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notebook.tex
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notebook.tex
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% Default to the notebook output style
% Inherit from the specified cell style.
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
% Nicer default font (+ math font) than Computer Modern for most use cases
\usepackage{mathpazo}
% Basic figure setup, for now with no caption control since it's done
% automatically by Pandoc (which extracts  syntax from Markdown).
\usepackage{graphicx}
% We will generate all images so they have a width \maxwidth. This means
% that they will get their normal width if they fit onto the page, but
% are scaled down if they would overflow the margins.
\makeatletter
\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth
\else\Gin@nat@width\fi}
\makeatother
\let\Oldincludegraphics\includegraphics
% Set max figure width to be 80% of text width, for now hardcoded.
\renewcommand{\includegraphics}[1]{\Oldincludegraphics[width=.8\maxwidth]{#1}}
% Ensure that by default, figures have no caption (until we provide a
% proper Figure object with a Caption API and a way to capture that
% in the conversion process - todo).
\usepackage{caption}
\DeclareCaptionLabelFormat{nolabel}{}
\captionsetup{labelformat=nolabel}
\usepackage{adjustbox} % Used to constrain images to a maximum size
\usepackage{xcolor} % Allow colors to be defined
\usepackage{enumerate} % Needed for markdown enumerations to work
\usepackage{geometry} % Used to adjust the document margins
\usepackage{amsmath} % Equations
\usepackage{amssymb} % Equations
\usepackage{textcomp} % defines textquotesingle
% Hack from http://tex.stackexchange.com/a/47451/13684:
\AtBeginDocument{%
\def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code
}
\usepackage{upquote} % Upright quotes for verbatim code
\usepackage{eurosym} % defines \euro
\usepackage[mathletters]{ucs} % Extended unicode (utf-8) support
\usepackage[utf8x]{inputenc} % Allow utf-8 characters in the tex document
\usepackage{fancyvrb} % verbatim replacement that allows latex
\usepackage{grffile} % extends the file name processing of package graphics
% to support a larger range
% The hyperref package gives us a pdf with properly built
% internal navigation ('pdf bookmarks' for the table of contents,
% internal cross-reference links, web links for URLs, etc.)
\usepackage{hyperref}
\usepackage{longtable} % longtable support required by pandoc >1.10
\usepackage{booktabs} % table support for pandoc > 1.12.2
\usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment)
\usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout)
% normalem makes italics be italics, not underlines
% Colors for the hyperref package
\definecolor{urlcolor}{rgb}{0,.145,.698}
\definecolor{linkcolor}{rgb}{.71,0.21,0.01}
\definecolor{citecolor}{rgb}{.12,.54,.11}
% ANSI colors
\definecolor{ansi-black}{HTML}{3E424D}
\definecolor{ansi-black-intense}{HTML}{282C36}
\definecolor{ansi-red}{HTML}{E75C58}
\definecolor{ansi-red-intense}{HTML}{B22B31}
\definecolor{ansi-green}{HTML}{00A250}
\definecolor{ansi-green-intense}{HTML}{007427}
\definecolor{ansi-yellow}{HTML}{DDB62B}
\definecolor{ansi-yellow-intense}{HTML}{B27D12}
\definecolor{ansi-blue}{HTML}{208FFB}
\definecolor{ansi-blue-intense}{HTML}{0065CA}
\definecolor{ansi-magenta}{HTML}{D160C4}
\definecolor{ansi-magenta-intense}{HTML}{A03196}
\definecolor{ansi-cyan}{HTML}{60C6C8}
\definecolor{ansi-cyan-intense}{HTML}{258F8F}
\definecolor{ansi-white}{HTML}{C5C1B4}
\definecolor{ansi-white-intense}{HTML}{A1A6B2}
% commands and environments needed by pandoc snippets
% extracted from the output of `pandoc -s`
\providecommand{\tightlist}{%
\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
\DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}}
% Add ',fontsize=\small' for more characters per line
\newenvironment{Shaded}{}{}
\newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
\newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}}
\newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\FloatTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\CharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\StringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\CommentTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textit{{#1}}}}
\newcommand{\OtherTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{{#1}}}
\newcommand{\AlertTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
\newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.02,0.16,0.49}{{#1}}}
\newcommand{\RegionMarkerTok}[1]{{#1}}
\newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
\newcommand{\NormalTok}[1]{{#1}}
% Additional commands for more recent versions of Pandoc
\newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.53,0.00,0.00}{{#1}}}
\newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\VerbatimStringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\SpecialStringTok}[1]{\textcolor[rgb]{0.73,0.40,0.53}{{#1}}}
\newcommand{\ImportTok}[1]{{#1}}
\newcommand{\DocumentationTok}[1]{\textcolor[rgb]{0.73,0.13,0.13}{\textit{{#1}}}}
\newcommand{\AnnotationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\CommentVarTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\VariableTok}[1]{\textcolor[rgb]{0.10,0.09,0.49}{{#1}}}
\newcommand{\ControlFlowTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
\newcommand{\OperatorTok}[1]{\textcolor[rgb]{0.40,0.40,0.40}{{#1}}}
\newcommand{\BuiltInTok}[1]{{#1}}
\newcommand{\ExtensionTok}[1]{{#1}}
\newcommand{\PreprocessorTok}[1]{\textcolor[rgb]{0.74,0.48,0.00}{{#1}}}
\newcommand{\AttributeTok}[1]{\textcolor[rgb]{0.49,0.56,0.16}{{#1}}}
\newcommand{\InformationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\WarningTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
% Define a nice break command that doesn't care if a line doesn't already
% exist.
