1. Math 285 // Professor B L Lee
Mini Project
Death Proof
Austin Powell, Francisco J Perez-Leon, Rongdan Liu Wenjing Tang
Abstract
We studied the problem of being able to predict the number of deaths per hour in Quentin Tarantino. In our
prediction, we wanted to incorporate our prior belief on deaths/hr based on expert belief along with actual death
counts for 8 of his movies into our calculation. Bayesian analysis allowed us to do this. We found that in a two
hour movie we believe that there will be 60 deaths based on our conclusion that the most likely number of deaths
per hour is 30.
Contents
Introduction 1
1 Data Characteristics 1
2 Model Selection and Interpretation 1
2.1 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2.2 Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.3 Prior Hyper-Parameters . . . . . . . . . . . . . . . . . . 2
3 Methods 2
4 Results and Discussion 3
References 3
Introduction
Figure 1. Prior Information
Orientation It is relatively well-known that QT movies have
a high death rate. The intent of this project is to make a pre-
dictive statement about the rate of deaths for a future Quentin
Tarantino (QT) movies.
Key Aspects Our approach was to use Bayesian methods
in the prediction of death rates based on prior expert belief
and given data about deaths in all QT movies.
Plan We will start with our prior belief based on expert
opinion and then update this belief with actual death count
data from QT movies. Our intent was to use our updated
model to then make an statement about our updated belief
in the death count for his next movie. In our model of this
problem, we chose as our ”expert” opinion Austin since he
had the most experience and strength of knowledge on deaths
in Quentin Tarantino’s (QT) movies.
1. Data Characteristics
The data that we used to form our likelihood was constructed
by counting the total number of deaths per movie produced
by Quentin Tarantino as well as the total exposure time per
movie and in total run time of all movies. Because of the
lack of knowledge about the producer, the information used
to find the hyper-parameters of our prior distribution came in
the form of an illustration provided in the assignment.
Movie Body Count Hours Kill Rate
Jackie Brown 4 2.67 1.50
Death Proof 6 2.12 2.83
Pulp Fiction 7 2.80 2.50
Reservoir Dogs 11 1.65 6.67
Kill Bill Vol. 2 13 2.3 5.65
Kill Bill Vol. 1 62 1.86 33.16
Django Unchained 64 3 21.34
Inglorious Bastards 396 2.51 157.77
TOTALS 563 18.91
Table 1. The data used was comprised of counts of the total
number of deaths in each movie produced by Quentin
Tarantino.
2. Model Selection and Interpretation
2.1 Likelihood
Since we expected the data to come in the form of counts
per movie, we thought it would be best to choose the Poisson
distribution as our Sampling Distribution. Then we start to
check its appropriation. For Poisson model: the rate at which
the event occurs is a constant; the counts among all small
intervals of exposure are exchangeable. We assume the kill
1

2. Death Proof — 2/3
rate parameter to be constant which we use in our distribu-
tion. The order of movies shows no difference to us, so it is
exchangeable.
2.2 Prior
We want the prior to have little effect on our posterior which
is desirable due to our lack of knowledge about Quentin
Tarantino. Our choice of prior belief about the rate of deaths
per hour began by drawing what we thought would be an ideal
shape for our prior belief about the death rate which looked
somewhat positively skewed. We narrowed this down to the
Gamma distribution which also happens to be the conjugate
prior of our sampling distribution and conjugate priors have
less effect on the posterior.
2.3 Prior Hyper-Parameters
Our conclusion of the hyper-parameters of our prior belief
about the death rate per hour was based on our observation
of the picture provided and Austin’s opinion. We concluded
that the kill rate per hour should be almost certainly between
10 and as many as 60 deaths per hour with probability of
.95 with equal tails. The exact hyper-parameters were calcu-
lated via parameter solver provided by MD Anderson[3]. Our
prior belief turned out to follow a Gamma distribution with
α = 5.25, β = 0.178 The Following is a Probability Distri-
bution Function of our prior belief with the aforementioned
parameters:
Calculated parameters based on expert opinion based on
Pr(X < 10) = 0.025 & Pr(X < 60) = 0.975
Shape 5.25799
Rate= 1
Scale
1
5.6531 = 0.1769
Figure 2. Prior Distribution
3. Methods
After selecting our prior and sampling distribution, we con-
cluded in the following manner that our posterior distribution
will take the following form:
p(xi|λ) ∝
λxi e−λ
xi!
⇒
p(model|data) ∝
n
∏
i=1
p(xi|λ)p(λ)
p(model|data) ∝ λ∑xi e−nλ
∗
βα
Γ(α)
λα−1
e−βλ
∝ λ∑xi+α−1
e−λ(n+β)
p(model|data) =
λ∑xi+α−1e−λ(n+β)(n+β)∑xi+α
Γ(∑xi +α)
= Gamma(∑xi +α,n+β)
= Gamma(α∗
,β∗
)
xi: total number of deaths in a movie, or the body count.
λ: The kill rate, or the average number of characters killed
per hour.
n: Total exposure time in all movies(total movie time)
α: Our prior shape parameter
β: Our prior rate parameter
Figure 3. Posterior vs Prior
Figure 4. Prior Information
Our posterior belief about the kill rate per hour given the
data and our prior belief follows a Gamma distribution and
has updated shape parameter = 568.25 and rate parameter =
19.098. Fig. 4 depicts our posterior versus prior beliefs about
the rate of deaths per hour.
4. Results and Discussion
Assumption 1) The kill rate is a constant in all Quentin
Tarantino movies. 2) Quentin Tarantino’s movie are exchange-
able to us.
Results The mode of our posterior belief about the death
rate per hour λ based on our data and prior belief is 29.70.
This is almost 30 deaths per hour in a movie produced by

3. Death Proof — 3/3
Quentin Tarantino. Based on this result, if Quentin Tarantino
will produce a 2 hour movie, we believe 60 people are most
likely to get killed. Also, note that our variance or uncertainty
on our belief about the updated death rate has been reduced
greatly after our analysis. This is most likely due to the fact
that our prior was very naive and uncertain to begin with. A
95% Highest Posterior Density (HPD) Credible Interval is
(27.3, 32.2) for our updated belief about the death rates given
our data and prior belief. Also ,notice that the median for
the posterior shifted to the right; we believe that is due to the
highly skewed counts especially the movie with 300+ deaths
in it.
References
[1] Bayesian Data Analysis, Andrew Gelman, John
B. Carlin, Hal S. Stern, David B. Dunson, Aki Ve-
htari, Donald B. Rubin, Taylor & Francis Group
2014
[2] BAYESIAN ANALYSIS: Week 3
and 4 - Poisson Distribution
http://ramlegacy.marinebiodiversity.ca/courses/church-
of-bayes/notes/week3-notes.pdf
[3] MD Anderson Cen-
ter,https://biostatistics.mdanderson.org