## Innumeracy

From testimony by Michael Gove, British Secretary of State for Education, before their Education Committee:

Q98 Chair: One is: if “good” requires pupil performance to exceed the national average, and if all schools must be good, how is this mathematically possible?

Michael Gove: By getting better all the time.

Q99 Chair: So it is possible, is it?

Michael Gove: It is possible to get better all the time.

Q100 Chair: Were you better at literacy than numeracy, Secretary of State?

Michael Gove: I cannot remember.

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## Expectation: Useful properties and inequalities

If ${X \geq 0}$ is a random variable on ${(\Omega, \mathcal{F}, P)}$. The expected value of ${X }$ is defined as

$\displaystyle \mathbb{E}(X) \equiv \int_{\Omega} X dP = \int_{\Omega} X(\omega) P (d \omega)$

Inequalities

• Jensen’s inequality. If ${\varphi}$ is convex and ${E|X|, E|\varphi(X)| < \infty}$

$\displaystyle \mathbb{E} (\varphi(X)) \geq \varphi(\mathbb{E}X)$

• Holder’s inequality. If ${p,q \in [1, \infty]}$ with ${1/p + 1/q =1}$ then

$\displaystyle \mathbb{E}|XY| \leq \|X\|_p \|Y\|_q$

• Cauchy-Schwarz Inequality: For ${p=q=2}$

$\displaystyle \mathbb{E}|XY| \leq \left( \mathbb{E}(X^2) \mathbb{E}(Y^2) \right)^{1/2}$

Let ${Y \geq 0}$ with ${\mathbb{E} (Y^2) < \infty}$. $\displaystyle \mathbb{E}(Y)^2 \leq \mathbb{E}(Y \mathbb{I}_{Y>0})^2 \leq \mathbb{E}(Y^2) P(Y>0)$
hence

$\displaystyle P(Y>0) \geq E(Y)^2/E(Y^2)$

• Markov’s/Chebyshev’s inequality. Suppose ${\varphi: \mathbb{R} \rightarrow \mathbb{R}}$ has ${\varphi \geq 0}$, let ${\mathcal{A} \in \mathcal{R}}$ and let ${i_A = \inf \{\varphi(y) : y \in A\}}$

$\displaystyle i_A P(X \in A) \leq \mathbb{E}(\varphi(X); X \in A) \leq \mathbb{E}(\varphi(X))$

where

$\displaystyle \mathbb{E}(X;A) = \int_A X dP, \qquad A \subset \Omega$

Alternatively suppose ${X \in m \mathcal{F}}$ and that ${\varphi: \mathbb{R} \rightarrow [0,\infty]}$ is ${\mathcal{B}}$-measurable and non-decreasing (note ${\varphi (X) = \varphi \circ X \in (m \mathcal{F})^+}$ ). Then

$\displaystyle \varphi(a)P(X \geq A) \leq \mathbb{E}(\varphi(X); X \geq a) \leq \mathbb{E}(\varphi(X))$

## Very brief notes on measures: From σ-fields to Carathéodory’s Theorem

Definition 1. A ${\sigma}$-field ${\mathcal{F}}$ is a non-empty collection of subsets of the sample space ${\Omega}$ closed under the formation of complements and countable unions (or equivalently of countable intesections – note ${\bigcap_{i} A_i = (\bigcup_i A_i^c)^c}$). Hence ${\mathcal{F}}$ is a ${\sigma}$-field if

$1. {A^c \in \mathcal{F}}$ whenever ${A \in \mathcal{F}}$
$2. {\bigcup_{i=1}^{\infty} A_i \in \mathcal{F}}$ whenever ${A_i \in \mathcal{F}, n \geq 1}$

Definition 2. Set functions and measures. Let ${S}$ be a set and ${\Sigma_0}$ be an algebra on ${S}$, and let ${\mu_0}$ be a non-negative set function

$\displaystyle \mu_0: \Sigma_0 \rightarrow [0, \infty]$

• ${\mu_0}$ is additive if ${\mu_0 (\varnothing) =0}$ and, for ${F,G \in \Sigma_0}$,

