exercises_rlab_04.Rmd

---
title: "Exercises Laboratory Session 04"
author: "Nicola Zomer"
date: "`r format(Sys.time(), '%d %B, %Y')`"
output:
  html_document: default
---

```{r setup-chunk}
knitr::opts_chunk$set(dev = "ragg_png")
options(digits=5) # set number of digits equal to 5

```

# Packages and functions

```{r, message=FALSE}

# tidyverse
library(tidyverse)

# others
library(kableExtra)
library(patchwork)
library(ggpubr)
library(latex2exp)
library(glue)
library(lubridate)
#library(gridExtra)

```


# Exercise 1 - Community Mobility Open Data
Community Mobility Reports have been created with the aim to provide insights into what has changed in response to policies aimed at combating COVID-19. Data can be found at https://www.google.com/covid19/mobility/.

Download and analyze the following data sets:

* https://www.gstatic.com/covid19/mobility/Global_Mobility_Report.csv 
* https://www.gstatic.com/covid19/mobility/Region_Mobility_Report_CSVs.zip

Select a couple of European countries of your choice and analyze the trends in the variables over time: produce a plot of the data by averaging the observable over a period of one week (hint: convert the data field to `lubridate::week`) and one month and quantify the impact of COVID-19 restrictions on mobility situations.

```{r}

# DEFINE THE FILEPATH (folder containing the files with the data by country)
filepath='../../data/'

```

```{r}

df_country_processed <- function(df_original, type='week'){
  # type can be 'week' or 'month'
  
  year <- year(df_original$date[1])
  country_code <- df_original$country_region_code[1]
  
  
  if (type=='week') df_original$date <- week(df_original$date)
  else if (type=='month') df_original$date <- month(df_original$date)
  else stop('Insert a correct conversion type!')
  
  return(df_original %>%
    subset(is.na(sub_region_1)&is.na(sub_region_2)) %>%
    group_by(date) %>%
    summarize(
       retail_and_recreation = mean(retail_and_recreation_percent_change_from_baseline, na.rm=TRUE),
       grocery_and_pharmacy  = mean(grocery_and_pharmacy_percent_change_from_baseline,  na.rm=TRUE),
       parks                 = mean(parks_percent_change_from_baseline, na.rm=TRUE),
       transit_stations      = mean(transit_stations_percent_change_from_baseline, na.rm=TRUE), 
       workplaces            = mean(workplaces_percent_change_from_baseline, na.rm=TRUE),
       residential           = mean(residential_percent_change_from_baseline, na.rm=TRUE)
    ) %>%
    add_column(country_code, .after=0) %>%
    add_column(year, .after=1) 
  )
  
}

