# Probability and an Introduction to Jupyter, Python and Pandas¶

## Data Science School, Nyeri, Kenya¶

### 15th June 2015 Neil Lawrence¶

Welcome to the Data Science School in Nyeri, Kenya. In this school we will introduce the basic concepts of machine learning and data science. In particular we will look at tools and techniques that describe how to model. An integrated part of that is how we approach data with the computer. We are choosing to do that with the tool you see in front of you: the Jupyter Notebook.

The notebook provides us with a way of interacting with the data that allows us to give the computer instructions and explore the nature of a data set. It is different to normal coding, but it is related. In this course you will, through intensive practical sessions and labs, develop your understanding of the interaction between data and computers.

The first thing we are going to do is ask you to forget a bit about what you think about normal programming, or 'classical software engineering'. Classical software engineering demands a large amount of design and testing. In data analysis, testing remains very important, but the design is often evolving. The design evolves through a process known as exploratory data analysis. You will learn some of the techniques of exploratory data analysis in this course.

A particular difference between classical software engineering and data analysis is the way in which programs are run. Classically we spend a deal of time working with a text editor, writing code. Compilations are done on a regular basis and aspects of the code are tested (perhaps with unit tests).

Data analysis is more like coding in a debugger. In a debugger (particularly a visual debugger) you interact with the data stored in the memory of the computer to try and understand what is happening in the computer, you need to understand exactly what your bug is: you often have a fixed idea of what the program is trying to do, you are just struggling to find out why it isn't doing it.

Naturally, debugging is an important part of data analysis also, but in some sense it can be seen as its entire premise. You load in a data set into a computer that you don't understand, your entire objective is to understand the data. This is best done by interrogating the data to visualise it or summarize it, just like in a power visual debugger. However, for data science the requirements for visualization and summarization are far greater than in a regular program. When the data is well understood, the actual number of lines of your program may well be very few (particularly if you disregard commands that load in the data and commands which plot your results). If a powerful data science library is available, you may be able to summarize your code with just two or three lines, but the amount of intellectual energy that is expended on writing those three lines is far greater than in standard code.

In the first lecture we will think a little about 'how we got here' in terms of computer science. In the lecture itself, this will be done by taking a subjective perspective, that of my own 'data autobiography'.

## Assumed Knowledge¶

### Linear Algebra, Probability and Differential Calculus¶

We will be assuming that you have good background in maths. In particular we will be making use of linear algebra (matrix operations including inverse, inner products, determinant etc), probability (sum rule of probability, product rule of probability), and the calculus of differentiation (and integration!). A new concept for the course is multivariate differentiation and integration. This combines linear algebra and differential calculus. These techniques are vital in understanding probability distributions over high dimensional distributions.

### Choice of Language¶

In this course we will be using Python for our programming language. A prerequisite of attending this course is that you have learnt at least one programming language in the past. It is not our objective to teach you python. At Level 4 and Masters we expect our students to be able pick up a language as they go. If you have not experienced python before it may be worth your while spending some time understanding the language. There are resources available for you to do this here that are based on the standard console. An introduction to the Jupyter notebook (formerly known as the IPython notebook) is available here.

## Choice of Environment¶

We are working in the Jupyter notebook (formerly known as the IPython notebook). It provides an environment for interacting with data in a natural way which is reproducible. We will be learning how to make use of the notebook throughout the course. The notebook allows us to combine code with descriptions, interactive visualizations, plots etc. In fact it allows us to do many of the things we need for data science. Notebooks can also be easily shared through the internet for ease of communication of ideas. The box this text is written in is a markdown box. Below we have a code box.

In [3]:
print "This is the Jupyter notebook"
print "It provides a platform for:"
words = ['Open', 'Data', 'Science']
from random import shuffle
for i in range(3):
shuffle(words)
print ' '.join(words)

This is the Jupyter notebook
It provides a platform for:
Data Open Science
Science Open Data
Science Open Data


Have a play with the code in the above box. Think about the following questions: what is the difference between CTRL-enter and SHIFT-enter in running the code? What does the command shuffle do? Can you find out by typing shuffle? in a code box? Once you've had a play with the code we can load in some data using the pandas library for data analysis.