\def\br{\hspace*{\fill} \\* }
% Math Jax compatability definitions
\def\gt{>}
\def\lt{<}
% Document parameters
\title{1\_Introduction-to-Bayes}
% Pygments definitions
\makeatletter
\def\PY@reset{\let\PY@it=\relax \let\PY@bf=\relax%
\let\PY@ul=\relax \let\PY@tc=\relax%
\let\PY@bc=\relax \let\PY@ff=\relax}
\def\PY@tok#1{\csname PY@tok@#1\endcsname}
\def\PY@toks#1+{\ifx\relax#1\empty\else%
\PY@tok{#1}\expandafter\PY@toks\fi}
\def\PY@do#1{\PY@bc{\PY@tc{\PY@ul{%
\PY@it{\PY@bf{\PY@ff{#1}}}}}}}
\def\PY#1#2{\PY@reset\PY@toks#1+\relax+\PY@do{#2}}
\expandafter\def\csname PY@tok@w\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.73,0.73}{##1}}}
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\def\PYZbs{\char`\\}
\def\PYZus{\char`\_}
\def\PYZob{\char`\{}
\def\PYZcb{\char`\}}
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% Exact colors from NB
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\definecolor{outcolor}{rgb}{0.545, 0.0, 0.0}
% Prevent overflowing lines due to hard-to-break entities
\sloppy
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breaklinks=true, % so long urls are correctly broken across lines
colorlinks=true,
urlcolor=urlcolor,
linkcolor=linkcolor,
citecolor=citecolor,
}
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\begin{document}
\maketitle
Table of Contents{}
{{1~~}An Introduction to Bayesian Statistical Analysis}
{{2~~}What is Bayesian Statistical Analysis?}
{{3~~}What is Probability?}
{{3.1~~}1. Classical probability}
{{3.2~~}2. Frequentist probability}
{{3.3~~}3. Subjective probability}
{{4~~}Bayesian vs Frequentist Statistics: What's the difference?}
{{4.1~~}The Frequentist World View}
{{4.2~~}The Bayesian World View}
{{5~~}Bayes' Formula}
{{5.1~~}Example: Genetic probabilities}
{{6~~}More on Bayesian Terminology}
{{6.1~~}Marginal}
{{6.2~~}Prior}
{{6.3~~}Likelihood}
{{6.4~~}Posterior}
{{7~~}Why be Bayesian?}
{{7.1~~}Example: confidence vs.~credible intervals}
{{8~~}Bayesian Inference, in 3 Easy Steps}
{{8.1~~}Step 1: Specify a probability model}
{{8.2~~}Step 2: Calculate a posterior distribution}
{{8.3~~}Step 3: Check your model}
{{9~~}References}
\hypertarget{an-introduction-to-bayesian-statistical-analysis}{%
\subsection{An Introduction to Bayesian Statistical
Analysis}\label{an-introduction-to-bayesian-statistical-analysis}}
Before we jump in to model-building and using MCMC to do wonderful
things, it is useful to understand a few of the theoretical
underpinnings of the Bayesian statistical paradigm. A little theory (and
I do mean a \emph{little}) goes a long way towards being able to apply
the methods correctly and effectively.