```


```{r}

gg_observable <- function(observable, country_region_code='', type='week'){
  
  country_20 <- read_csv(paste(filepath, '2020_', country_region_code, '_Region_Mobility_Report.csv', sep=''), col_names=TRUE)
  country_21 <- read_csv(paste(filepath, '2021_', country_region_code, '_Region_Mobility_Report.csv', sep=''), col_names=TRUE)
  country_22 <- read_csv(paste(filepath, '2022_', country_region_code, '_Region_Mobility_Report.csv', sep=''), col_names=TRUE)
  
  df_list <- list(country_20, country_21, country_22)
  
  ggplot()+
    geom_line(data=df_country_processed(df_list[[1]], type=type), aes(x=date, y=get(observable), color='2020'), size=1)+
    geom_line(data=df_country_processed(df_list[[2]], type=type), aes(x=date, y=get(observable), color='2021'), size=1)+
    geom_line(data=df_country_processed(df_list[[3]], type=type), aes(x=date, y=get(observable), color='2022'), size=1)+
    labs(
      title = names(observable),
      x = type,
      y = 'Perc. change from baseline' 
    )+
    {if (type=='week') scale_x_continuous(breaks=seq(0, 54, by=6))}+
    #{if (type=='month') scale_x_continuous(breaks=seq(1, 12, by=1), minor_breaks=seq(1, 12, by=1))}+
    {if (type=='month') scale_x_continuous(breaks=seq(1, 12, by=1), minor_breaks=seq(1, 12, by=1), labels=month.abb[1:12])}+
    scale_color_manual(name = "", values = c('2020' = 'steelblue', 
                                             '2021' = 'darkgreen', 
                                             '2022' = 'firebrick'))+
    theme_bw()+
    theme(legend.title = element_blank(), 
          plot.title  = element_text(size=20, hjust=0.5), 
          axis.text   = element_text(size=14),
          axis.title  = element_text(size=14), 
          legend.text = element_text(size=20))
}


gg_tot_obs <- function(cr_code, country_name, type='week'){
  
  observables <- c("retail_and_recreation", "grocery_and_pharmacy", "parks", "transit_stations", "workplaces", "residential")
  names(observables) <- c("Retail and recreation", "Grocery and pharmacy", "Parks ", "Transit stations", "Workplaces", "Residential")
  
  (gg_observable(observables[1], cr_code, type=type) + gg_observable(observables[2], cr_code, type=type) + gg_observable(observables[3], cr_code, type=type)) /
  (gg_observable(observables[4], cr_code, type=type) + gg_observable(observables[5], cr_code, type=type)) +
  plot_layout(guides = "collect") +
  plot_annotation(
    title = paste('Community Mobility Reports of ', country_name, ' by ', type, sep=''),
    caption = 'Source: https://www.gstatic.com/covid19/mobility/Region_Mobility_Report_CSVs.zip', 
    theme = theme(plot.title = element_text(size = 26, hjust = 0.5), 
                  plot.caption = element_text(size = 14))
  ) &
  theme(legend.position = "bottom")
}


```

### Italy

```{r, fig.height=12, fig.width=18, message=FALSE}

gg_tot_obs('IT', 'Italy', 'week')
gg_tot_obs('IT', 'Italy', 'month')

```

### United Kingdom

```{r, fig.height=12, fig.width=18, message=FALSE}

gg_tot_obs('GB', 'United Kingdom', 'week')
gg_tot_obs('GB', 'United Kingdom', 'month')

```

### France

```{r, fig.height=12, fig.width=18, message=FALSE}

gg_tot_obs('FR', 'France', 'week')
gg_tot_obs('FR', 'France', 'month')

```

### Germany

```{r, fig.height=12, fig.width=18, message=FALSE}

gg_tot_obs('GE', 'Germany', 'week')
gg_tot_obs('GE', 'Germany', 'month')

```

### Results

In all countries shown and in all categories it can be seen how the curve of 2021 is above that of 2020, apart from the first months of the year (from January to March). The time spent in the categorized places therefore grows in general from 2020 to 2021, almost surely due to a relaxation of the mobility restrictions. The behavior in the first months of the year is explained by the fact that in 2020 there were still no policies in place aimed at combating COVID-19, which then led to a decrease in the time spent in these places in 2021. With regard to 2022, it is possible to see that the curve for now appears to be above that of 2020, in all states and in all categories, a sign of further easing of COVID restrictions. 

Finally, notice that the important drop in April 2020 appears precisely in conjunction with the initial period of very strong restrictions. 

# Exercise 2 - Random number generators
One of the first random number generator was proposed by von Neumann, the so-called _middle square_ algorithm. 