## Movie Body Count Example¶

There is a crisis in the movie industry, deaths are occurring on a massive scale. In every feature film the body count is tolling up. But what is the cause of all these deaths? Let's try and investigate.

For our first example of data science, we take inspiration from work by researchers at NJIT. They researchers were comparing the qualities of Python with R (my brief thoughts on the subject are available in a Google+ post here: https://plus.google.com/116220678599902155344/posts/5iKyqcrNN68). They put together a data base of results from the the "Internet Movie Database" and the Movie Body Count website which will allow us to do some preliminary investigation.

We will make use of data that has already been 'scraped' from the Movie Body Count website. Code and the data is available at a github repository. Git is a version control system and github is a website that hosts code that can be accessed through git. By sharing the code publicly through github, the authors are licensing the code publicly and allowing you to access and edit it. As well as accessing the code via github you can also download the zip file. But let's do that in python

In [5]:
import pods
import pandas as pd
data = pods.datasets.movie_body_count()
film_deaths = data['Y']

---------------------------------------------------------------------------
ImportError                               Traceback (most recent call last)
<ipython-input-5-54d660a12817> in <module>()
----> 1 import tffutfjytfvjtuyfgkutc
2 import pandas as pd
3 data = pods.datasets.movie_body_count()
4 film_deaths = data['Y']

ImportError: No module named tffutfjytfvjtuyfgkutc

Once the data is downloaded we can unzip it into the same directory where we are running the lab class.

Once it is loaded in the data can be summarized using the describe method in pandas.