There are several introductory references to Bayesian statistics that go
well beyond what we will cover here. Some suggestions:
\href{https://www.kobo.com/us/en/ebook/schaum-s-outline-of-probability-and-statistics-4th-edition}{Chapter
11 of Schaum's Outline of Probability and Statistics}
\href{https://www.stat.auckland.ac.nz/~brewer/stats331.pdf}{Introduction
to Bayesian Statistics}
\href{https://www.york.ac.uk/depts/maths/histstat/pml1/bayes/book.htm}{Bayesian
Statistics}
\hypertarget{what-is-bayesian-statistical-analysis}{%
\subsection{\texorpdfstring{What \emph{is} Bayesian Statistical
Analysis?}{What is Bayesian Statistical Analysis?}}\label{what-is-bayesian-statistical-analysis}}
Though many of you will have taken a statistics course or two during
your undergraduate (or graduate) education, most of those who have will
likely not have had a course in \emph{Bayesian} statistics. Most
introductory courses, particularly for non-statisticians, still do not
cover Bayesian methods at all, except perhaps to derive Bayes' formula
as a trivial rearrangement of the definition of conditional probability.
Even today, Bayesian courses are typically tacked onto the curriculum,
rather than being integrated into the program.
In fact, Bayesian statistics is not just a particular method, or even a
class of methods; it is an entirely different paradigm for doing
statistical analysis.
\begin{quote}
Practical methods for making inferences from data using probability
models for quantities we observe and about which we wish to learn.
\emph{-- Gelman et al.~2013}
\end{quote}
A Bayesian model is described by parameters, uncertainty in those
parameters is described using probability distributions.
All conclusions from Bayesian statistical procedures are stated in terms
of \emph{probability statements}
\includegraphics{images/prob_model.png}
This confers several benefits to the analyst, including:
\begin{itemize}
\tightlist
\item
ease of interpretation, summarization of uncertainty
\item
can incorporate uncertainty in parent parameters
\item
easy to calculate summary statistics
\end{itemize}
\hypertarget{what-is-probability}{%
\subsection{What is Probability?}\label{what-is-probability}}
\begin{quote}
\emph{Misunderstanding of probability may be the greatest of all
impediments to scientific literacy.} --- Stephen Jay Gould
\end{quote}
It is useful to start with defining what probability is. There are three
main categories:
\hypertarget{classical-probability}{%
\subsubsection{1. Classical probability}\label{classical-probability}}
\[Pr(X=x) = \frac{\text{# x outcomes}}{\text{# possible outcomes}}\]
Classical probability is an assessment of \textbf{possible} outcomes of
elementary events. Elementary events are assumed to be equally likely.
\hypertarget{frequentist-probability}{%
\subsubsection{2. Frequentist
probability}\label{frequentist-probability}}
\[Pr(X=x) = \lim_{n \rightarrow \infty} \frac{\text{# times x has occurred}}{\text{# independent and identical trials}}\]
This interpretation considers probability to be the relative frequency
``in the long run'' of outcomes.
\hypertarget{subjective-probability}{%
\subsubsection{3. Subjective probability}\label{subjective-probability}}
\[Pr(X=x)\]
Subjective probability is a measure of one's uncertainty in the value of
\(X\). It characterizes the state of knowledge regarding some unknown
quantity using probability.
It is not associated with long-term frequencies nor with
equal-probability events.
For example:
\begin{itemize}
\tightlist
\item
X = the true prevalence of diabetes in Austin is \textless{} 15\%
\item
X = the blood type of the person sitting next to you is type A
\item
X = the Nashville Predators will win next year's Stanley Cup
\item
X = it is raining in Nashville
\end{itemize}
\hypertarget{bayesian-vs-frequentist-statistics-whats-the-difference}{%
\subsection{Bayesian vs Frequentist Statistics: What's the
difference?}\label{bayesian-vs-frequentist-statistics-whats-the-difference}}
See the
\href{http://conference.scipy.org/proceedings/scipy2014/pdfs/vanderplas.pdf}{VanderPlas
paper and video}.
\begin{figure}
\centering
\includegraphics{images/can-of-worms.jpg}
\caption{can of worms}
\end{figure}
Any statistical paradigm, Bayesian or otherwise, involves at least the
following:
\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
Some \textbf{unknown quantities} about which we are interested in
learning or testing. We call these \emph{parameters}.
