Subjective probability
Types of Probability
There are several different types of probability, including:
- Classical probability
- Empirical probability
- Subjective probability
- Axiomatic probability
- Frequentist probability
- Bayesian probability
- Constructive probability
- Algorithmic probability
- Theoretical probability
- Conditional probability
- Joint probability
Subjective probability, also known as personal probability or Bayesian probability, is a type of probability that is based on an individual’s personal belief or knowledge about the likelihood of an event occurring. It is a measure of an individual’s degree of belief or confidence in a certain outcome.
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In contrast to classical probability, which is based on the concept of equally likely outcomes and can be calculated using objective methods such as counting or experimentation, subjective probability is based on an individual’s personal beliefs or knowledge and can be influenced by factors such as personal experience, intuition, and expert opinion.
For example, a meteorologist might use a combination of historical data and current weather patterns to assign a probability of 0.8 (or 80%) that it will rain tomorrow, while a layperson might assign a probability of 0.5 (or 50%) based on the fact that the sky is currently overcast. Both values represent a subjective probability that is based on the individual’s knowledge and belief about the likelihood of rain.
Bayesian probability is a way of updating a probability based on new information, it’s a mathematical framework to update the belief in the light of new data.
Subjective probability can be useful in situations where there is limited information or data available, and it can also be used to incorporate expert opinion or other forms of knowledge that are not easily quantifiable. However, it should be used with caution, as it can be influenced by personal biases or cognitive errors.
In summary, subjective probability is a measure of an individual’s belief or knowledge about the likelihood of an event occurring, and it is based on personal experience, intuition, and expert opinion rather than objective data. Bayesian probability is a way of updating probability based on new information.
Subjective probability formula
The formula for subjective probability is typically represented as:
P(A) = degree of belief or confidence in event A occurring
Where P(A) is the probability of event A occurring, and the degree of belief or confidence is a value between 0 and 1, with 0 indicating that the event is impossible and 1 indicating that the event is certain.
In Bayesian probability, it is common to use the following formula to update the prior probability after new data or information is obtained:
P(A|B) = P(B|A) * P(A) / P(B)
Where P(A|B) is the updated probability of event A occurring given new information B, P(B|A) is the likelihood of observing information B given that event A has occurred, P(A) is the prior probability of event A occurring and P(B) is the probability of observing information B regardless of whether event A has occurred or not.
It’s important to note that these formulas are mathematical representation of how we update our belief based on new information, but the actual probability of an event happening is not known and is uncertain.
Subjective probability, unlike classical probability, does not rely on the notion of equally likely outcomes, it’s based on the belief of the individual.
Subjective probability example with solution
Example: A weather forecaster is asked to predict the probability of rain tomorrow. They have historical data that shows that it has rained on 20 out of the last 50 days. They also know that the forecast for tomorrow is for cloudy skies and a high chance of precipitation.
Solution:
- Prior probability:
The prior probability of rain tomorrow is based on the historical data. The weather forecaster knows that it has rained on 20 out of the last 50 days, so they can calculate the prior probability as: P(Rain) = 20/50 = 0.4 or 40%
- Likelihood:
The likelihood of observing cloudy skies and a high chance of precipitation given that it will rain tomorrow is high. The weather forecaster assigns a likelihood of 0.8 or 80%.
- Updated probability:
Using the Bayesian formula, the updated probability of rain tomorrow can be calculated as:
P(Rain|Cloudy Skies and High Chance of Precipitation) = P(Cloudy Skies and High Chance of Precipitation|Rain) * P(Rain) / P(Cloudy Skies and High Chance of Precipitation)
= 0.8 * 0.4 / P(Cloudy Skies and High Chance of Precipitation)
As P(Cloudy Skies and High Chance of Precipitation) is not known, it can be ignored, and the updated probability becomes:
P(Rain|Cloudy Skies and High Chance of Precipitation) = 0.8 * 0.4 = 0.32 or 32%
So, the weather forecaster’s prediction of the probability of rain tomorrow is 32% based on the historical data and the forecast for cloudy skies and a high chance of precipitation.
It’s important to note that this is an example and the actual probability of rain tomorrow is not known. The forecast is an estimation of the chance of it raining based on the information available.
