In the previous three units we covered

was first, a mathematical account

of how to model the arrival time of

taxicabs in New York, and then I tried

to generalized and give a sense of how

maximum entropy methods are used in

the real world, in particular

how they are used or might be used

to described the open Safer ecosystem

I did by analogy by a set

of foundational work people have done

in the study of ecosystems

And I showed how, for example,

the maximum entropy model might be

intention with a simpler mechanistic

model

and currently we dont have the ability

to distinguish between the two

functional forms

MaxEnt predicts one functional form

its mechanistic probabalistic accrual

of language aderant has slightly

functional form and they

look too similiar to us to

decide right now.

In this next part of the talk

or next part of this unit,

I'm going to try to show you

another kind of argument that gets made

about social systems and biological

systems (of course in this case) a

social system and I'm going to show you

how those arguments get made in a

MaxEnt form and the kinds of insights

you might be able to derive

So this is a story that focuses

on a really interesting part of

Americana, it's the Sears-Roebuck

catalogue --so the Sears-Roebuck

company invented this idea of selling

large amounts of consumer goods

not directly through a store, through a

printed catalogue

that was then distributed all across the

country

So if you were a farmer in

the fall of 1909, you weren't able

necessarily to get to Chicago

to go buy the things you need to buy

to get by--needles and thread

and clothes pins and buggy whips

and Remington shavers

So what you did instead

you consulted the Sears-Roebuck & Co.

catalogue

and you able to order, by mail

all the things that you needed and this

revolutionized, of course,

consumer buying, sort of

the Amazon Prime or

Amazon.com of

the early Twentieth Century

In fact the Sears and Roebuck

catalogue ran before, all

the way from the 1800s

all the way through to the end of

the Twentieth Century

It may still exist in some form, today

buying things through the mail

has declined somewhat

So I'm going to talk in particular

about a paper that was written in

1981, apparently Montroll

called "On the Entropy Function

of Socio-Technical Systems"

and it's interesting in part because

it's one of the first times that

somebody tried to build

and argument about social systems,

about living systems,

by using Maximum Entropy arguments

So here's what Montroll did

Montroll looked at the prices

of goods in the

Sears-Roebuck catalogue

(and in fact he took data from another

source) and what he

plots here is year-by-year..

This is 1916,

this is 1924,

this is 1974

And what he does--he plots

the distribution of prices

the probability of that a "good"

in the Sears catalogue has

some cost seed.

He plots this on a log scale

This is log price

and, in fact, he uses log base 2

and this ranges from -6

that is 1 over 64,

to +6, 64 $ dollars

in the 1916 case,

and he plots the distribution

of goods. So here, for example,

the 60 percent chance you pull

an item out of the Sears catalogue

in 1916 at random

that it costs roughly log2

dollars equal zero, or in other words

it cost about a dollar.

So sixty percent of all the goods

in the catalogue cost a dollar

and you can see that on the

extremes the distribution dies out.

There are very few goods that cost more

than 60 dollars and very few here that

cost on the order of 10.

So the first thing he notes

is this distribution looks roughly

Gaussian. ...or normal.

And if you paid attention to the previous

unit, you noticed that

this here at the log price, in fact,

the distribution

of prices in the Sears catalogue

is log-normal

In other words, if you take the

logarithm of the price

to show the distribution, you get a

Gaussian.

So let's dig a little into the log-normal

distribution ...it looks like

P of X is proportional

To E to the negative X minus

Mu squared over

2(Sigma) squared.

I called mu x_ there but mu

is the mean of the distribution

(We call it the mean)

And sigma is something we call the

variant.

Let's expand this a little bit more

I'm going to write this as e to

the negative x squared over

2 sigma squared

plus 2X mu over 2 sigma squared

Minus mu squared over

2 sigma squared.

All I have done is expand the x

minus mu squared over 2 sigma squared

term.

So, I'm going to rewrite this as e

to the negative lambda 1 X squared plus

lambda 2 X plus lambda 3

And when I write it in this form you

realized the log-normal distribution

is just a maximum entropy

distribution if we constrain two things

One: we constrain Xsquared

And the other: we constrain x

And of course, constraining these two

things, is equivalent to

constraining the variant, which is

X-the expectatiion value of Xsquared

and the mean.

Constraining these two is the equivalent

of constraining these two.

And of course you can just expand

this here.....so fixing this

and this, to some set of values

is the same as fixing this to a value and

this to a value.

So the log-normal distribution is

secretly, and has been secretly

all along another

MaxEnt distribution.

Once you write it like this,

and realize that all these sort of

constants...here....and....here,

all these constants are really just

LaGrange multipliers that people

figured out the right answers for

You realize that the log-normal

distribution constrains these two

quantities.