We live in an uncertain and complex world, yet we continually have to make decisions in the present with uncertain future outcomes. Indeed, we should be on the look-out for "black swans" - low-probability high-impact events. To study, or not to study? To invest, or not to invest? To marry, or not to marry? While uncertainty makes decision-making difficult, it does at least make life exciting! If the entire future was known in advance, there would never be an element of surprise. Whether a good future or a bad future, it would be a known future. In this course we consider many useful tools to deal with uncertainty and help us to make informed (and hence better) decisions - essential skills for a lifetime of good decision-making. Key topics include quantifying uncertainty with probability, descriptive statistics, point and interval estimation of means and proportions, the basics of hypothesis testing, and a selection of multivariate applications of key terms and concepts seen throughout the course.
1.4 Simplicity vs. Complexity - The Need for Models
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Из курса от партнера University of London
Probability and Statistics: To p or not to p?
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Dealing with Uncertainty and Complexity in a Chaotic World
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Dr James AbdeyAssistant Professorial Lecturer, LSE
University of London
In this section, we're going to consider
the perennial trade-off between simplicity and complexity.
So, we've already made reference to the world being a very complex place.
Now, other things equal, we dislike uncertainty.
If I asked you to describe the world,
let's consider a simple binary choice.
Is the real world, A,
nice simple and easy,
or B, big horrible and complicated?
Well clearly, it's the latter at being B.
However, it will be much nicer,
if I asked if it was nice simple and easy i.e.
our option A.
So, whenever we try to understand the real world around us,
what we would like to try and do is to simplify things as much as possible,
because there's an inherent appeal for simplicity over complexity.
We like things to be as easy as possible for us to follow and to understand.
So, within a learning environment,
we're not necessarily restricted to say statistics,
we could extend this to mathematics, economics, finance,
and I'm sure many qualitative disciplines as well.
But our goal is to try and model the real world.
So, what is a model?
Perhaps, many definitions we could offer here,
but a version I like is to say,
"A model is a deliberate simplification of reality."
Such that, we are trying to model some real world phenomenon, whereby,
a good model is one which captures the most important features of the real world,
and ignores some less important, some trivial details.
So, do keep in mind,
that a model does not equal reality.
However, a good model is one which is very close to reality.
So, the real world,
highly complex, we dislike it.
So, if we work with the model, we gain simplicity.
So, we reduce the complexity,
improve the simplicity, as we go from the real world to a model.
But of course, that model is a departure from reality.
So, we do lose a little bit of realism.
So, the question is,
is this trade off acceptable and how do we resolve this trade-off?
Well, ideally, we would like the benefit i.e.
that gain in simplicity,
to be enough to offset the slight loss of reality that we inevitably get with a model.
So, as an interesting example,
I'll offer you the London Tube map.
This is an iconic map,
whether or not you've visited London before,
you may well have seen a snapshot of the world famous Tube map.
Now, arguably, I could have chosen any metro system around the world, for this example.
They will all have similar maps which people use,
but this is the University of London,
so let's be London-centric for a moment.
So, is this a good model?
Well, let's revisit my definition of a model,
a deliberate simplification of reality.
So, is this a good model or a bad model?
While arguably, the answer to this question is, it depends.
It depends, who we are,
and what we wish to use this model for.
Now, if you are, let say,
a tourist visiting London,
I offer this up as an excellent example of a model.
So, what do we see on the Tube map?
Well, we see different colors representing the different London underground lines.
Now, if your goal is simply to get from point A to point B,
and provided you're not color blind,
and can read the station names,
this tube map is excellent for achieving that simple objective,
from point A to point B.
Why? Because this map offers the need to know information for achieving that objective.
It tells you the order of stations on a particular line,
you have the different colors to distinguish the different lines,
and it also shows the points of intersection,
where you would change from one London Underground line to another.
Frankly, if your goal is to get from point A to point B,
that's the essential information you need to know.
Now, I'm not sure how familiar you are with the London Underground map,
but those different colors do represent different lines.
Let's just pick one of them, the yellow line.
Now, this is called the Circle Line.
I'm sure you all know what a circle is, something like that.
Now, if you have a look at the London Underground map,
you will see that the so-called,
Circle Line is in fact a lie because the line is not a true circle.