Write R code to implement this type of generator and, given a fixed digit number input, square it an remove the leading and trailing digits, in order to return a number with the same number of digits as the original number.

```{r}

middle_square <- function(x.start=5772156649, n){
  input.digits <- length(unlist(strsplit(as.character(x.start),"")))
  
  # print(paste('Number of digits of the input number:', input.digits))
  if (x.start%%1 != 0) stop('The starting number must be integer')
  
  x.squared.char <- unlist(strsplit(as.character(x.start^2),""))  
  half.output.digits <- length(x.squared.char)%/%2
  
  if (input.digits%%2 ==0){ # even number of digits
    x.i <- x.squared.char[(half.output.digits-(input.digits/2-1)):(half.output.digits+input.digits/2)]
  } else{ # odd number of digits
    x.i <- x.squared.char[(half.output.digits-(input.digits%/%2)):(half.output.digits+input.digits%/%2)]
  }
  
  # check if the first digit is zero: in this case shift to the left
  i<-1
  while (x.i[1]=="0"){
    print(x.i)
    if (input.digits%%2 ==0){ # even number of digits
      x.i <- x.squared.char[(half.output.digits-(input.digits/2-1)-i):(half.output.digits+input.digits/2-i)]
    } else{ # odd number of digits
      x.i <- x.squared.char[(half.output.digits-(input.digits%/%2)-i):(half.output.digits+input.digits%/%2-i)]
    }
    i<-i+1
  }
  return(as.numeric(paste(x.i, collapse="")))
}

```

Generate 10 numbers using the _middle square_ algorithm:
```{r}

numbers <- numeric(11)
numbers[1] <- 5772156649

for (i in 1:10){
  numbers[i+1] <- middle_square(numbers[i])
}

data.frame(Iteration=0:10, Generated.Number=numbers) %>%
  kable() %>%
  kable_styling(full_width=FALSE)

```

# Exercise 3 - Bayesian Inference

A publishing company has recently launched a new journal. In order to determine how effective it is in reaching its possible audience, a market survey company selects a random sample of people from a possible target audience and interviews them. Out of 150 interviewed people, 29 have read the last issue of the journal.

### a) What kind of distribution would you assume for y, the number of people that have seen the last issue of the journal?
Since I have no prior knowledge about the data, I assume a uniform distribution for $y$. I could make a different choice, such as a beta distribution, if for example I had an idea of the expected value of $y$, or I could cut the uniform prior if I thought it is highly unlikely that everyone or no one has read the journal. In the latter case I would choose a trapezoidal distribution. Having no information, the best choice seems to me a uniform distribution, which corresponds to the least amount of bias.

### b) Assuming a uniform prior, what is the posterior distribution for y ?
Assuming a uniform prior, the posterior distribution is simply proportional to the likelihood, which in this case is a Binomial distribution with parameters $n=150$, $r=29$. So
$$
P(\pi|r=29, n=100, M) = \begin{pmatrix} 100\\ 29\end{pmatrix} \pi^{29}(1-\pi)^{100-29} = Binom(n=100, r=29, p=\pi)
$$

### c) Plot both posterior and likelihood ditributions functions

```{r}

p <- seq(0, 1, length=1000)
delta.p <- 1/1000
r <- 29
n <- 150

journal.prior <- dunif(p, min=0, max=1)
journal.likelihood <- dbinom(r, n, p)
journal.posterior.norm <- journal.likelihood/(delta.p*sum(journal.likelihood))

ggplot()+
  geom_line(aes(x=p, y=journal.prior), color='steelblue', size=0.8)+
  labs(
    title='Journal readings prior',
    x=TeX('$p$'), 
    y=TeX('$P(\\pi|M)$$')
  )+
  scale_y_continuous(breaks=seq(0, 1.2, by=0.2), limits=c(0, 1.1))+
  theme_bw()

ggplot()+
  geom_line(aes(x=p, y=journal.likelihood), color='steelblue', size=0.8)+
  labs(
    title='Journal readings likelihood',
    x=TeX('$p$'), 
    y=TeX('$P(\\pi|r, n, M)$$')
  )+
  theme_bw()

ggplot()+
  geom_line(aes(x=p, y=journal.posterior.norm), color='steelblue', size=0.8)+
  geom_vline(xintercept=r/n, linetype='dashed')+
  annotate('text', x=r/n+0.09, y=12.5, label=paste('mode = ', round(r/n, 2), sep=''))+
  labs(
    title='Journal readings posterior',
    x=TeX('$p$'), 
    y=TeX('$P(\\pi|r, n, M)$$')
  )+
  theme_bw()


```


# Exercise 4 - Bayesian Inference
A coin is flipped n = 30 times with the following outcomes:

T, T, T, T, T, H, T, T, H, H, T, T, H, H, H, T, H, T, H, T, H, H, T, H, T, H, T, H, H, H

Notice that there are 15 T and 15 H.

