## Exercises

From Problem Set 2.

### Exercise 1

1. 2/3
2. 1/2
1. Prosecutor says $$P(BT(z)=x|NotGuilty(z))= P(BT(z)=x, NotGuilty(z))/P(NotGuilty(z)) = 0.01$$, but we are given P(BT(z)=x)=0.01. Then goes onto conclude P(Guilty(z)|BT(z)=x)=1-P(BT(z)=x|NotGuilty(z))$and there is no such relation. $\sum_{z, BT(z)=x} P(Guilty(z) | BT(z)=x) =1,$ but the prosecutorâ€™s argument applies to each of these yielding a sum that is greater than 1. Assuming the killer is from the population. The defenderâ€™s argument holds if the police chose the defendant uniformly at random from the subpopulation with the specified bloodtype. Under the reasonable, but unstated assumption, that the police chose the defendant out of a smaller population without regard to the blood type this argument is false. 1. $$P(T=P|D=P)=0.99$$, and $$P(T=N|D=N)=0.99$$ and $$P(D=P)=10^-4$$. $P(D=P|T=P) = \frac{P(T=P|D=P)P(D=P)}{P(T=P)} = \frac{P(T=P|D=P)P(D=P)}{P(T=P|D=P)P(D=P)+(1-P(T=N|D=N))(1-P(D=P))}.$ This yields $$0.009803921568627442$$. 1.$P(H|e_1,e_2)= so ii. is sufficient. I believe the first is not, but I donâ€™t have a counterexample. If it isnâ€™t the third is not sufficent, because it includes strictly less information.
2. Assuming $$P(e_1,e_2|H)=P(e_1|H)P(e_2|H)$$. And $$P(e_1,e_2)=\sum_{H=Z} P(e_1,e_2|Z)$$ so we can derive the probabilities in ii from those in iii (and therefore also i). So all three are sufficient.

## Monte Carlo estimation of $$\pi$$

Letâ€™s set our random seed:

set.seed(42)

Letâ€™s write a function that takes in a number of iterations and returns a data frame with all of the relevant information for our Monte Carlo simulation.

library(tidyverse)
mc_pi = function(n) {
df = tibble(x = runif(n)*2-1, y = runif(n)*2-1)
df = df %>% mutate(r = x^2+y^2) %>%
mutate(incirc = ifelse(x^2+y^2 <= 1, 1, 0)) %>%
mutate(perc_inside = cummean(incirc)) %>%
mutate(pi_est = perc_inside*4) %>%
mutate(err = pi-pi_est) %>%
mutate(abs_err = abs(err))
return(df)
}

Test this out:

test = mc_pi(10^6)
tail(test$pi_est) ## [1] 3.140900 3.140901 3.140901 3.140902 3.140903 3.140904 Graph our error: test %>% slice(seq(1,length(test$y),1000)) %>%
ggplot() + geom_point(aes(x=1:length(x), y=abs_err), size = 0.1) + scale_y_log10() + xlab("Iteration") + ylab("Log error")

## Kaggle example

Label factors:

hr = hr %>% mutate(number_project = ordered(number_project)) %>%
mutate(time_spend_company = ordered(time_spend_company)) %>%
mutate(work_accident = factor(Work_accident)) %>%
mutate(left = factor(left)) %>%
mutate(sales = factor(sales)) %>%
mutate(salary = factor(salary))

Drop the extra column with inconsistent naming:

hr = hr %>% select(-Work_accident)

Letâ€™s shuffle the dataframe:

sh_hr = slice(hr, sample(nrow(hr), replace = FALSE))
## # A tibble: 6 x 3
##   satisfaction_level last_evaluation number_project
##                <dbl>           <dbl>          <ord>
## 1               0.38            0.53              2
## 2               0.80            0.86              5
## 3               0.11            0.88              7
## 4               0.72            0.87              5
## 5               0.37            0.52              2
## 6               0.41            0.50              2
## # A tibble: 6 x 3
##   satisfaction_level last_evaluation number_project
##                <dbl>           <dbl>          <ord>
## 1               0.36            0.57              2
## 2               0.09            0.79              6
## 3               0.65            0.96              2
## 4               0.56            0.79              4
## 5               0.99            0.73              3
## 6               0.78            0.89              4

Split the dataset:

hr_train = slice(sh_hr,1:10000)
hr_test = slice(sh_hr, seq(10001, nrow(sh_hr)))

Examine some summary statistics:

summary(hr_train)
##  satisfaction_level last_evaluation  number_project average_montly_hours
##  Min.   :0.0900     Min.   :0.3600   2:1547         Min.   : 96.0
##  1st Qu.:0.4400     1st Qu.:0.5600   3:2708         1st Qu.:156.0
##  Median :0.6500     Median :0.7200   4:2911         Median :201.0
##  Mean   :0.6147     Mean   :0.7174   5:1859         Mean   :201.7
##  3rd Qu.:0.8200     3rd Qu.:0.8700   6: 806         3rd Qu.:246.0
##  Max.   :1.0000     Max.   :1.0000   7: 169         Max.   :310.0
##
##  time_spend_company left     promotion_last_5years         sales
##  3      :4304       0:7627   Min.   :0.0000        sales      :2740
##  2      :2116       1:2373   1st Qu.:0.0000        technical  :1856
##  4      :1725                Median :0.0000        support    :1489
##  5      : 983                Mean   :0.0222        IT         : 829
##  6      : 478                3rd Qu.:0.0000        marketing  : 588
##  10     : 147                Max.   :1.0000        product_mng: 577
##  (Other): 247                                      (Other)    :1921
##     salary     work_accident
##  high  : 824   0:8574
##  low   :4845   1:1426
##  medium:4331
##
##
##
##
library(GGally)
prs = ggpairs(hr_train)
prs