch03_sampling.qmd

# Sampling the Imaginary {#sampling}

```{r }
#| include: false
library(assertthat)
library(rethinking)
library(brms)
library(dplyr)
library(tidyr)
library(ggplot2)
library(ggdist)
library(paletteer)
library(simpr)
```

## Sampling from grid-approximation posterior

We use the example from chapter 2.

The grid of $p$ values has a grid size of $grid\_size$.

The prior is uniformly distributed and so, as discussed in Overthinking box of section 2.3.3, p. 35, $P(p)=\frac{1}{1-0}=1$.

```{r}
the_grid <- data.frame(
 prob = seq(from = 0, to = 1, length.out = 1000),
 prior = 1)
```

and we calculate the posterior using the data. We compute the likelihood using the grid of priors from above, then compute the average likelihood which is the sum of the likelihood.

The posterior is defined in detailed in section 2.3.4, p. 37.

$$
\text{Posterior} = 
 \frac{\text{Likelihood} \times \text{Prior}}
  {\text{Average Likelihood}}
$$

Note the code `stopifnot(sum(d$posterior) == 1)`, it is always a good idea to 
verify this.

```{r}
# the data, see page 28
data <- c("W","L","W","W","W","L","W","L","W")
n_success <- sum(data == "W")
n_trials <- length(data)

# compute the likelihood each value in the grid
the_grid <- the_grid |>
    mutate(
        likelihood = dbinom(x = n_success, size = n_trials, prob = prob),
        posterior = likelihood * prior / sum(likelihood))
assert_that(sum(the_grid$posterior) == 1,
            msg = "The total posterior prob. must equal 1.")
```

which gives the estimated posterior probability $p$ conditional on the data for 
each point of a grid.

Generate and visualize `n_samples` samples from the grid with the $p$ values 
with their respective posterior probability $p$ computed above.

> Note: We use `dplyr::slice_sample` because `dplyr::sample_n` is deprecated.

```{r}
set.seed(1223)
the_samples <- the_grid |>
  slice_sample(n = 1e4, weight_by = posterior, replace = TRUE) |>
  # this distance from the mean is used for coloring
  mutate(dist = abs(prob - mean(prob)))
the_samples$id <- seq_len(nrow(the_samples))
# str(the_samples)
```

visualize the sample of water proportion

```{r}
ggplot(data = the_samples, mapping = aes(x = id, y = prob, color = dist)) +
  geom_point(size = 0.75, alpha = 0.9) +
  scale_color_gradientn(colors = paletteer_d(palette="Manu::Kotare")) +
  theme_minimal() +
  theme(legend.position = "none") +
  labs(title = sprintf("%d samples", nrow(the_samples)))
```

visualize the density

```{r}
# show the p density
ggplot(data = the_samples, aes(x = prob)) +
    geom_density(aes(y=..scaled..), color = "blue", size = 1, fill = "lightblue") +
    theme_light() +
    labs(title = sprintf("%d samples", nrow(the_samples)))
```

## Sampling to summarize

### Intervals of defined boundaries

```{r}
the_grid |>
  filter(prob < 0.5) |>
  summarize(sum = sum(posterior))
```

and you can obtain the same result using the sampling data by counting the rows

```{r}
the_samples |>
  filter(prob < 0.5) |>
  count() |>
  mutate(pct = n / nrow(the_samples)) |>
  identity()
```

### Intervals of defined mass

Beside the base R `quantile` function, the `mean_qi` function from the package `ggdist` will be used extensively in this project. The benefits of using this package in conjonction with `posterior`, `tidybayes` etc. will become obvious in later chapters.

```{r}
the_samples |>
  ggdist::mean_qi(prob, .width = 0.8)
```

And if we redo the sampling with observing 3 $W$ in 3 tosses we have the grid

```{r}
the_grid <- data.frame(
  prob = seq(from = 0, to = 1, length.out = 1000),
  prior = 1) |>
  mutate(
    likelihood = dbinom(x = 3, size = 3, prob = prob),
    posterior = likelihood * prior / sum(likelihood))
assert_that(sum(the_grid$posterior) == 1,
            msg = "The total posterior prob. must equal 1.")

```

and we use it to resample

```{r}
set.seed(1223)
the_samples <- the_grid |>
  slice_sample(n = 1e4, weight_by = posterior, replace = TRUE) |>
  # this distance from the mean is used for coloring
  mutate(dist = abs(prob - mean(prob)))
the_samples$id <- seq_len(nrow(the_samples))
```

```{r}
the_samples |>
  ggdist::mean_qi(prob, .width = 0.5)
```

```{r}
the_samples |>
  ggdist::mean_hdi(prob, .width = 0.5)
```

and we can illustrate the intervals with `ggdist` as follows

```{r}
qtl <- c(0.5, 0.8, 0.95, 1)
x_breaks <- ggdist::mean_qi(.data = the_samples$prob, 
                            .width = qtl) |>
  select(y, ymin, ymax) |>
  pivot_longer(cols = c("y", "ymin", "ymax")) |>
  distinct(value) |>
  arrange(value) |>
  round(digits = 2) |>
  pull()
ggplot(the_samples, aes(x=prob)) +
         stat_halfeye(aes(fill=stat(cut_cdf_qi(
           cdf,
           .width = qtl,
           labels = scales::percent_format()
           )))) +
  scale_x_continuous(breaks = x_breaks) +
  scale_fill_paletteer_d(palette = "Manu::Takahe", direction = -1,
                         na.translate = FALSE) +
  theme_ggdist() +
  theme(legend.position = c(0.1, 0.75)) +
  labs(title = "Intervals of defined mass",
       x = "p_grid", y = "prob of p_grid",fill = "quantiles")
```

### Point estimates (loss function)

The **linex loss function** can be very useful in business analysis.
This is to be investigated later.

## Sampling to simulate prediction

## Summary