Math 285 // Professor B L LeeMini Project

Death ProofAustin Powell, Francisco J Perez-Leon, Rongdan Liu Wenjing Tang

AbstractWe studied the problem of being able to predict the number of deaths per hour in Quentin Tarantino. In ourprediction, we wanted to incorporate our prior belief on deaths/hr based on expert belief along with actual deathcounts for 8 of his movies into our calculation. Bayesian analysis allowed us to do this. We found that in a twohour movie we believe that there will be 60 deaths based on our conclusion that the most likely number of deathsper hour is 30.

Contents

Introduction 1

1 Data Characteristics 1

2 Model Selection and Interpretation 12.1 Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 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 havea high death rate. The intent of this project is to make a pre-dictive statement about the rate of deaths for a future QuentinTarantino (QT) movies.

Key Aspects Our approach was to use Bayesian methodsin the prediction of death rates based on prior expert beliefand given data about deaths in all QT movies.

Plan We will start with our prior belief based on expertopinion and then update this belief with actual death countdata from QT movies. Our intent was to use our updatedmodel to then make an statement about our updated beliefin the death count for his next movie. In our model of this

problem, we chose as our ”expert” opinion Austin since hehad the most experience and strength of knowledge on deathsin Quentin Tarantino’s (QT) movies.

1. Data CharacteristicsThe data that we used to form our likelihood was constructedby counting the total number of deaths per movie producedby Quentin Tarantino as well as the total exposure time permovie and in total run time of all movies. Because of thelack of knowledge about the producer, the information usedto find the hyper-parameters of our prior distribution came inthe form of an illustration provided in the assignment.

Movie Body Count Hours Kill Rate

Jackie Brown 4 2.67 1.50Death Proof 6 2.12 2.83Pulp Fiction 7 2.80 2.50Reservoir Dogs 11 1.65 6.67Kill Bill Vol. 2 13 2.3 5.65Kill Bill Vol. 1 62 1.86 33.16Django Unchained 64 3 21.34Inglorious Bastards 396 2.51 157.77

TOTALS 563 18.91

Table 1. The data used was comprised of counts of the totalnumber of deaths in each movie produced by QuentinTarantino.

2. Model Selection and Interpretation

2.1 LikelihoodSince we expected the data to come in the form of countsper movie, we thought it would be best to choose the Poissondistribution as our Sampling Distribution. Then we start tocheck its appropriation. For Poisson model: the rate at whichthe event occurs is a constant; the counts among all smallintervals of exposure are exchangeable. We assume the kill

1

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 isexchangeable.

2.2 PriorWe want the prior to have little effect on our posterior whichis desirable due to our lack of knowledge about QuentinTarantino. Our choice of prior belief about the rate of deathsper hour began by drawing what we thought would be an idealshape for our prior belief about the death rate which lookedsomewhat positively skewed. We narrowed this down to theGamma distribution which also happens to be the conjugateprior of our sampling distribution and conjugate priors haveless effect on the posterior.

2.3 Prior Hyper-ParametersOur conclusion of the hyper-parameters of our prior beliefabout the death rate per hour was based on our observationof the picture provided and Austin’s opinion. We concludedthat the kill rate per hour should be almost certainly between10 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]. Ourprior belief turned out to follow a Gamma distribution withα = 5.25, β = 0.178 The Following is a Probability Distri-bution Function of our prior belief with the aforementionedparameters:

Calculated parameters based on expert opinion based onPr(X < 10) = 0.025 & Pr(X < 60) = 0.975

Shape 5.25799Rate= 1

Scale1

5.6531 = 0.1769

Figure 2. Prior Distribution

3. MethodsAfter selecting our prior and sampling distribution, we con-cluded in the following manner that our posterior distributionwill take the following form:

p(xi|λ ) ∝λ xie−λ

xi!⇒

p(model|data) ∝

n

∏i=1

p(xi|λ )p(λ )

p(model|data) ∝

(λ ∑xie−nλ

)∗(

β α

Γ(α)λ

α−1e−βλ

)∝ λ ∑xi+α−1e−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 parameterβ : Our prior rate parameter

Figure 3. Posterior vs Prior

Figure 4. Prior Information

Our posterior belief about the kill rate per hour given thedata and our prior belief follows a Gamma distribution andhas updated shape parameter = 568.25 and rate parameter =19.098. Fig. 4 depicts our posterior versus prior beliefs aboutthe rate of deaths per hour.

4. Results and DiscussionAssumption 1) The kill rate is a constant in all QuentinTarantino movies. 2) Quentin Tarantino’s movie are exchange-able to us.

Results The mode of our posterior belief about the deathrate 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

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Quentin Tarantino. Based on this result, if Quentin Tarantinowill produce a 2 hour movie, we believe 60 people are mostlikely to get killed. Also, note that our variance or uncertaintyon our belief about the updated death rate has been reducedgreatly after our analysis. This is most likely due to the factthat our prior was very naive and uncertain to begin with. A95% Highest Posterior Density (HPD) Credible Interval is(27.3, 32.2) for our updated belief about the death rates givenour data and prior belief. Also ,notice that the median forthe posterior shifted to the right; we believe that is due to thehighly skewed counts especially the movie with 300+ deathsin 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 Group2014

[2] BAYESIAN ANALYSIS: Week 3and 4 - Poisson Distributionhttp://ramlegacy.marinebiodiversity.ca/courses/church-of-bayes/notes/week3-notes.pdf

[3] MD Anderson Cen-ter,https://biostatistics.mdanderson.org