---
title: "Sequential Bayes: Utilizing the Posterior Distribution"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Sequential Bayes: Utilizing the Posterior Distribution}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

```{r setup, include = FALSE}
library(Rnaught)
```

In the Sequential Bayes method, the probability distribution of R0 is updated
sequentially from one case count to the next, starting from a (discretized)
uniform prior. By default, the function `seq_bayes()` returns the mean of the
last updated posterior distribution as its estimate of R0. However, by setting
the parameter `post` to `TRUE`, it is possible to return the final distribution
itself:

```{r}
# Daily case counts.
cases <- c(1, 4, 10, 5, 3, 4, 19, 3, 3, 14, 4)

posterior <- seq_bayes(cases, mu = 8, kappa = 7, post = TRUE)
```

First, the distribution can be used to retrieve the original estimate (had
`post` been left to its default value of `FALSE`) by calculating its mean:

```{r}
# `supp` is the support of the distribution, and `pmf` is its probability mass
# function.
post_mean <- sum(posterior$supp * posterior$pmf)
post_mean

# Verify that the following is true:
post_mean == seq_bayes(cases, mu = 8, kappa = 7)
```

Another use of the posterior is to obtain an alternative estimate of R0. For
instance, the following extracts the posterior mode rather than the mean:

```{r}
post_mode <- posterior$supp[which.max(posterior$pmf)]
post_mode
```

Returning the posterior is suitable for visualization purposes. Below is a graph
containing the uniform prior, final posterior distribution, posterior mean and
posterior mode:

```{r, dpi = 192, echo = FALSE}
par(mar = c(4.1, 4.1, 0.5, 0.5))

# Posterior.
plot(posterior$supp, posterior$pmf, xlab = "x", ylab = "p(x)",
  col = "black", lty = 1, type = "l"
)
# Uniform prior.
segments(x0 = 0, x1 = 7, y0 = 1 / (7 / 0.01 + 1), y1 = 1 / (7 / 0.01 + 1),
  col = "orange"
)
# Posterior mean.
abline(v = post_mean, col = "blue", lty = 2)
# Posterior mode.
abline(v = post_mode, col = "red", lty = 2)

legend("topright",
  legend = c("Prior", "Posterior", "Posterior mean", "Posterior mode"),
  col = c("orange", "black", "blue", "red"),
  lty = c(1, 1, 2, 2), cex = 0.5
)
```