### a) Assuming a flat prior, and a beta prior, plot the likelihood, prior and posterior distributions for the data set.

Flat prior
```{r}

n <- 30
r <- 15

nsamples <- 1000
p <- seq(0, 1, length = nsamples)
delta.p <- 1/nsamples

unif.prior <- dunif(p)
unif.post <- dbinom(x=r, size=n, prob=p)
unif.post.norm <- unif.post/(delta.p*sum(unif.post))

ggplot()+
  geom_line(aes(x=p, y=unif.post.norm, color='Posterior and likelihood'), size=0.7)+
  geom_line(aes(x=p, y=unif.prior, color='Prior'), size=0.7)+
  labs(
    title='Flat prior', 
    x=TeX('$p$'), 
    y='Density'
  )+
  scale_color_manual(name = "", values = c("Posterior and likelihood" = "steelblue", "Prior" = "firebrick"))+
  theme(legend.title= element_blank())+
  theme_bw()

```
Beta prior
```{r, fig.height=18, fig.width=18}

nsamples <- 1000
p <- seq(0, 1, length = nsamples)
delta.p <- 1/nsamples

n <- 30
r <- 15

gg_beta <- function(alpha, beta){
  ggplot()+
    geom_line(aes(x=p, y=dbeta(x=p, alpha+r, beta+n-r), color='Posterior'), size=1)+
    geom_line(aes(x=p, y=dbinom(x=r, size=n, prob=p), color='Likelihood'), size=1)+
    geom_line(aes(x=p, y=dbeta(x=p, alpha , beta), color='Prior'), size=1)+
    labs(
      title=paste('Beta prior (', alpha, ', ', beta, ')', sep=''), 
      x='p', 
      y='Density' #TeX('$P(p|r,n, M)$')
    )+
    scale_color_manual(name = "", values = c("Posterior" = "steelblue", "Likelihood"= "darkgreen", "Prior" = "firebrick"))+
    theme_bw()+
    theme(legend.title = element_blank(), 
          plot.title  = element_text(size=22), 
          axis.text   = element_text(size=18),
          axis.title  = element_text(size=18), 
          legend.text = element_text(size=22))
}

combined <- (gg_beta(1,1) +ylim(0,6) + gg_beta(1,5) +ylim(0,6) + gg_beta(1,10) +ylim(0,10) )/
            (gg_beta(5,1) +ylim(0,6) + gg_beta(5,5) +ylim(0,6) + gg_beta(5,10) +ylim(0,6))/ 
            (gg_beta(10,1)+ylim(0,10)+ gg_beta(10,5)+ylim(0,6) + gg_beta(10,10)+ylim(0,6)) & theme(legend.position = "bottom")

combined + plot_layout(guides = "collect")

```


### b) Evaluate the most probable value for the coin probability p and, integrating the posterior probability distribution, give an estimate for a 95% credibility interval.

```{r}

argmax <- function(x){
  max.x <- max(x)
  found <- FALSE
  
  i <- 1
  while (!found & i<=length(x)){
    if (x[i]==max.x){
      found <- TRUE
      return(i)
    }
    i <- i+1
  }
}

# uniform
mpv.unif <- p[argmax(unif.post.norm)]
glue('Most probable value with an uniform prior: p={format(mpv.unif, digits=4)}')

# beta (10, 10)
beta.prior.10 <- dbeta(x=p, 10, 10)
beta.post.10 <- dbeta(x=p, 10+r, 10+n-r)
mpv.beta <- p[argmax(beta.post.10)]
glue('Most probable value with a beta prior (alpha=10, beta=10): p={format(mpv.beta ,digits=4)}')

# plot both posterior
ggplot()+
  geom_line(aes(x=p, y=unif.post.norm, color='Uniform prior'), size=0.7)+
  geom_line(aes(x=p, y=beta.post.10, color='Beta(10, 10) prior'), size=0.7)+
  geom_vline(xintercept=mpv.beta, linetype='dashed', size=0.6)+
  annotate('text', x=mpv.beta+0.06, y=0.5, label=paste(round(mpv.beta, 3), sep=''))+
  labs(
    title='Coin tossing posterior', 
    x=TeX('$p$'), 
    y=TeX('$P(p|r,n, M)$')
  )+
  scale_color_manual(name = "", values = c("Beta(10, 10) prior" = "steelblue", "Uniform prior" = "darkgray"))+
  theme(legend.title= element_blank())+
  theme_bw()