In [2]:
film_deaths.describe()
film_deaths

Out[2]:
Film Year Body_Count MPAA_Rating Genre Director Actors Length_Minutes IMDB_Rating
0 24 Hour Party People 2002 7 R Biography|Comedy|Drama|Music Michael Winterbottom Steve Coogan|John Thomson|Paul Popplewell|Lenn... 117 7.4
1 3:10 to Yuma 2007 45 R Adventure|Crime|Drama|Western James Mangold Russell Crowe|Christian Bale|Logan Lerman|Dall... 122 7.8
2 300 2006 0 R Action|Fantasy|History|War Zack Snyder Gerard Butler|Lena Headey|Dominic West|David W... 117 7.8
3 8MM 1999 7 R Crime|Mystery|Thriller Joel Schumacher Nicolas Cage|Joaquin Phoenix|James Gandolfini|... 123 6.4
4 The Abominable Dr. Phibes 1971 10 PG-13 Fantasy|Horror Robert Fuest Vincent Price|Joseph Cotten|Hugh Griffith|Terr... 94 7.2
5 Above the Law 1988 18 NaN Action|Crime|Drama|Thriller Andrew Davis Steven Seagal|Pam Grier|Henry Silva|Ron Dean|D... 99 5.9
6 Action Jackson 1988 17 NaN Action|Comedy|Crime|Thriller Craig R. Baxley Carl Weathers|Craig T. Nelson|Vanity|Sharon St... 96 5.0
7 The Adventures of Ford Fairlane 1990 7 NaN Action|Adventure|Comedy|Music Renny Harlin Andrew Dice Clay|Wayne Newton|Priscilla Presle... 104 6.2
8 ?on Flux 2005 58 PG-13 Action|Sci-Fi Karyn Kusama Charlize Theron|Marton Csokas|Jonny Lee Miller... 93 5.5
9 Akira 1988 119 R Animation|Action|Adventure|Drama|Horror|Myster... Katsuhiro Ohtomo Mitsuo Iwata|Nozomu Sasaki|Mami Koyama|Tesshô ... 124 8.1
10 Ali G Indahouse 2002 11 R Comedy Mark Mylod Sacha Baron Cohen|Emilio Rivera|Gina La Piana|... 85 6.2
11 Alien 1979 9 R Horror|Sci-Fi Ridley Scott Tom Skerritt|Sigourney Weaver|Veronica Cartwri... 117 8.5
12 AVPR: Aliens vs Predator - Requiem 2007 115 R Action|Horror|Sci-Fi|Thriller Colin Strause|Greg Strause Steven Pasquale|Reiko Aylesworth|John Ortiz|Jo... 94 4.7
13 Alpha Dog 2006 3 R Biography|Crime|Drama Nick Cassavetes Bruce Willis|Matthew Barry|Emile Hirsch|Fernan... 122 6.9
14 Altered States 1980 5 NaN Drama|Fantasy|Horror|Sci-Fi|Thriller Ken Russell William Hurt|Blair Brown|Bob Balaban|Charles H... 102 6.9
15 American Gangster 2007 15 R Biography|Crime|Drama Ridley Scott Denzel Washington|Russell Crowe|Chiwetel Ejiof... 157 7.8
16 American Ninja 1985 114 NaN Action|Adventure|Romance|Sport Sam Firstenberg Michael Dudikoff|Steve James|Judie Aronson|Gui... 95 5.2
17 American Pop 1981 61 NaN Animation|Drama|Music Ralph Bakshi Ron Thompson|Mews Small|Jerry Holland|Lisa Jan... 96 7.1
18 American Psycho 2000 18 R Crime|Drama Mary Harron Christian Bale|Justin Theroux|Josh Lucas|Bill ... 102 7.6
19 American Yakuza 1993 53 NaN Action|Crime|Drama|Thriller Frank A. Cappello Viggo Mortensen|Ryo Ishibashi|Michael Nouri|Fr... 96 5.7
20 Another Day in Paradise 1998 9 R Crime|Drama|Thriller Larry Clark James Woods|Melanie Griffith|Vincent Kartheise... 101 6.5
21 Apocalypse Now 1979 62 R Drama|War Francis Ford Coppola Marlon Brando|Martin Sheen|Robert Duvall|Frede... 153 8.5
22 Apocalypto 2006 114 R Action|Adventure|Drama|Thriller Mel Gibson Rudy Youngblood|Dalia Hernández|Jonathan Brewe... 139 7.8
23 Appaloosa 2008 10 R Crime|Drama|Western Ed Harris Robert Jauregui|Jeremy Irons|Timothy V. Murphy... 115 6.8
24 Armageddon 1998 16 PG-13 Action|Adventure|Sci-Fi|Thriller Michael Bay Bruce Willis|Billy Bob Thornton|Ben Affleck|Li... 151 6.6
25 Army of Darkness 1992 107 R Comedy|Fantasy|Horror Sam Raimi Bruce Campbell|Embeth Davidtz|Marcus Gilbert|I... 81 7.6
26 Assault on Precinct 13 1976 39 NaN Action|Crime|Thriller John Carpenter Austin Stoker|Darwin Joston|Laurie Zimmer|Mart... 91 7.4
27 Assault on Precinct 13 2005 21 R Action|Drama|Crime|Thriller Jean-François Richet Ethan Hawke|Laurence Fishburne|Gabriel Byrne|M... 109 6.3
28 Atonement 2007 34 R Drama|Mystery|Romance|War Joe Wright Saoirse Ronan|Ailidh Mackay|Brenda Blethyn|Jul... 123 7.8
29 Austin Powers in Goldmember 2002 6 PG-13 Action|Comedy|Crime Jay Roach Mike Myers|Beyoncé Knowles|Seth Green|Michael ... 94 6.2
