Getting to Grips with Bayes' Formula: Exploring the Problem & Questions

In summary: This is a probability of an event occurring in a probability space. We can call this space "C" or we can call it "O" or we can call it "I" or we can call it "IC" or we can call it "OVERSEAS". But the probability space is not an event. It is a set of events.So, we have an event space, and it has some events in it. We know the probabilities of some of these events, and we want to know the probabilities of some other events. These other events are "conditional" on the events we know the probabilities of. For example, we know the probability of the corporation increasing
  • #1
Vital
108
4
Hello!

I am trying to get to grips with the Bayes' formula by developing an intuition about the formula itself, and on how to use it, and how to interpret.

Please, take a problem, and my questions written within them - I will highlight my questions and will post them as I add the information.

The problem:
A corporation plans to announce an expansion into overseas markets. The expansion will occur only if the overseas demand is sufficient to support the necessary sales. And if demand is sufficient and overseas expansion occurs, the Corporation is likely to raise its prices.
O represents the event of overseas expansion; I represents a price increase; IC represents no price increase.
The unconditional and conditional probabilities are estimated as follows:
P(I) = 0.3
P(IC) = 0.7
P(O | I) = 0.6
P(O | IC) = 0.4

question1: Do I understand correctly that by giving P(I) and P(IC) we assume that the corporation might increase/not increase its prices in any case, independently of any changes in the business model, i.e. independently whether it expanses overseas, or it doesn't?

question2: If the first assumption is correct, then we must determine probabilities that the corporation might, or might not, expand its business overseas in case it first raises its prices (or doesn't raise them), correct?
Therefore, that should become our base case probability assumptions
.

The problem asks to solve P(I | O) - the probability of price increase if the corporation announces the overseas expansion.
Then the textbook proceeds in the following way.
Given the multiplication rule and the joint probability rule:
P(O | I) = P(IO) / P(I) and P(IO) = P( I | O) x P(O)

hence, from the second formula P(I | O) = P(IO) / P(O)
and from the first formula P(IO) = P(O | I) x P(I)

=> P(I | O) = [P(O | I) x P(I)] / P(O)
Using the total probability rule we determine P(O) = P(O | I) x P(I) + P(O | IC) x P(IC)) =
0.6 x 0.3 + 0.4 x 0.7 = 0.46

=> P(I | O) = [0.6 x 0.3] / 0.46 = 0.3913
=> if the new information expand overseas is announced, then the prior probability estimate of P(I) = 0.30 should be increased to 0.3913.
The problem is solved.

Question3: It all sounds so artificial to me, and I have a trouble understanding why we are using such approach. To find the probability of "increased prices given the expansion overseas" we first shall estimate the probability of increased prices on its own, independently of any event; then estimate if any expansion can occur in case prices are increased prior to the expansion; and then based on these estimates find the probability of a reverse situation - increased prices given the expansion occurred.
Sounds so bewildering.

I will be grateful for your help.
Thank you so much.
 
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  • #4
I wish examples and explanations of statistics would be more positive ones - I don't understand why the majority of examples are based on medical themes, using truly horrific ones. I can't study math on these examples.
 
  • #5
Vital said:
Hello!

question1: Do I understand correctly that by giving P(I) and P(IC) we assume that the corporation might increase/not increase its prices in any case, independently of any changes in the business model, i.e. independently whether it expanses overseas, or it doesn't?

When we apply probability theory to real life problems ( or "word problems") it is natural to think about the probability of events described in words without mathematical formalities. However, this usually leads to utter confusion. You should begin by thinking about the "probability spaces" (or "event spaces") that are involved. And you should be careful about using words like "independently" , which can have one meaning in common speech and a different meaning mathematically.

Mathematically, if ##I## was independent of ##O## then, by definition, we would have ##P(I|0) = P(I)##, So when you ask if we are assuming that price increase is independent of overseas expansion, you are asking if this mathematical relation is assumed. It is not directly assumed by the numbers you gave. If it could be proven from the numbers you gave, then we might say it is implicitly assumed. However, if the numbers you gave imply ##P(I|0) = 0.46 \ne P(I) ## then the independence is obviously not implicitly assumed.
question2: If the first assumption is correct, then we must determine probabilities that the corporation might, or might not, expand its business overseas in case it first raises its prices (or doesn't raise them), correct?
Therefore, that should become our base case probability assumptions
.

You can't make sense of conditional probability by thinking of a single probability space that has "base case" assumptions. Conditional probability involves at least two probability spaces.

A probability space has "points" or "outcomes" that define mutually exclusive "atomic" events. The outcomes are not described in this problem. You can imagine them as events like "The corporation implements overseas expansion on Feb. 28, 2019 and does not increase prices" or "The corporation does not implement overseas expansion and increases prices on July 23, 2020". ( For dealing with a real life situation, the information given about outcomes in this problem is too vague .)