```

```{r}

setClass(Class="CredInterval",
        representation(
              x1="numeric",
              x2="numeric"
          )
)

cred.interval <- function(p.post.funct, nsamples=1000, perc=0.95){
  
    p <- seq(0, 1, length=nsamples)
    cred.range <- (1-perc)/2
    
    x.1.found <- FALSE
    x.2.found <- FALSE
    
    i<-1
    while (!x.1.found){
      int.i <- integrate(p.post.funct, 0, p[i])$value
      int.iplus <- integrate(p.post.funct, 0, p[i+1])$value
      
      if (between(cred.range, int.i, int.iplus)){
        x.1.found <- TRUE
        x.1 <- (p[i]+p[i+1])/2
      }
      i <- i+1 
    }
    i <- nsamples
    while (!x.2.found){
      int.i <- integrate(p.post.funct, p[i], tail(p, 1))$value
      int.iminus <- integrate(p.post.funct, p[i-1], tail(p, 1))$value
      
      if (between(cred.range, int.i, int.iminus)){
        x.2.found <- TRUE
        x.2 <- (p[i]+p[i-1])/2
      }
      i <- i-1 
    }
    return(new("CredInterval", x1 = x.1, x2 = x.2))
}

credlim.unif <- cred.interval(function(x){dbeta(x, 1+r, 1+n-r)}) # beta distribution with alpha=1, beta=1
credlim.beta.10 <- cred.interval(function(x){dbeta(x, 10+r, 10+n-r)})

glue('Credibility interval limits, with uniform prior: \nx1={format(credlim.unif@x1, digits=3)}, x2={format(credlim.unif@x2, digits=3)}')
glue('Credibility interval limits, with Beta(10,10) prior: \nx1={format(credlim.beta.10@x1, digits=3)}, x2={format(credlim.beta.10@x2, digits=3)}')
```

```{r}

range.95.unif <- seq(credlim.unif@x1, credlim.unif@x2, length=200)
range.95.beta <- seq(credlim.beta.10@x1, credlim.beta.10@x2, length=200)

gg_credlim <- function(alpha, beta, range95, limits, Title='Posterior'){
  ggplot()+
    geom_line(aes(x=p, y=dbeta(x=p, alpha+r, beta+n-r)), size=0.7, color='steelblue')+
    geom_area(aes(x=range95, y=dbeta(x=range95, alpha+r, alpha+n-r)), fill='lightblue', color='steelblue', size=0.7)+
    labs(
      title=Title, 
      x='p', 
      y=TeX('$P(p|r,n, M)$')
    )+
    ylim(0, 6)+
    geom_vline(xintercept=limits, linetype='dashed', size=0.6)+
    geom_vline(xintercept=0.5, linetype='dashed', size=0.6, color='firebrick')+
    annotate('text', x =limits[1]-0.06, y = 5.2, label=paste('x1=', round(limits[1], 2), sep=''))+
    annotate('text', x =limits[2]+0.06, y = 5.2, label=paste('x2=', round(limits[2], 2), sep=''))+
    annotate('text', x =0.5+0.045, y = 6, label='mode', color='firebrick')+
    theme_bw()
}

gg_credlim(1, 1, range.95.unif, c(credlim.unif@x1, credlim.unif@x2), 'Posterior with uniform prior')
gg_credlim(10, 10, range.95.beta, c(credlim.beta.10@x1, credlim.beta.10@x2), 'Posterior with beta (10,10) prior')


```


### c) Repeat the same analysis assuming a sequential analysis of the data. Show how the most probable value and the credibility interval change as a function of the number of coin tosses (i.e. from 1 to 30).