\item
Some \textbf{data} which have been observed, and hopefully contain
information about (1).
\item
One or more \textbf{models} that relate the data to the parameters,
and is the instrument that is used to learn.
\end{enumerate}
\hypertarget{the-frequentist-world-view}{%
\subsubsection{The Frequentist World
View}\label{the-frequentist-world-view}}
\begin{figure}
\centering
\includegraphics{images/fisher.png}
\caption{Fisher}
\end{figure}
\begin{itemize}
\tightlist
\item
The data that have been observed are considered \textbf{random},
because they are realizations of random processes, and hence will vary
each time one goes to observe the system.
\item
Model parameters are considered \textbf{fixed}. The parameters' values
are unknown, but they are fixed, and so we \emph{condition} on them.
\end{itemize}
In mathematical notation, this implies a (very) general model of the
following form:
\[f(y | \theta)\]
Here, the model \(f\) accepts data values \(y\) as an argument,
conditional on particular values of \(\theta\).
Frequentist inference typically involves deriving \textbf{estimators}
for the unknown parameters. Estimators are formulae that return
estimates for particular estimands, as a function of data. They are
selected based on some chosen optimality criterion, such as
\emph{unbiasedness}, \emph{variance minimization}, or \emph{efficiency}.
\begin{quote}
For example, lets say that we have collected some data on the prevalence
of autism spectrum disorder (ASD) in some defined population. Our sample
includes \(n\) sampled children, \(y\) of them having been diagnosed
with autism. A frequentist estimator of the prevalence \(p\) is:
\end{quote}
\begin{quote}
\[\hat{p} = \frac{y}{n}\]
\end{quote}
\begin{quote}
Why this particular function? Because it can be shown to be unbiased and
minimum-variance.
\end{quote}
It is important to note that new estimators need to be derived for every
estimand that is introduced.
\hypertarget{the-bayesian-world-view}{%
\subsubsection{The Bayesian World View}\label{the-bayesian-world-view}}
\begin{figure}
\centering
\includegraphics{images/bayes.png}
\caption{Bayes}
\end{figure}
\begin{itemize}
\tightlist
\item
Data are considered \textbf{fixed}. They used to be random, but once
they were written into your lab notebook/spreadsheet/IPython notebook
they do not change.
\item
Model parameters themselves may not be random, but Bayesians use
probability distribtutions to describe their uncertainty in parameter
values, and are therefore treated as \textbf{random}. In some cases,
it is useful to consider parameters as having been sampled from
probability distributions.
\end{itemize}
This implies the following form:
\[p(\theta | y)\]
This formulation used to be referred to as \textbf{\emph{inverse
probability}}, because it infers from observations to parameters, or
from effects to causes.
Bayesians do not seek new estimators for every estimation problem they
encounter. There is only one estimator for Bayesian inference:
\textbf{Bayes' Formula}.
\hypertarget{bayes-formula}{%
\subsection{Bayes' Formula}\label{bayes-formula}}
Given two events A and B, the conditional probability of A given that B
is true is expressed as follows:
\[Pr(A|B) = \frac{Pr(B|A)Pr(A)}{Pr(B)}\]
where P(B)\textgreater{}0. Although Bayes' theorem is a fundamental
result of probability theory, it has a specific interpretation in
Bayesian statistics.
In the above equation, A usually represents a proposition (such as the
statement that a coin lands on heads fifty percent of the time) and B
represents the evidence, or new data that is to be taken into account
(such as the result of a series of coin flips). P(A) is the
\textbf{prior} probability of A which expresses one's beliefs about A
before evidence is taken into account. The prior probability may also
quantify prior knowledge or information about A.
P(B\textbar{}A) is the \textbf{likelihood}, which can be interpreted as
the probability of the evidence B given that A is true. The likelihood
quantifies the extent to which the evidence B supports the proposition
A.
P(A\textbar{}B) is the \textbf{posterior} probability, the probability
of the proposition A after taking the evidence B into account.
Essentially, Bayes' theorem updates one's prior beliefs P(A) after
considering the new evidence B.
P(B) is the \textbf{marginal likelihood}, which can be interpreted as
the sum of the conditional probability of B under all possible events
\$A\_i\# in the sample space
\begin{itemize}
\tightlist
\item
For two events \(P(B) = P(B|A)P(A) + P(B|\bar{A})P(\bar{A})\)
\end{itemize}
\hypertarget{example-genetic-probabilities}{%
\subsubsection{Example: Genetic
probabilities}\label{example-genetic-probabilities}}
Let's put Bayesian inference into action using a very simple example.