It does go around in a bit of a loop,
but it's not the exact geometric circle that you
would have been familiar with from school maths.
Nonetheless though, it is a simplification of what we observe in reality,
and you see that loop,
and circle seems a reasonable enough name.
Perhaps to pick another one, the red line.
This is the central line,
passing through really the main part of central London,
roughly from East to West.
Now, if, let's say, you look at the center of the central line,
you will see a straight red line on that London Underground map.
You'll also see, according to the map,
that those stations are equidistant from one another.
The gap from one station to another seems to be exactly the same.
Well, if any of you have been on the London Underground line,
or any of those lines,
I'm sure you can attest to these tunnels,
not being dead straight under the streets of London.
They are often have to hold onto the handrail as the train
meanders through these curved tunnels under the streets.
So, the map does not clearly represent those features.
That straight part of the red line there,
the central line, is simply used to simplify reality.
Frankly, if your goal is simply to go from point A to point B,
do you really need to know exactly how that tunnel meanders underneath the streets?
Clearly, not.
So, that map for let's say,
a tourist someone just visiting London,
is indeed fit for purpose.
It tells you the essential information that you need to know.
But of course, it is not a perfect representation of the real world.
Indeed, we know those tunnels do.
So, you move in curves and there are many bends along the way.
Also, not represented on the map,
would be the depth of those tunnels.
Some, indeed, are actually on ground level,
some of them very deep underlying.
Indeed, they're the black line, the Northern line,
I believe is the deepest of the London Underground Lines.
Now, depth is not presented on that map,
but frankly to get from point A to point B,
you don't need to know that piece of information.
On the other hand, if you were,
lets say, an engineer,
you're responsible for maintaining the tunnel network,
maybe doing some repairs to the rails say,
then clearly, that geo map would not be fit for purpose.
You would need to know the exact distance between the stations,
which is typically not equidistant as the map seems to suggest,
you will need to know where the actual access points are,
you'll need to know exactly how that tunnel bends around under the streets,
you will need to know the depth information,
and other things as well.
So, depending on who you are and what you need that map for,
you could argue it's a good model or it's a bad model.
But of course, we know it as this world famous iconic map,
primarily for its use,
not by those engineers,
but for the general public and tourists visiting London.
So, in that sense, I uphold this as a great example of a model.
It retains the most important features that one needs for decision-making i.e.
how to get from point A to point B,
and ignores less important details.
However, it can be a little bit misleading.
So, you'll also find access through
this course to what I call "The real London Underground map",
which is in fact more geographically accurate.
So, perhaps as a takeaway exercise for yourselves,
is for you to compare the familiar looking Tube map,
with those nice straight lines,
very easy for people to interpret,
with the geographically accurate Tube map,
and see a comparison of the two.
Now, both are examples of models.
Even the geographically accurate one,
still omits some information.
For example, the depth of those tunnels.
You can't see those on that geographically accurate representation.
Now, perhaps as a question,
an open question to you to consider,
is of those two maps,
both are models, they're clearly different models.
One geographically accurate, the other not,
which one of those is the better model?
I'll give it as an open-ended question,
I would perhaps offer,
maybe a subjective answer here namely, it depends.
There's clearly a trade-off between these two models.
The geographically accurate one has the advantage of being geographically accurate.
It's perhaps using that one,
it's clearer to see the relative distances between one point and another,
and to perhaps, decide on the optimal route in terms of the quickest route to undertake.
Whereas, the more familiar looking Tube map is geographically inaccurate,
and can give very misleading indications,
as to the distance between stations.
Just as perhaps, a simple example of that,
if you compare say,
Edgware Road station with Marble Arch,
across these two maps,
and judge for yourselves what might be
the optimal route to take from one point to the other?
I'll give you the answer here.
In fact, the quickest thing to do is to actually walk it.
But these are both models.
There is a trade-off.
Geographically accurate is a good thing,
but you might see, looking at it,
it's a little bit more confusing to use than the more familiar form.
So, potentially, we could offer that up as an explanation for why, transport for London,
The Transport Authority here,
for getting people from A to B through public transport,
opt for that more iconic map than the geographically accurate version.
So, I'm not saying these are academic models,
but are familiar things,
which really emphasize the point about what a good model should try and be.
Namely, a deliberate simplification of reality.
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