... ... ... ... ... ... ... ... ... ...
391 The Usual Suspects 1995 39 R Crime|Mystery|Thriller Bryan Singer Stephen Baldwin|Gabriel Byrne|Benicio Del Toro... 106 8.7
392 V for Vendetta 2005 59 R Action|Sci-Fi|Thriller James McTeigue Natalie Portman|Hugo Weaving|Stephen Rea|Steph... 132 8.2
393 Valkyrie 2008 18 PG-13 Drama|History|Thriller|War Bryan Singer Tom Cruise|Kenneth Branagh|Bill Nighy|Tom Wilk... 121 7.1
394 Vampires: The Turning 2005 53 R Action|Horror|Thriller Marty Weiss Colin Egglesfield|Stephanie Chao|Roger Yuan|Pa... 84 3.3
395 Versus 2000 127 R Action|Fantasy|Horror Ryûhei Kitamura Tak Sakaguchi|Hideo Sakaki|Chieko Misaka|Kenji... 119 6.6
396 Videodrome 1983 8 NaN Horror|Sci-Fi David Cronenberg James Woods|Sonja Smits|Deborah Harry|Peter Dv... 87 7.3
397 A View to a Kill 1985 37 NaN Action|Adventure|Crime|Thriller John Glen Roger Moore|Christopher Walken|Tanya Roberts|G... 131 6.3
398 Waist Deep 2006 6 R Action|Crime|Drama|Thriller Vondie Curtis-Hall Tyrese Gibson|Shawn Parr|Henry Hunter Hall|Joh... 97 5.9
399 Walk Hard: The Dewey Cox Story 2007 5 R Comedy|Drama|Music Jake Kasdan Nat Faxon|John C. Reilly|Tim Meadows|Conner Ra... 96 6.7
400 Walking Tall 2004 6 PG-13 Action|Crime Kevin Bray Michael Bowen|Johnny Knoxville|Dwayne Johnson|... 86 6.2
401 War 2007 97 R Action|Crime|Thriller Philip G. Atwell Jet Li|Jason Statham|John Lone|Devon Aoki|Luis... 103 6.2
402 War Inc. 2008 73 R Action|Comedy|Thriller Joshua Seftel John Cusack|Hilary Duff|Marisa Tomei|Joan Cusa... 107 5.7
403 War of the Worlds 2005 52 PG-13 Adventure|Sci-Fi|Thriller Steven Spielberg Tom Cruise|Dakota Fanning|Miranda Otto|Justin ... 116 6.5
404 Wasabi 2001 22 R Action|Drama|Comedy|Crime|Thriller Gérard Krawczyk Jean Reno|Ryôko Hirosue|Michel Muller|Carole B... 94 6.6
405 The Way of the Gun 2000 18 R Action|Crime|Drama|Thriller Christopher McQuarrie Ryan Phillippe|Benicio Del Toro|Juliette Lewis... 119 6.7
406 We Were Soldiers 2002 305 R Action|Drama|History|War Randall Wallace Mel Gibson|Madeleine Stowe|Greg Kinnear|Sam El... 138 7.1
407 Where Eagles Dare 1968 100 NaN Action|Adventure|War Brian G. Hutton Richard Burton|Clint Eastwood|Mary Ure|Patrick... 158 7.7
408 The Wild Bunch 1969 145 NaN Western Sam Peckinpah William Holden|Ernest Borgnine|Robert Ryan|Edm... 145 8.1
409 X-Men 2000 6 PG-13 Action|Adventure|Sci-Fi Bryan Singer Hugh Jackman|Patrick Stewart|Ian McKellen|Famk... 104 7.4
410 X2 2003 26 PG-13 Action|Adventure|Sci-Fi|Thriller Bryan Singer Patrick Stewart|Hugh Jackman|Ian McKellen|Hall... 133 7.5
411 X-Men: The Last Stand 2006 57 PG-13 Action|Adventure|Sci-Fi|Thriller Brett Ratner Hugh Jackman|Halle Berry|Ian McKellen|Patrick ... 104 6.8
412 xXx 2002 75 PG-13 Action|Thriller Rob L. Cohen Vin Diesel|Asia Argento|Marton Csokas|Samuel L... 124 5.8
413 xXx: State of the Union 2005 66 PG-13 Action|Crime|Adventure|Thriller Lee Tamahori Willem Dafoe|Samuel L. Jackson|Ice Cube|Scott ... 101 4.2
414 The Yakuza 1974 31 NaN Action|Crime|Drama|Thriller Sydney Pollack Robert Mitchum|Ken Takakura|Brian Keith|Herb E... 123 7.3
415 The Yards 2000 2 R Crime|Drama|Romance|Thriller James Gray Mark Wahlberg|Joaquin Phoenix|Charlize Theron|... 115 6.4
416 You Kill Me 2007 10 R Comedy|Crime|Romance|Thriller John Dahl Ben Kingsley|Téa Leoni|Luke Wilson|Dennis Fari... 93 6.6
417 You Only Live Twice 1967 91 NaN Action|Adventure|Crime|Thriller Lewis Gilbert Sean Connery|Akiko Wakabayashi|Mie Hama|Tetsur... 117 6.9
418 Zodiac 2007 3 R Crime|Drama|Mystery|Thriller David Fincher Jake Gyllenhaal|Mark Ruffalo|Anthony Edwards|R... 157 7.7
419 Zoolander 2001 4 PG-13 Comedy Ben Stiller Ben Stiller|Owen Wilson|Christine Taylor|Will ... 89 6.6
420 Zulu 1964 140 NaN Drama|History|War Cy Endfield Stanley Baker|Jack Hawkins|Ulla Jacobsson|Jame... 138 7.8