I've chosen this example because it is one of the rare occasions where
the posterior can be calculated by hand. We will show how data can be
used to update our belief in competing hypotheses.
Hemophilia is a rare genetic disorder that impairs the ability for the
body's clotting factors to coagualate the blood in response to broken
blood vessels. The disease is an \textbf{x-linked recessive} trait,
meaning that there is only one copy of the gene in males but two in
females, and the trait can be masked by the dominant allele of the gene.
This implies that males with 1 gene are \emph{affected}, while females
with 1 gene are unaffected, but \emph{carriers} of the disease. Having 2
copies of the disease is fatal, so this genotype does not exist in the
population.
In this example, consider a woman whose mother is a carrier (because her
brother is affected) and who marries an unaffected man. Let's now
observe some data: the woman has two consecutive (non-twin) sons who are
unaffected. We are interested in determining \textbf{if the woman is a
carrier}.
\begin{figure}
\centering
\includegraphics{images/hemophilia.png}
\caption{hemophilia}
\end{figure}
To set up this problem, we need to set up our probability model. The
unknown quantity of interest is simply an indicator variable \(W\) that
equals 1 if the woman is affected, and zero if she is not. We are
interested in the probability that the variable equals one, given what
we have observed:
\[Pr(W=1 | s_1=0, s_2=0)\]
Our prior information is based on what we know about the woman's
ancestry: her mother was a carrier. Hence, the prior is
\(Pr(W=1) = 0.5\). Another way of expressing this is in terms of the
\textbf{prior odds}, or:
\[O(W=1) = \frac{Pr(W=1)}{Pr(W=0)} = 1\]
Now for the likelihood: The form of this function is:
\[L(W | s_1=0, s_2=0)\]
This can be calculated as the probability of observing the data for any
passed value for the parameter. For this simple problem, the likelihood
takes only two possible values:
\[\begin{aligned}
L(W=1 &| s_1=0, s_2=0) = (0.5)(0.5) = 0.25 \cr
L(W=0 &| s_1=0, s_2=0) = (1)(1) = 1
\end{aligned}\]
With all the pieces in place, we can now apply Bayes' formula to
calculate the posterior probability that the woman is a carrier:
\[\begin{aligned}
Pr(W=1 | s_1=0, s_2=0) &= \frac{L(W=1 | s_1=0, s_2=0) Pr(W=1)}{L(W=1 | s_1=0, s_2=0) Pr(W=1) + L(W=0 | s_1=0, s_2=0) Pr(W=0)} \cr
&= \frac{(0.25)(0.5)}{(0.25)(0.5) + (1)(0.5)} \cr
&= 0.2
\end{aligned}\]
Hence, there is a 0.2 probability of the woman being a carrier.
Its a bit trivial, but we can code this in Python:
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}13}]:} \PY{n}{prior} \PY{o}{=} \PY{l+m+mf}{0.5}
\PY{n}{p} \PY{o}{=} \PY{l+m+mf}{0.5}
\PY{n}{L} \PY{o}{=} \PY{k}{lambda} \PY{n}{w}\PY{p}{,} \PY{n}{s}\PY{p}{:} \PY{n}{np}\PY{o}{.}\PY{n}{prod}\PY{p}{(}\PY{p}{[}\PY{p}{(}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{n}{i}\PY{p}{,} \PY{n}{p}\PY{o}{*}\PY{o}{*}\PY{n}{i} \PY{o}{*} \PY{p}{(}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{n}{p}\PY{p}{)}\PY{o}{*}\PY{o}{*}\PY{p}{(}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{n}{i}\PY{p}{)}\PY{p}{)}\PY{p}{[}\PY{n}{w}\PY{p}{]} \PY{k}{for} \PY{n}{i} \PY{o+ow}{in} \PY{n}{s}\PY{p}{]}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}14}]:} \PY{n}{s} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{]}
\PY{n}{post} \PY{o}{=} \PY{n}{L}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,} \PY{n}{s}\PY{p}{)} \PY{o}{*} \PY{n}{prior} \PY{o}{/} \PY{p}{(}\PY{n}{L}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,} \PY{n}{s}\PY{p}{)} \PY{o}{*} \PY{n}{prior} \PY{o}{+} \PY{n}{L}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{n}{s}\PY{p}{)} \PY{o}{*} \PY{p}{(}\PY{l+m+mi}{1} \PY{o}{\PYZhy{}} \PY{n}{prior}\PY{p}{)}\PY{p}{)}
\PY{n}{post}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}14}]:} 0.2
\end{Verbatim}
Now, what happens if the woman has a third unaffected child? What is our
estimate of her probability of being a carrier then?