$\displaystyle F \cap G = \varnothing \qquad \Rightarrow \qquad \mu_0(F \cup G ) = \mu_0(F) + \mu_0(G)$

• The map ${\mu_0}$ is called countably additive (or ${\sigma}$-additive) if ${\mu (\varnothing)=0}$ and whenever ${(F_n: n \in \mathbb{N})}$ is a sequence of disjoint sets in ${\Sigma_0}$ with union ${F = \cup F_n}$ in ${\Sigma_0}$, then

$\displaystyle \mu_0 (F) = \sum_{n}\mu_0 (F_n)$

• Let ${(S, \Sigma)}$ be a measurable space, so that ${\Sigma}$ is a ${\sigma}$-algebra on ${S}$.
• A map $\displaystyle \mu: \Sigma \rightarrow [0,\infty].$ is called a measure on ${(S, \Sigma)}$ if ${\mu}$ is countable additive. The triple ${(S, \Sigma, \mu)}$ is called a measure space.
• The measure ${\mu}$ is called finite if

$\displaystyle \mu(S) < \infty,$

and ${\sigma}$-finite if

${\exists \{S_n\} \in \Sigma}$, (${n \in \mathbb{N}}$) s.th.$\displaystyle \mu(S_n)< \infty, \forall n \in \mathbb{N} \text{ and } \cup S_n = S.$

• Measure ${\mu}$ is called a probability measure if $\displaystyle \mu(S) = 1,$ and ${(S, \Sigma, \mu)}$ is then called a probability triple.
• An element ${F}$ of ${\Sigma}$ is called ${\mu}$-null if ${\mu(F)=0}$.
• A statement ${\mathcal{S}}$ about points ${s}$ of ${\mathcal{S}}$ is said to hold almost everywhere (a.s.) if

$\displaystyle F \equiv \{ s: \mathcal{S}(s) \text{ is false} \} \in \Sigma \text{ and } \mu(F)=0.$

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## Survivorship Bias

Originally posted on You Are Not So Smart:

The Misconception: You should focus on the successful if you wish to become successful.

The Truth: When failure becomes invisible, the difference between failure and success may also become invisible.

Illustration by Brad Clark at http://www.plus3video.com

In New York City, in an apartment a few streets away from the center of Harlem, above trees reaching out over sidewalks and dogs pulling at leashes and conversations cut short to avoid parking tickets, a group of professional thinkers once gathered and completed equations that would both snuff and spare several hundred thousand human lives.

People walking by the apartment at the time had no idea that four stories above them some of the most important work in applied mathematics was tilting the scales of a global conflict as secret agents of the United States armed forces, arithmetical soldiers, engaged in statistical combat. Nor could people today know as they open umbrellas and…

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Photograph by Jake Slagle @ flickr

“A swami puts m dollars in one envelope and 2m dollars in another. You and your opponent each get one of the envelopes (at random). You open your envelope and find x dollars, and then the swami asks you if you want to trade envelopes. You reason that if you switch, you will get either x/2 or 2x dollars, each with probability 1/2. This makes the expected value of a switch equal to (1/2)(x/2) + (1/2)(2x)=5x/4, which is greater than the x dollars that you hold in your hand. So you offer to trade.

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## Larry Laudan. Why Presuming Innocence is Not a Bayesian Prior

“…the presumption (of innocence) is not (or at least should not be) an instruction about whether jurors believe defendant did or did not commit the crime. It is, rather, an instruction about their probative attitudes.”

Originally posted on Error Statistics Philosophy:

“Why presuming innocence has nothing to do with assigning low prior probabilities to the proposition that defendant didn’t commit the crime”

by Professor Larry Laudan
Philosopher of Science*

Several of the comments to the July 17 post about the presumption of innocence suppose that jurors are asked to believe, at the outset of a trial, that the defendant did not commit the crime and that they can legitimately convict him if and only if they are eventually persuaded that it is highly likely (pursuant to the prevailing standard of proof) that he did in fact commit it. Failing that, they must find him not guilty. Many contributors here are conjecturing how confident jurors should be at the outset about defendant’s material innocence.

That is a natural enough Bayesian way of formulating the issue but I think it drastically misstates what the presumption of innocence amounts to.  In my view, the…

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