421 rows × 9 columns

In ipython and the jupyter notebook it is possible to see a list of all possible functions and attributes by typing the name of the object followed by . for example in the above case if we type film_deaths. it show the columns available (these are attributes in pandas dataframes) such as Body_Count, and also functions, such as .describe().

For functions we can also see the documentation about the function by following the name with a question mark. This will open a box with documentation at the bottom which can be closed with the x button.

In [4]:
film_deaths.describe?


The film deaths data is stored in an object known as a 'data frame'. Data frames come from the statistical family of programming languages based on S, the most widely used of which is R). The data frame gives us a convenient object for manipulating data. The describe method summarizes which columns there are in the data frame and gives us counts, means, standard deviations and percentiles for the values in those columns. To access a column directly we can write

In [5]:
film_deaths['Year']
#print film_deaths['Body_Count']

Out[5]:
0     2002
1     2007
2     2006
3     1999
4     1971
5     1988
6     1988
7     1990
8     2005
9     1988
10    2002
11    1979
12    2007
13    2006
14    1980
...
406    2002
407    1968
408    1969
409    2000
410    2003
411    2006
412    2002
413    2005
414    1974
415    2000
416    2007
417    1967
418    2007
419    2001
420    1964
Name: Year, Length: 421, dtype: int64

This shows the number of deaths per film across the years. We can plot the data as follows.

In [6]:
# this ensures the plot appears in the web browser
%matplotlib inline
import pylab as plt # this imports the plotting library in python

plt.plot(film_deaths['Year'], film_deaths['Body_Count'], 'rx')

Out[6]:
[<matplotlib.lines.Line2D at 0x10df91e50>]

You may be curious what the arguments we give to plt.plot are for, now is the perfect time to look at the documentation

In [7]:
plt.plot?


We immediately note that some films have a lot of deaths, which prevent us seeing the detail of the main body of films. First lets identify the films with the most deaths.

In [8]:
film_deaths[film_deaths['Body_Count']>200]

Out[8]:
Film Year Body_Count MPAA_Rating Genre Director Actors Length_Minutes IMDB_Rating
60 Dip huet gaai tau 1990 214 NaN Crime|Drama|Thriller John Woo Tony Leung Chiu Wai|Jacky Cheung|Waise Lee|Sim... 136 7.7
117 Equilibrium 2002 236 R Action|Drama|Sci-Fi|Thriller Kurt Wimmer Christian Bale|Dominic Purcell|Sean Bean|Chris... 107 7.6
154 Grindhouse 2007 310 R Action|Horror|Thriller Robert Rodriguez|Eli Roth|Quentin Tarantino|Ed... Kurt Russell|Zoë Bell|Rosario Dawson|Vanessa F... 191 7.7
159 Lat sau san taam 1992 307 R Action|Crime|Drama|Thriller John Woo Yun-Fat Chow|Tony Leung Chiu Wai|Teresa Mo|Phi... 128 8.0
193 Kingdom of Heaven 2005 610 R Action|Adventure|Drama|History|War Ridley Scott Martin Hancock|Michael Sheen|Nathalie Cox|Eriq... 144 7.2
206 The Last Samurai 2003 558 R Action|Drama|History|War Edward Zwick Ken Watanabe|Tom Cruise|William Atherton|Chad ... 154 7.7
222 The Lord of the Rings: The Two Towers 2002 468 PG-13 Action|Adventure|Fantasy Peter Jackson Bruce Allpress|Sean Astin|John Bach|Sala Baker... 179 8.8
223 The Lord of the Rings: The Return of the King 2003 836 PG-13 Action|Adventure|Fantasy Peter Jackson Noel Appleby|Alexandra Astin|Sean Astin|David ... 201 8.9
291 Rambo 2008 247 R Action|Thriller|War Sylvester Stallone Sylvester Stallone|Julie Benz|Matthew Marsden|... 92 7.1
317 Saving Private Ryan 1998 255 R Action|Drama|War Steven Spielberg Tom Hanks|Tom Sizemore|Edward Burns|Barry Pepp... 169 8.6
349 Starship Troopers 1997 256 R Action|Sci-Fi Paul Verhoeven Casper Van Dien|Dina Meyer|Denise Richards|Jak... 129 7.2
375 Titanic 1997 307 PG-13 Drama|Romance James Cameron Leonardo DiCaprio|Kate Winslet|Billy Zane|Kath... 194 7.7
382 Troy 2004 572 R Adventure|Drama Wolfgang Petersen Julian Glover|Brian Cox|Nathan Jones|Adoni Mar... 163 7.2
406 We Were Soldiers 2002 305 R Action|Drama|History|War Randall Wallace Mel Gibson|Madeleine Stowe|Greg Kinnear|Sam El... 138 7.1