Bayes' formula makes it easy to update analyses with new information, in
a sequential fashion. We simply assign the posterior from the previous
analysis to be the prior for the new analysis, and proceed as before:
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}15}]:} \PY{n}{L}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,} \PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}15}]:} 0.5
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}16}]:} \PY{n}{s} \PY{o}{=} \PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}
\PY{n}{prior} \PY{o}{=} \PY{n}{post}
\PY{n}{L}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,} \PY{n}{s}\PY{p}{)} \PY{o}{*} \PY{n}{prior} \PY{o}{/} \PY{p}{(}\PY{n}{L}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,} \PY{n}{s}\PY{p}{)} \PY{o}{*} \PY{n}{prior} \PY{o}{+} \PY{n}{L}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{n}{s}\PY{p}{)} \PY{o}{*} \PY{p}{(}\PY{l+m+mi}{1} \PY{o}{\PYZhy{}} \PY{n}{prior}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}16}]:} 0.11111111111111112
\end{Verbatim}
Thus, observing a third unaffected child has further reduced our belief
that the mother is a carrier.
\hypertarget{more-on-bayesian-terminology}{%
\subsection{More on Bayesian
Terminology}\label{more-on-bayesian-terminology}}
Replacing Bayes' Formula with conventional Bayes terms:
\begin{figure}
\centering
\includegraphics{images/bayes_formula.png}
\caption{bayes formula}
\end{figure}
The equation expresses how our belief about the value of \(\theta\), as
expressed by the \textbf{prior distribution} \(P(\theta)\) is
reallocated following the observation of the data \(y\), as expressed by
the posterior distribution the posterior distribution.
\hypertarget{marginal}{%
\subsubsection{Marginal}\label{marginal}}
The denominator \(P(y)\) is the likelihood integrated over all
\(\theta\):
\[Pr(\theta|y) = \frac{Pr(y|\theta)Pr(\theta)}{\int Pr(y|\theta)Pr(\theta) d\theta}\]
This usually cannot be calculated directly. However it is just a
normalization constant which doesn't depend on the parameter; the act of
summation washes out whatever info we had about the parameter. Hence it
can often be ignored;The normalising constant makes sure that the
resulting posterior distribution is a true probability distribution by
ensuring that the sum of the distribution is equal to 1.
In some cases we don't care about this property of the distribution. We
only care about where the peak of the distribution occurs, regardless of
whether the distribution is normalised or not
Unfortunately sometimes we are obliged to calculate it. The
intractability of this integral is one of the factors that has
contributed to the under-utilization of Bayesian methods by
statisticians.
\hypertarget{prior}{%
\subsubsection{Prior}\label{prior}}
Once considered a controversial aspect of Bayesian analysis, the prior
distribution characterizes what is known about an unknown quantity
before observing the data from the present study. Thus, it represents
the information state of that parameter. It can be used to reflect the
information obtained in previous studies, to constrain the parameter to
plausible values, or to represent the population of possible parameter
values, of which the current study's parameter value can be considered a
sample.
\hypertarget{likelihood}{%
\subsubsection{Likelihood}\label{likelihood}}
The likelihood represents the information in the observed data, and is
used to update prior distributions to posterior distributions. This
updating of belief is justified becuase of the \textbf{likelihood
principle}, which states:
\begin{quote}
Following observation of \(y\), the likelihood \(L(\theta|y)\) contains
all experimental information from \(y\) about the unknown \(\theta\).
\end{quote}
Bayesian analysis satisfies the likelihood principle because the
posterior distribution's dependence on the data is only through the
likelihood. In comparison, most frequentist inference procedures violate
the likelihood principle, because inference will depend on the design of
the trial or experiment.