First, I store the data assigning $0$ to $H$ and $1$ to $T$. Here I assume a uniform prior. 

```{r}

H<-0
T<-1
coin.toss.data <- c(T, T, T, T, T, H, T, T, H, H, T, T, H, H, H, T, H, T, H, T, H, H, T, H, T, H, T, H, H, H)

# vector of probabilities
npoints <- 1000
p <- seq(0, 1, length = npoints)
delta.p <- 1/npoints

# initialize outputs
unif.mode <- numeric(30)
unif.x1 <- numeric(30)
unif.x2 <- numeric(30)

beta10.mode <- numeric(30)
beta10.x1 <- numeric(30)
beta10.x2 <- numeric(30)

# initialize priors
unif.prior.i <- dunif(p)
beta10.prior.i <- dbeta(p, 10, 10)

for (i in 1:30){
  r = coin.toss.data[i] # toss one coin
  n = 1                 # n is fixed and equal to 1
  
  # likelihood
  likelihood <- dbinom(r, n, p)
  
  # posterior
  unif.post.i <- likelihood*unif.prior.i
  unif.post.i <- unif.post.i/(delta.p*sum(unif.post.i))
  
  beta10.post.i <- likelihood*beta10.prior.i
  beta10.post.i <- beta10.post.i/(delta.p*sum(beta10.post.i))

  # store results
  unif.mode[i] <- p[argmax(unif.post.i)]
  beta10.mode[i] <- p[argmax(beta10.post.i)]
  
  # to get the credibility interval I can't used the function defined above because it requires the analytical form of the posterior
  unif.x1[i] <- p[cumsum(unif.post.i/npoints)>0.025][1]
  unif.x2[i] <- p[cumsum(unif.post.i/npoints)>=0.975][1]
  beta10.x1[i] <- p[cumsum(beta10.post.i/npoints)>0.025][1]
  beta10.x2[i] <- p[cumsum(beta10.post.i/npoints)>=0.975][1]
  
  # update the priors
  unif.prior.i <- unif.post.i
  beta10.prior.i <- beta10.post.i

}

```

Plot the final posterior distributions, including the most probable value and credibility interval.
```{r}

cred_range_unif <- p[cumsum(unif.post.i/npoints)>0.025 & cumsum(unif.post.i/npoints)<0.975] 
cred_range_unif.post <- unif.post.i[cumsum(unif.post.i/npoints)>0.025 & cumsum(unif.post.i/npoints)<0.975] 

cred_range_beta10 <- p[cumsum(beta10.post.i/npoints)>0.025 & cumsum(beta10.post.i/npoints)<0.975] 
cred_range_beta10.post <- beta10.post.i[cumsum(beta10.post.i/npoints)>0.025 & cumsum(beta10.post.i/npoints)<0.975] 

# PLOT
ggplot()+
  geom_line(aes(x=p, y=unif.post.i), size=0.7, color='steelblue')+
  geom_area(aes(x=cred_range_unif, y=cred_range_unif.post),  fill='lightblue', color='steelblue', size=0.7)+
  labs(
    title='Final posterior with uniform prior, sequential analysis', 
    x='p', 
    y=TeX('$P(p|r,n, M)$')
  )+
  ylim(0, 6)+
  geom_vline(xintercept=tail(unif.x1, 1), linetype='dashed', size=0.6)+
  geom_vline(xintercept=tail(unif.x2, 1), linetype='dashed', size=0.6)+
  geom_vline(xintercept=tail(unif.mode, 1), linetype='dashed', size=0.6, color='firebrick')+
  annotate('text', x =tail(unif.x1, 1)-0.06, y = 5.2, label=paste('x1=', round(tail(unif.x1, 1), 2), sep=''))+
  annotate('text', x =tail(unif.x2, 1)+0.06, y = 5.2, label=paste('x2=', round(tail(unif.x2, 1), 2), sep=''))+
  annotate('text', x =0.5+0.045, y = 6, label='mode', color='firebrick')+
  theme_bw()

ggplot()+
  geom_line(aes(x=p, y=beta10.post.i), size=0.7, color='steelblue')+
  geom_area(aes(x=cred_range_beta10, y=cred_range_beta10.post),  fill='lightblue', color='steelblue', size=0.7)+
  labs(
    title='Final posterior with beta(10, 10) prior, sequential analysis', 
    x='p', 
    y=TeX('$P(p|r,n, M)$')
  )+
  ylim(0, 6)+
  geom_vline(xintercept=tail(beta10.x1, 1), linetype='dashed', size=0.6)+
  geom_vline(xintercept=tail(beta10.x2, 1), linetype='dashed', size=0.6)+
  geom_vline(xintercept=tail(beta10.mode, 1), linetype='dashed', size=0.6, color='firebrick')+
  annotate('text', x =tail(beta10.x1, 1)-0.06, y = 5.2, label=paste('x1=', round(tail(beta10.x1, 1), 2), sep=''))+
  annotate('text', x =tail(beta10.x2, 1)+0.06, y = 5.2, label=paste('x2=', round(tail(beta10.x2, 1), 2), sep=''))+
  annotate('text', x =0.5+0.045, y = 6, label='mode', color='firebrick')+
  theme_bw()

