3.11 More conditional probability, decision trees and Bayes' Law

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來自 University of Amsterdam 的課程

统计基础

1707 個評分

University of Amsterdam

统计基础

1707 個評分

課程 3（共 5 門，Specialization 社会科学的方法和统计）

Understanding statistics is essential to understand research in the social and behavioral sciences. In this course you will learn the basics of statistics; not just how to calculate them, but also how to evaluate them. This course will also prepare you for the next course in the specialization - the course Inferential Statistics. In the first part of the course we will discuss methods of descriptive statistics. You will learn what cases and variables are and how you can compute measures of central tendency (mean, median and mode) and dispersion (standard deviation and variance). Next, we discuss how to assess relationships between variables, and we introduce the concepts correlation and regression. The second part of the course is concerned with the basics of probability: calculating probabilities, probability distributions and sampling distributions. You need to know about these things in order to understand how inferential statistics work. The third part of the course consists of an introduction to methods of inferential statistics - methods that help us decide whether the patterns we see in our data are strong enough to draw conclusions about the underlying population we are interested in. We will discuss confidence intervals and significance tests. You will not only learn about all these statistical concepts, you will also be trained to calculate and generate these statistics yourself using freely available statistical software.

從本節課中

Probability

This module introduces concepts from probability theory and the rules for calculating with probabilities. This is not only useful for answering various kinds of applied statistical questions but also to understand the statistical analyses that will be introduced in subsequent modules. We start by describing randomness, and explain how random events surround us. Next, we provide an intuitive definition of probability through an example and relate this to the concepts of events, sample space and random trials. A graphical tool to understand these concepts is introduced here as well, the tree-diagram.Thereafter a number of concepts from set theory are explained and related to probability calculations. Here the relation is made to tree-diagrams again, as well as contingency tables. We end with a lesson where conditional probabilities, independence and Bayes rule are explained. All in all, this is quite a theoretical module on a topic that is not always easy to grasp. That's why we have included as many intuitive examples as possible.

3.08 Joint and marginal probabilities6:02

3.09 Conditional probability4:49

3.10 Independence between random events5:13

3.11 More conditional probability, decision trees and Bayes' Law8:12

與講師見面

Matthijs Rooduijn
Dr.
Department of Political Science
Emiel van Loon
Assistant Professor
Institute for Biodiversity and Ecosystem Dynamics

In this video, I'll explain the relation between conditional probability,

decision trees, and

an equation that relates different conditional probabilities, Bayes' law.

Let's look at conditional probability as a way by which a sample space is reversed.

You may decide that if Event B occurs before Event A,

the sample space is effectively reduced to a new smaller sample space.

When calculating conditional probabilities,

you are considering the probabilities for

Event A, with a reduced sample space, rather than a complete sample space.

The important point is that all the properties that you know to be

applicable for

events in the complete sample space, apply to this reduced sample space as well.

For example, if A, in reduced sample space, that is, A given B,

could take three values, the union of these three probabilities should be one.

Let's apply this to an example.

You've made a count of different activities by the people on your beach,

which you've distinguished by gender.

And you have turned it into this table with probabilities.

The conditional probabilities for

the three activities given that the person were a male

would be given by dividing the respective joint probabilities with 0.45.

And for females, the conditional probabilities for

the activities would be given by dividing with 0.55.

The resulting conditional probabilities for

the activities given gender are shown here.

And the same can be done for the conditional probabilities of

gender given activity.

You can see that a rule stating that the sum of all conditional

probability should be one does apply.

Sometimes, you deal with conditional probabilities without really noticing.

You may have done so already while making calculations with a tree diagram.

Let's turn the table with joint and

marginal probabilities into a tree diagram.

You can imagine that at the beach you would first look at all the people

resting, and subsequently count the number of male and female persons in that group.

And the same for the other two activities.

So your tree would have to first split over activity and

second split over gender within each activity branch.

Now you'd find these probabilities along each branch.

The marginal probabilities for the activities would be placed at the first note.

In the second step, they're splitted by gender.

Here, you would deal with conditional probabilities gender,

given that an activity is known.

You would calculate these probabilities by dividing the count for male and

female per activity by the total people in that activity,

not the total at the entire beach.

The joint probabilities at the end would result from multiplying the probability

per activity within the conditional probabilities of gender, given activity.

So, in this tree diagram, the probabilities that you encounter at

the first node are marginal probabilities, for the activities, in this case.

The probabilities at the second node are conditional probabilities, for

gender, given the outcome of a certain activity.

And the probabilities at the end of a tree are the joint probabilities.

To get the marginal probabilities for gender,

you would have to add the respective joint probabilities.

If you would interchange gender and activity in a decision tree,

the marginal and conditional probabilities would change.

But the joint probabilities would remain the same.

This leads to an interesting insight.

The joint probability of A and B equals the multiplication of conditional

probability of event A, given B, times the probability of B.

But it also equals the multiplication of conditional probability, or

event B, given A, times the probability of A.

Therefore, we can just as well say that the conditional probability

of A given B is related to B given A.

This equation is known as Bayes' Law.

It is, in fact, the shortened version, because often the marginal

probability of B is not known directly, but has to be computed from the sum

of the different conditional probabilities for B, given an outcome of A.

If we return to our example once more,

Bayes' law allows to calculate the probability that, for example,

a person will be swimming when you know that the person is female.

While we have only the reversed conditional probability

to our availability, a person is female given that this person is swimming.

The conditional probability that someone is female

given the person is swimming is multiplied with the probability of swimming

and this is divided by the sum of the probabilities for

the three activities times their conditional probabilities.

In simple, discrete probability calculations, Bayes' law

is a convenient function, which follows from the axioms of probability calculus.

However, it's often interpreted in a more abstract way, where the left-hand side of

the equation is understood as a degree of belief in hypothesis A, after observing B.

And the right hand side gives to believe in A,

prior to knowing B times the support that B provides for A.

In this context, the probability of A is called prior probability

because it's your knowledge about A prior to observing B.

And the conditional probability of A given B is called posterior probability.

And in this wider context,

there are no simple frequency counts to inform you about the belief in A.

So, it needs to be assessed qualitatively.

Considering probability is a belief,

rather than a frequency based quantity, is quite a different viewpoint, and

has led to a lot of philosophical debates among the different proponents.

But in practical situations,

the different perspectives appear not to be too far apart.

In any realistic analysis there is prior knowledge involved as well as assumptions

and subjective choices.

While wherever possible one would use observations to estimate probabilities.

Let me summarize what I have explained in this video.

The probability of A conditional on B can be considered as the probability of

A in the reduced sample space where B occurred.

A tree diagram contains different probabilities.

At the first node, it has marginal probabilities and

for any node further on, it has conditional probabilities.

The joint probabilities at the end result from multiplying marginal and

conditional probabilities along a branch.

The fact that the joint probability can be calculated

based on a condition of probability of A given B, as well as B given A,

leads to Bayes' law which relates these two conditional probabilities.

Bayes' law is also used to express

how a prior belief in hypothesis A can be updated by new evidence B.

In that context, the probability of A is called the prior probability and

the conditional probability of A given B is called posterior probability.

Considering probability as a belief rather than a frequency based entity

is a different theoretical starting point.

However, in practice, elements of the two are often combined.

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