Here we are using the command film_deaths['Kill_Count']>200 to index the films in the pandas data frame which have over 200 deaths. To sort them in order we can also use the sort command. The result of this command on its own is a data series of True and False values. However, when it is passed to the film_deaths data frame it returns a new data frame which contains only those values for which the data series is True. We can also sort the result. To sort the result by the values in the Kill_Count column in descending order we use the following command.

In [9]:
film_deaths[film_deaths['Body_Count']>200].sort(columns='Body_Count', ascending=False)

Out[9]:
Film Year Body_Count MPAA_Rating Genre Director Actors Length_Minutes IMDB_Rating
223 The Lord of the Rings: The Return of the King 2003 836 PG-13 Action|Adventure|Fantasy Peter Jackson Noel Appleby|Alexandra Astin|Sean Astin|David ... 201 8.9
193 Kingdom of Heaven 2005 610 R Action|Adventure|Drama|History|War Ridley Scott Martin Hancock|Michael Sheen|Nathalie Cox|Eriq... 144 7.2
382 Troy 2004 572 R Adventure|Drama Wolfgang Petersen Julian Glover|Brian Cox|Nathan Jones|Adoni Mar... 163 7.2
206 The Last Samurai 2003 558 R Action|Drama|History|War Edward Zwick Ken Watanabe|Tom Cruise|William Atherton|Chad ... 154 7.7
222 The Lord of the Rings: The Two Towers 2002 468 PG-13 Action|Adventure|Fantasy Peter Jackson Bruce Allpress|Sean Astin|John Bach|Sala Baker... 179 8.8
154 Grindhouse 2007 310 R Action|Horror|Thriller Robert Rodriguez|Eli Roth|Quentin Tarantino|Ed... Kurt Russell|Zoë Bell|Rosario Dawson|Vanessa F... 191 7.7
159 Lat sau san taam 1992 307 R Action|Crime|Drama|Thriller John Woo Yun-Fat Chow|Tony Leung Chiu Wai|Teresa Mo|Phi... 128 8.0
375 Titanic 1997 307 PG-13 Drama|Romance James Cameron Leonardo DiCaprio|Kate Winslet|Billy Zane|Kath... 194 7.7
406 We Were Soldiers 2002 305 R Action|Drama|History|War Randall Wallace Mel Gibson|Madeleine Stowe|Greg Kinnear|Sam El... 138 7.1
349 Starship Troopers 1997 256 R Action|Sci-Fi Paul Verhoeven Casper Van Dien|Dina Meyer|Denise Richards|Jak... 129 7.2
317 Saving Private Ryan 1998 255 R Action|Drama|War Steven Spielberg Tom Hanks|Tom Sizemore|Edward Burns|Barry Pepp... 169 8.6
291 Rambo 2008 247 R Action|Thriller|War Sylvester Stallone Sylvester Stallone|Julie Benz|Matthew Marsden|... 92 7.1
117 Equilibrium 2002 236 R Action|Drama|Sci-Fi|Thriller Kurt Wimmer Christian Bale|Dominic Purcell|Sean Bean|Chris... 107 7.6
60 Dip huet gaai tau 1990 214 NaN Crime|Drama|Thriller John Woo Tony Leung Chiu Wai|Jacky Cheung|Waise Lee|Sim... 136 7.7

We now see that the 'Lord of the Rings' is a large outlier with a very large number of kills. We can try and determine how much of an outlier by histograming the data.