The events ##I,I^c,O,O^c## are events in a probability space. These events presumably consist of more than one outcome. For example, events ##I## and ##O## may both contain some common outcomes.

A probability space has a probability "measure" that assigns probabilities to events. For one probability space, we are given the information ##P(I) = .3,\ P(I^c) = .7##.

In the given information, we hear about a second probability space whose probability measures are denoted by "##P(...| I)##" and a third probability space who probability measures are denoted by ""##P(...|I^c)##". The problem asks us to find a probability from a fourth probability space whose probability measures are denoted by "##P(..|O)##".
Question3: It all sounds so artificial to me, and I have a trouble understanding why we are using such approach.

It is artificial. You have to think of the problem as a riddle or puzzle, not as a real life practical problem. To invent a sensible real life situation that fits the problem, you would have to invent an example that has plausible outcomes and examine the real life interpretations of probability spaces with measures like ##P(...| O)##. That would involve considerable mental exertion. It is very easy to utter phrases like "the probability of a price increase given there is overseas expansion", but to be specific about what this means in practical terms requires much more detail.

The key to understanding Bayes formula is to realize that ##P(A \cap B)## and ##P(A|B)## are both probabilities for
the same set of outcomes ( namely ##A \cap B##, which you denote as "##AB##") but they are probabilities given by different probability measures. So they are probabilities from different probability spaces .

The relation between the probability measure ##P(...)## and ##P(...|B)## is intuitively explained by saying that ##P(...|B)## is computed by revising our estimates of ##P(X)## by adding the assumption that event ##B## must happen. If we think of ##P(X)## as "the fraction of times X happens" then we think of ##P(X|B)## as "the fraction of times that X happens out of those times that B happens". Hence we have the mathematical definition ##P(X|B) = P(X \cap B)/ P(B)##, which says the probability measure ##P(...|B)## is computed by "renormalizing" the probability measure ##P(...)##

Perhaps you can invent an example that makes practical sense of probability spaces like ##P(...| I^c)## or ##P(...| O)##. However, the efficient way to understand Bayes formula by examples, is to consider examples where sufficient detail is given about the outcomes. It's very round-about to take a problem where the outcomes are not precisely given and then be forced to invent a plausible definition of the outcomes before considering Bayes formula.
 
  • #6
Vital said:
question1: Do I understand correctly that by giving P(I) and P(IC) we assume that the corporation might increase/not increase its prices in any case, independently of any changes in the business model, i.e. independently whether it expanses overseas, or it doesn't?
Independence is a different concept. This is what is called marginalized. Basically P(I) = P(I|O)*P(O)+P(I|~O)*P(~O)

The idea is that it is the overall probability for I that you get from averaging over all of the O possibilities.

More importantly for the purpose of Bayesian probability, P(I) is the prior probability of I. It is your subjective belief, based on whatever prior experience or knowledge you have, but before seeing the data. The purpose of Bayes theorem is to tell you how much you should rationally change that prior belief in the face of new data. It is common to irrationally change your beliefs too much when given medical test data, and it is common to change your beliefs too little when given data informing political beliefs. Bayesian probabilities avoid both errors

Vital said:
I don't understand why the majority of examples are based on medical themes, using truly horrific ones. I can't study math on these examples.
“I dislike studying math on these examples” does not mean “I can’t study math on these examples”. You certainly can.
 
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1. What is Bayes' formula and why is it important?

Bayes' formula is a mathematical formula used to calculate conditional probabilities, or the probability of an event occurring given some prior knowledge or information. It is important in many fields, including statistics, machine learning, and data science, as it allows us to update our beliefs or predictions based on new evidence or information.

2. Can you explain the components of Bayes' formula?

Bayes' formula consists of four components: P(A), the prior probability of event A; P(B), the prior probability of event B; P(A|B), the conditional probability of event A given event B has occurred; and P(B|A), the conditional probability of event B given event A has occurred. These components are used to calculate the posterior probability, or the updated probability of event A given event B has occurred.

3. How is Bayes' formula used in real-world applications?

Bayes' formula is used in many real-world applications, such as medical diagnosis, spam filtering, and weather forecasting. In medical diagnosis, it can help determine the probability of a person having a disease given their symptoms and medical history. In spam filtering, it can help identify the probability of an email being spam based on its content. In weather forecasting, it can help predict the probability of rain based on historical data and current weather conditions.

4. What are some common misconceptions about Bayes' formula?

One common misconception about Bayes' formula is that it provides absolute certainty or proof of an event occurring. In reality, it is still based on probabilities and can be affected by the quality of the prior information or assumptions. Another misconception is that it can only be used for binary events, when in fact it can be applied to events with multiple outcomes.

5. How can one improve their understanding of Bayes' formula?

To improve understanding of Bayes' formula, it is important to have a solid foundation in probability and statistics. It can also be helpful to practice using the formula with different examples and scenarios. Additionally, familiarizing oneself with the different applications and real-world uses of Bayes' formula can also aid in understanding its concepts and principles.

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