```


```{r, fig.height=7, fig.width=18}

gg_coin_unif <- ggplot()+
  geom_ribbon(aes(x=1:30, ymin=unif.x1, ymax=unif.x2), fill = "lightsteelblue", alpha=0.4) +
  geom_line(aes(x=1:30, y = unif.mode, color='Mode'), size=1) + 
  geom_line(aes(x=1:30, y = unif.x1, color='Credibility Bands'), size=1) + 
  geom_line(aes(x=1:30, y = unif.x2, color='Credibility Bands'), size=1) + 
  labs(
    title = 'Uniform prior', 
    x='Iteration step', 
    y='p'
  )+
  xlim(1, 30)+
  ylim(0, 1)+
  scale_colour_manual("",values=c("Mode"="navyblue", "Credibility Bands"="steelblue"))+
  theme(plot.title  = element_text(size=20, hjust = 0.5), 
          axis.text   = element_text(size=18),
          axis.title  = element_text(size=18), 
          legend.text = element_text(size=16))
  
gg_coin_beta10 <- ggplot()+
  geom_ribbon(aes(x=1:30, ymin=beta10.x1, ymax=beta10.x2), fill = "lightgoldenrod", alpha=0.4) +
  geom_line(aes(x=1:30, y = beta10.mode, color='Mode'), size=1) + 
  geom_line(aes(x=1:30, y = beta10.x1, color='Credibility Bands'), size=1) + 
  geom_line(aes(x=1:30, y = beta10.x2, color='Credibility Bands'), size=1) +  
  labs(
    title = 'Beta(10, 10) prior', 
    x='Iteration step', 
    y='p'
  )+
  xlim(1, 30)+
  ylim(0, 1)+
  scale_colour_manual("",values=c("Mode"="firebrick", "Credibility Bands"="darkorange"))+
  theme(plot.title  = element_text(size=20, hjust = 0.5), 
          axis.text   = element_text(size=18),
          axis.title  = element_text(size=18), 
          legend.text = element_text(size=16))

(gg_coin_unif+gg_coin_beta10) +
  plot_annotation(
    title = 'Coin tossing, sequential analysis',
    subtitle = 'Filled bands correspond to the 95% credibility interval',
    theme = theme(plot.title = element_text(size = 24, hjust = 0.5),
                  plot.subtitle = element_text(size = 20, hjust = 0.5),
                  plot.caption = element_text(size = 14))
  )

```

### d) Do you get a different result, by analyzing the data sequentially with respect to a one-step analysis (i.e. considering all the data as a whole)?
No, as expected I get exactly the same results.