### Plotting the Data¶

In [10]:
film_deaths['Body_Count'].hist(bins=20) # histogram the data with 20 bins.
plt.title('Histogram of Film Kill Count')

Out[10]:
<matplotlib.text.Text at 0x10e2622d0>

We could try and remove these outliers, but another approach would be plot the logarithm of the counts against the year.

In [11]:
plt.plot(film_deaths['Year'], film_deaths['Body_Count'], 'rx')
ax = plt.gca() # obtain a handle to the current axis
ax.set_yscale('log') # use a logarithmic death scale
# give the plot some titles and labels
plt.title('Film Deaths against Year')
plt.ylabel('deaths')
plt.xlabel('year')

Out[11]:
<matplotlib.text.Text at 0x10e2bfa50>

Note a few things. We are interacting with our data. In particular, we are replotting the data according to what we have learned so far. We are using the progamming language as a scripting language to give the computer one command or another, and then the next command we enter is dependent on the result of the previous. This is a very different paradigm to classical software engineering. In classical software engineering we normally write many lines of code (entire object classes or functions) before compiling the code and running it. Our approach is more similar to the approach we take whilst debugging. Historically, researchers interacted with data using a console. A command line window which allowed command entry. The notebook format we are using is slightly different. Each of the code entry boxes acts like a separate console window. We can move up and down the notebook and run each part in a different order. The state of the program is always as we left it after running the previous part.

## Probabilities¶

We are now going to do some simple review of probabilities and use this review to explore some aspects of our data.

A probability distribution expresses uncertainty about the outcome of an event. We often encode this uncertainty in a variable. So if we are considering the outcome of an event, $Y$, to be a coin toss, then we might consider $Y=1$ to be heads and $Y=0$ to be tails. We represent the probability of a given outcome with the notation: $$P(Y=1) = 0.5$$ The first rule of probability is that the probability must normalize. The sum of the probability of all events must equal 1. So if the probability of heads ($Y=1$) is 0.5, then the probability of tails (the only other possible outcome) is given by $$P(Y=0) = 1-P(Y=1) = 0.5$$

Probabilities are often defined as the limit of the ratio between the number of positive outcomes (e.g. heads) given the number of trials. If the number of positive outcomes for event $y$ is denoted by $n$ and the number of trials is denoted by $N$ then this gives the ratio $$P(Y=y) = \lim_{N\rightarrow \infty}\frac{n_y}{N}.$$ In practice we never get to observe an event infinite times, so rather than considering this we often use the following estimate $$P(Y=y) \approx \frac{n_y}{N}.$$ Let's use this rule to compute the approximate probability that a film from the movie body count website has over 40 deaths.

In [12]:
deaths = (film_deaths.Body_Count>40).sum()  # number of positive outcomes (in sum True counts as 1, False counts as 0)
total_films = film_deaths.Body_Count.count()

prob_death = float(deaths)/float(total_films)
print "Probability of deaths being greather than 40 is:", prob_death

Probability of deaths being greather than 40 is: 0.377672209026


# Conditioning¶

When predicting whether a coin turns up head or tails, we might think that this event is independent of the year or time of day. If we include an observation such as time, then in a probability this is known as condtioning. We use this notation, $P(Y=y|T=t)$, to condition the outcome on a second variable (in this case time). Or, often, for a shorthand we use $P(y|t)$ to represent this distribution (the $Y=$ and $T=$ being implicit). Because we don't believe a coin toss depends on time then we might write that $$P(y|t) = p(y).$$ However, we might believe that the number of deaths is dependent on the year. For this we can try estimating $P(Y>40 | T=2000)$ and compare the result, for example to $P(Y>40|2002)$ using our empirical estimate of the probability.