What is a likelihood function? It is closely related to the probability
density (or mass) function. Taking a common example, consider some data
that are binomially distributed (that is, they describe the outcomes of
\(n\) binary events). Here is the binomial sampling distribution:
\[p(Y|\theta) = {n \choose y} \theta^{y} (1-\theta)^{n-y}\]
We can code this easily in Python:
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}8}]:} \PY{k+kn}{from} \PY{n+nn}{scipy}\PY{n+nn}{.}\PY{n+nn}{special} \PY{k}{import} \PY{n}{comb}
\PY{n}{pbinom} \PY{o}{=} \PY{k}{lambda} \PY{n}{y}\PY{p}{,} \PY{n}{n}\PY{p}{,} \PY{n}{p}\PY{p}{:} \PY{n}{comb}\PY{p}{(}\PY{n}{n}\PY{p}{,} \PY{n}{y}\PY{p}{)} \PY{o}{*} \PY{n}{p}\PY{o}{*}\PY{o}{*}\PY{n}{y} \PY{o}{*} \PY{p}{(}\PY{l+m+mi}{1}\PY{o}{\PYZhy{}}\PY{n}{p}\PY{p}{)}\PY{o}{*}\PY{o}{*}\PY{p}{(}\PY{n}{n}\PY{o}{\PYZhy{}}\PY{n}{y}\PY{p}{)}
\end{Verbatim}
This function returns the probability of observing \(y\) events from
\(n\) trials, where events occur independently with probability \(p\).
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}9}]:} \PY{n}{pbinom}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,} \PY{l+m+mi}{10}\PY{p}{,} \PY{l+m+mf}{0.5}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}9}]:} 0.1171875
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}10}]:} \PY{n}{pbinom}\PY{p}{(}\PY{l+m+mi}{1}\PY{p}{,} \PY{l+m+mi}{25}\PY{p}{,} \PY{l+m+mf}{0.5}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}10}]:} 7.450580596923828e-07
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}11}]:} \PY{n}{yvals} \PY{o}{=} \PY{n+nb}{range}\PY{p}{(}\PY{l+m+mi}{10}\PY{o}{+}\PY{l+m+mi}{1}\PY{p}{)}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{yvals}\PY{p}{,} \PY{p}{[}\PY{n}{pbinom}\PY{p}{(}\PY{n}{y}\PY{p}{,} \PY{l+m+mi}{10}\PY{p}{,} \PY{l+m+mf}{0.5}\PY{p}{)} \PY{k}{for} \PY{n}{y} \PY{o+ow}{in} \PY{n}{yvals}\PY{p}{]}\PY{p}{,} \PY{l+s+s1}{\PYZsq{}}\PY{l+s+s1}{ro}\PY{l+s+s1}{\PYZsq{}}\PY{p}{)}\PY{p}{;}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_20_0.png}
\end{center}
{ \hspace*{\fill} \\}
What about the likelihood function?
The likelihood function is the exact same form as the sampling
distribution, except that we are now interested in varying the parameter
for a given dataset.
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}12}]:} \PY{n}{pvals} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linspace}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{1}\PY{p}{)}
\PY{n}{y} \PY{o}{=} \PY{l+m+mi}{4}
\PY{n}{plt}\PY{o}{.}\PY{n}{plot}\PY{p}{(}\PY{n}{pvals}\PY{p}{,} \PY{p}{[}\PY{n}{pbinom}\PY{p}{(}\PY{n}{y}\PY{p}{,} \PY{l+m+mi}{10}\PY{p}{,} \PY{n}{p}\PY{p}{)} \PY{k}{for} \PY{n}{p} \PY{o+ow}{in} \PY{n}{pvals}\PY{p}{]}\PY{p}{)}\PY{p}{;}
\end{Verbatim}
\begin{center}
\adjustimage{max size={0.9\linewidth}{0.9\paperheight}}{output_22_0.png}
\end{center}
{ \hspace*{\fill} \\}
So, though we are dealing with the same equation, these are entirely
different functions; the distribution is discrete, while the likelihood
is continuous; the distribtion's range is from 0 to 10, while the
likelihood's is 0 to 1; the distribution integrates (sums) to one, while
the likelhood does not.
\hypertarget{posterior}{%
\subsubsection{Posterior}\label{posterior}}
The mathematical form \(p(\theta | y)\) that we associated with the
Bayesian approach is referred to as a \textbf{posterior distribution}.
\begin{quote}
posterior /pos·ter·i·or/ (pos-tēr´e-er) later in time; subsequent.
\end{quote}
Why posterior? Because it tells us what we know about the unknown
\(\theta\) \emph{after} having observed \(y\).
