In jrminter/statshelpR: Convenience functions for statistical analysis

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Introduction

This vignette is a summary of a great video by Ian Olasov (CUNY) in this YouTube video.

We begin with a definition, Probability: The chance or likelihood that a proposition is true.

Examples using Bayes Theorem

Example 1

Example: One student in a class of 20 has the flu. You do not know who. The probability that one student in the class, Sally, has the flu is 1/20 or 5% (.05). This is called your prior probability because it is your estimate of the probability prior to finding out any new information.

$$P(Sally \ has \ the \ flu) = 0.05$$

Suppose that the class has 5 girls and the patient was a girl, your probability that Sally has the flu would go up to 1/5 (0.20). On the other hand if you found out that the patient was a boy, your probability that Sally has the flu would go down to 0. Since we do not yet know whether the patient is a boy or a girl, we call these conditional probabilities.

$$P(Sally \ has \ the \ flu\ |\ patient \ is \ a \ girl) = 0.20$$ $$P(Sally \ has \ the \ flu\ |\ patient \ is \ a \ boy) = 0 $$

The vertical line tells us that we are talking about conditional probabilities.

Bayes Theorem shows us how to calculate our conditional probabilities

Bayes theorem says that the probability of a hypothesis, H, conditional on a new piece of evidence, E, is

$$P(H | E) = \frac{ P(E|H) \ P(H)}{P(E)}$$ In other words it tells us the three ingredients that go into the probability of a hypothesis conditional on some evidence:

1. The probability of the evidence given the hypothesis
2. The prior probability of the hypothesis
3. The prior probability of the evidence

Example 2

Let's consider another example.

Imagine that one morning you do not feel well and go on WebMD to try to figure out what is wrong. You browse around until you find an illness that catches your eye, Hypothesitis.

$P(H)$ - You have hypothesitis.

As you read through all the symptoms that you would have if you have hypothesitis, you are pretty sure this describes your symptoms. We can then say that the probability of symptoms, given hypothesitis is 0.95.

$P(E \ |\ H) = 0.95$

You begin to freak out. But then you remember Bayes Theorem. It tells you that there are two more things that you need to know to figure out the probability that you have hypothesitis.

1. The prior probability that you would come down with hypothesitis, $P(H)$
2. The prior probability that you would have the symptoms that you actually have, $P(E)$

With a little more Googling you discover that the disease is extremely rare, only 1/100000 people have it, so $P(H) = 0.00001$.

Next we ask, "what kind of symptoms are they?" Let's say they are a headache and a runny nose. Lots of people have them. Google tells you 1 in 100. So $P(E)$, the prior probability of having these symptoms, is So $P(E) = 0.01$.

At last, you know everything you need to know to figure out the probability that you have hypothesitis, given your symptoms. Bayes theorem tells us that,

$P(H|E) = \frac{0.95 \times\ 0.00001}{.01} = 0.00095$

or a little less than 1 in 1000. What a relief!

A common error

Bayes Theorem is extremely helpful in figuring out what to make of some new piece of evidence. People often ignore the prior of the hypothesis or treat $P(H|E)$ as $P(E|H)$, which it is not. This mistake is sometimes called the base rate fallacy. Note that in our last example $P(H|E)$ and $P(E|H)$ differ by a factor of $10^3$.

Summary

1. Bayes Theorem shows us how to calculate conditional probabilities or the probability that you should assign to some hypothesis given a piece of evidence.

2. The conditional probability of a hypothesis given new evidence depends on three things:

3. The conditional probability of E given H: $P(E|H$
4. The prior probability of the hypothesis: $P(H)$
5. The prior probability of the evidence $P(E)$ If you leave one of those three things out, you don't have a complete picture.

jrminter/statshelpR documentation built on May 2, 2020, 12:08 a.m.