In [13]:
for year in [2000, 2002]:
deaths = (film_deaths.Body_Count[film_deaths.Year==year]>40).sum()
total_films = (film_deaths.Year==year).sum()

prob_death = float(deaths)/float(total_films)
print "Probability of deaths being greather than 40 in year", year, "is:", prob_death

Probability of deaths being greather than 40 in year 2000 is: 0.166666666667
Probability of deaths being greather than 40 in year 2002 is: 0.407407407407


### Rules of Probability¶

We've now introduced conditioning and independence to the notion of probability and computed some conditional probabilities on a practical example The scatter plot of deaths vs year that we created above can be seen as a joint probability distribution. We represent a joint probability using the notation $P(Y=y, T=t)$ or $P(y, t)$ for short. Computing a joint probability is equivalent to answering the simultaneous questions, what's the probability that the number of deaths was over 40 and the year was 2002? Or any other question that may occur to us. Again we can easily use pandas to ask such questions.

In [14]:
year = 2000
deaths = (film_deaths.Body_Count[film_deaths.Year==year]>40).sum()
total_films = film_deaths.Body_Count.count() # this is total number of films
prob_death = float(deaths)/float(total_films)
print "Probability of deaths being greather than 40 and year being", year, "is:", prob_death

Probability of deaths being greather than 40 and year being 2000 is: 0.00712589073634


### The Product Rule¶

This number is the joint probability, $P(Y, T)$ which is much smaller than the conditional probability. The number can never be bigger than the conditional probabililty because it is computed using the product rule. $$p(Y=y, T=t) = p(Y=y|T=t)p(T=t)$$ and $$p(T=t)$$ is a probability distribution, which is equal or less than 1, ensuring the joint distribution is typically smaller than the conditional distribution.

The product rule is a fundamental rule of probability, and you must remember it! It gives the relationship between the two questions: 1) What's the probability that a film was made in 2002 and has over 40 deaths? and 2) What's the probability that a film has over 40 deaths given that it was made in 2002?

In our shorter notation we can write the product rule as $$p(y, t) = p(y|t)p(t)$$ We can see the relation working in practice for our data above by computing the different values for $t=2000$.

In [15]:
p_t = float((film_deaths.Year==2002).sum())/float(film_deaths.Body_Count.count())
p_y_given_t = float((film_deaths.Body_Count[film_deaths.Year==2002]>40).sum())/float((film_deaths.Year==2002).sum())
p_y_and_t = float((film_deaths.Body_Count[film_deaths.Year==2002]>40).sum())/float(film_deaths.Body_Count.count())

print "P(t) is", p_t
print "P(y|t) is", p_y_given_t
print "P(y,t) is", p_y_and_t

P(t) is 0.0641330166271
P(y|t) is 0.407407407407
P(y,t) is 0.0261282660333


### The Sum Rule¶

The other fundamental rule of probability is the sum rule this tells us how to get a marginal distribution from the joint distribution. Simply put it says that we need to sum across the value we'd like to remove. $$P(Y=y) = \sum_{t} P(Y=y, T=t)$$ Or in our shortened notation $$P(y) = \sum_{t} P(y, t)$$

## Bayes' Rule¶

Bayes rule is a very simple rule, it's hardly worth the name of a rule at all. It follows directly from the product rule of probability. Because $P(y, t) = P(y|t)P(t)$ and by symmetry $P(y,t)=P(t,y)=P(t|y)P(y)$ then by equating these two equations and dividing through by $P(y)$ we have $$P(t|y) = \frac{P(y|t)P(t)}{P(y)}$$ which is known as Bayes' rule (or Bayes's rule, it depends how you choose to pronounce it). It's not difficult to derive, and its importance is more to do with the semantic operation that it enables. Each of these probability distributions represents the answer to a question we have about the world. Bayes rule (via the product rule) tells us how to invert the probability.

## More Fun on the Python Data Farm¶

If you want to explore more of the things you can do with movies and python you might be interested in the imdbpy python library.

You can try installing it using easy_install as follows.

In [ ]:
!easy_install -U IMDbPY


If this doesn't work on your machine, try following instructions on (http://imdbpy.sourceforge.net/)

Once you've installed imdbpy you can test it works with the following script, which should list movies with the word 'python' in their title. To run the code in the following box, simply click the box and press SHIFT-enter or CTRL-enter. Then you can try running the code below.

In [ ]:
from imdb import IMDb
ia = IMDb()

for movie in ia.search_movie('python'):
print movie

In [ ]:
from IPython.display import YouTubeVideo