14 Day 14 (March 6)

14.1 Announcements

  • Update on projects

  • Read Chs. 11 and 12

  • Selected questions/clarifications from journals

    • Didn’t have time to do this :(

14.2 The Bayesian Linear Model

14.3 Bayesian prediction

  • Prior prediction vs. posterior prediction

    • Prior prediction is used before fitting your model to data. It is used as a model checking tool (e.g., to see if priors are reasonable) or to do prediction if you have little or no data.
    • Posterior prediction is more typically what you think of as prediction. It is done after fitting your model to data.

  • Prior prediction of disease spread (here)

  • Prior prediction example: my retirement

    • Personal information
      • Obviously this isn’t my actual information, but it isn’t too far off!
      • Since I am a millennial/doge I don’t think social security will be around when I retire (i.e., assume social security contributes $0 to my retirement)
      • As of 1/1/25 I have $250,000 in 401k retirement in accounts
      • All of money is invested into an S&P 500 index fund (VOO to be exact)
      • I am 38 as of 1/1/25
      • I will contribute ~ $20k to my retirement each year
      • I want to know how much pre-tax money I will have at a given retirement age (e.g., 60, 65, 70, etc)
    • Example using a mathematical model
      • Whiteboard demonstration
      • What are the model assumptions?
      • In program R
    # The value of my 401k retirement account as of 1/1/25
y_2025 <- 250000 


# How much money will I add to my 401k each year
q <- 20000


# Rate of return for S&P 500 index fund
r <- 0.12


# How much $ will I have in 2026
y_2026 <- y_2025*(1+r)+q
y_2026
    ## [1] 3e+05
    # How much $ will I have in 2027
y_2027 <- y_2026*(1+r)+q
y_2027
    ## [1] 356000
    # How much $ will I have in 2028
y_2028 <- y_2027*(1+r)+q
y_2028
    ## [1] 418720
    # Using a for loop to calculate how much $ will I have 
year <- seq(2025,2025+22,by=1)
y <- matrix(,length(year),1)
rownames(y) <- year
y[1,1] <- 250000

for(t in 1:22){
  y[t+1,1] <- y[t,1]*(1+r)+q
}

plot(year,y/10^6,typ="b",pch=20,col="deepskyblue",xlab="Year",ylab="Pretax retirement amount ($ millions)")

    # How much $ will I have when I am 60? 
# Note that units are millions of $
retirement.year <- 2025+22
y[which(year==retirement.year)]/10^6
    ## [1] 4.875129
    • Example using a Bayesian statistical model
      • Whiteboard demonstration
      • S&P 500 return since inception in 1957
    # Download S&P 500 returns    
url <- "https://www.dropbox.com/s/81ccahyuaas1zpd/s%26p500.csv?dl=1"
df.sp500 <- read.csv(url)

head(df.sp500)
    ##   year  return
## 1 2022 -0.1811
## 2 2021  0.2871
## 3 2020  0.1840
## 4 2019  0.3149
## 5 2018 -0.0438
## 6 2017  0.2183
    mean(df.sp500$return)
    ## [1] 0.1152742
    hist(df.sp500$return,main="",col="grey",xlab=" Return rate of S&P 500")

    - Example using the prior predictive distribution

# Download S&P 500 returns
url <- "https://www.dropbox.com/s/81ccahyuaas1zpd/s%26p500.csv?dl=1"
df.sp500 <- read.csv(url)

# The value of my 401k retirement account as of 1/1/25
y_2025 <- 250000 

# How much money will I add to my 401k each year
q <- 20000


# Using a for loop to calculate how much $ will I have 
year <- seq(2025,2025+22,by=1)
Y <- matrix(,length(year),1000)
rownames(Y) <- year
Y[1,] <- y_2025

set.seed(3410)
for(m in 1:1000){
for(t in 1:22){
  r <- sample(df.sp500$return,1)
  Y[t+1,m] <- Y[t,m]*(1+r)+q
              }
                }


# Prior predictive distribution for a given year
retirement.year <- 2025+22
hist(Y[which(year==retirement.year),]/10^6,col="grey",freq=FALSE,xlab="Pretax retirement amount ($ millions)",main="")

# Summary of prior predictive distribution
mean(Y[which(year==retirement.year),]/10^6)
## [1] 4.393178
max(Y[which(year==retirement.year),]/10^6)
## [1] 21.66186
min(Y[which(year==retirement.year),]/10^6)
## [1] 0.3341111
quantile(Y[which(year==retirement.year),]/10^6,probs=c(0.025,0.975))
##      2.5%     97.5% 
##  0.901447 12.309287
# Plot some financial trajectories (i.e., random draws from the prior predictive distribution)
plot(year,Y[,1]/10^6,typ="l",lwd=1,ylim=c(0,max(Y/10^6)),col=rgb(0.1,0.1,0.1,.2),xlab="Year",ylab="Pretax retirement amount ($ millions)") 
for(i in 1:30){
points(year,Y[,i]/10^6,typ="l",lwd=1,col=rgb(0.1,0.1,0.1,.2))
}

# Plot of prior predictive distribution for all years
E.Y <- apply(Y/10^6,1,mean)
u.CI <- apply(Y/10^6,1,quantile,prob=0.975)
l.CI <- apply(Y/10^6,1,quantile,prob=0.025)
max.Y <- apply(Y/10^6,1,max)
min.Y <- apply(Y/10^6,1,min)

plot(year,E.Y,typ="l",lwd=3,ylim=c(0,max(max.Y)),xlab="Year",ylab="Pretax retirement amount ($ millions)") 
points(year,max.Y,typ="l",lwd=3,col="red")
points(year,min.Y,typ="l",lwd=3,col="red")
polygon(c(year,rev(year)),c(u.CI,rev(l.CI)),
        col=rgb(0.5,0.5,0.5,0.5),border=rgb(0.5,0.5,0.5,0.5))