\hypertarget{why-be-bayesian}{%
\subsection{Why be Bayesian?}\label{why-be-bayesian}}
At this point, it is worth addressing the question of why one might
consider an alternative statistical paradigm to the
classical/frequentist statistical approach. After all, it is not always
easy to specify a full probabilistic model, nor to obtain output from
the model once it is specified. So, why bother?
\begin{quote}
\ldots{} the Bayesian approach is attractive because it is useful. Its
usefulness derives in large measure from its simplicity. Its simplicity
allows the investigation of far more complex models than can be handled
by the tools in the classical toolbox.\\
\emph{-- Link and Barker 2010}
\end{quote}
We already noted that there is just one estimator in Bayesian inference,
which lends to its \textbf{\emph{simplicity}}. Moreover, Bayes affords a
conceptually simple way of coping with multiple parameters; the use of
probabilistic models allows very complex models to be assembled in a
modular fashion, by factoring a large joint model into the product of
several conditional probabilities.
Bayesian statistics is also attractive for its
\textbf{\emph{coherence}}. All unknown quantities for a particular
problem are treated as random variables, to be estimated in the same
way. Existing knowledge is given precise mathematical expression,
allowing it to be integrated with information from the study dataset,
and there is formal mechanism for incorporating new information into an
existing analysis.
Finally, Bayesian statistics confers an advantage in the
\textbf{\emph{interpretability}} of analytic outputs. Because models are
expressed probabilistically, results can be interpreted
probabilistically. Probabilities are easy for users (particularly
non-technical users) to understand and apply.
\hypertarget{example-confidence-vs.credible-intervals}{%
\subsubsection{Example: confidence vs.~credible
intervals}\label{example-confidence-vs.credible-intervals}}
A commonly-used measure of uncertainty for a statistical point estimate
in classical statistics is the \textbf{\emph{confidence interval}}. Most
scientists were introduced to the confidence interval during their
introductory statistics course(s) in college. Yet, a large number of
users mis-interpret the confidence interval.
Here is the mathematical definition of a 95\% confidence interval for
some unknown scalar quantity that we will here call \(\theta\):
\[Pr(a(Y) < \theta < b(Y) | \theta) = 0.95\]
how the endpoints of this interval are calculated varies according to
the sampling distribution of \(Y\), but for as an example, the
confidence interval for the population mean when \(Y\) is normally
distributed is calculated by:
\[Pr(\bar{Y} - 1.96\frac{\sigma}{\sqrt{n}}< \theta < \bar{Y} + 1.96\frac{\sigma}{\sqrt{n}}) = 0.95\]
It would be tempting to use this definition to conclude that there is a
95\% chance \(\theta\) is between \(a(Y)\) and \(b(Y)\), but that would
be a mistake.
Recall that for frequentists, unknown parameters are \textbf{fixed},
which means there is no probability associated with them being any value
except what they are fixed to. Here, the interval itself, and not
\(\theta\) is the random variable. The actual interval calculated from
the data is just one possible realization of a random process, and it
must be strictly interpreted only in relation to an infinite sequence of
identical trials that might be (but never are) conducted in practice.
A valid interpretation of the above would be:
\begin{quote}
If the experiment were repeated an infinite number of times, 95\% of the
calculated intervals would contain \(\theta\).
\end{quote}
This is what the statistical notion of ``confidence'' entails, and this
sets it apart from probability intervals.
Since they regard unknown parameters as random variables, Bayesians can
and do use probability intervals to describe what is known about the
value of an unknown quantity. These intervals are commonly known as
\textbf{\emph{credible intervals}}.
The definition of a 95\% credible interval is:
\[Pr(a(y) < \theta < b(y) | Y=y) = 0.95\]
Notice that we condition here on the data \(y\) instead of the unknown
\(\theta\). Thus, the endpoints are fixed and the variable is random.
We are allowed to interpret this interval as:
\begin{quote}
There is a 95\% chance \(\theta\) is between \(a\) and \(b\).
\end{quote}
Hence, the credible interval is a statement of what we know about the
value of \(\theta\) based on the observed data.
\hypertarget{bayesian-inference-in-3-easy-steps}{%
\subsection{Bayesian Inference, in 3 Easy
Steps}\label{bayesian-inference-in-3-easy-steps}}
We are now ready (and willing!) to apply Bayesian methods to our
problem. Gelman et al. (2013) describe the process of conducting
Bayesian statistical analysis in 3 steps: