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Milds # Posterior probability

In Bayesian statistics, the posterior probability of a random event or an uncertain proposition[clarification needed] is the conditional probability that is assigned[clarification needed] after the relevant evidence or background is taken into account. Similarly, the posterior probability distribution is the probability distribution of an unknown quantity, treated as a random variable, conditional on the evidence obtained from an experiment or survey. "Posterior", in this context, means after taking into account the relevant evidence related to the particular case being examined. For instance, there is a ("non-posterior") probability of a person finding buried treasure if they dig in a random spot, and a posterior probability of finding buried treasure if they dig in a spot where their metal detector rings.

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• ✪ Posterior Probabilities
• ✪ Decision Analysis 5: Posterior (Revised) Probability Calculations
• ✪ Prior and Posterior Distributions
• ✪ Posterior Probability Computation using Bayes Criteria
• ✪ Decision Analysis 4: EVSI - Expected Value of Sample Information

#### Transcription

A conditional probability is the probability of one event B, conditional on the occurrence of another event A. It is denoted by P(B|A), and is read as the probability of B given A, and it can be computed by the following formula. Take the probability of A and B occurring, divided by the probability of A. We can divide by the probability of A because the conditional probability is only used if A actually occurs. In other words, the probability of A is non-zero. Let's look at our first example. Consider the stock price per share of Hooli. The probability of the stock price going up is 90%. This probability will vary with general market conditions. The probability of a good market and the stock price going up is 67.5%. Find the probability of good market given that the stock price went up. So this means that we need to find the conditional probability of a good market given that the stock price went up. We will use the formula that we just discussed. The probability of a good market and the stock price going up is 67.5%. The probability of the stock price going up is 90%. So we have 0.675 divided by 0.9. Therefore, the probability of a good market given that the stock price went up is 75%. Continuing on from Example 1, suppose that the market is good 73.5% of the time. Find the probability of the stock price going up given a good market. Again, we will have to use the conditional probability formula. From Example 1, we saw that the probability of a good market and the stock price going up is 67.5%. The probability of a good market is 73.5%. So we have 0.675 divided by 0.735. Therefore, the probability of the stock price going up given a good market is 91.8%. Now let's have a look at the chain rule which can be used to compute the joint probability of both A and B occurring. The probability of A and B is the probability of A multiplied by the probability of B given that A occurs. Re-arranging this equation, we have the formula for the conditional probability. In the event that B actually occurs, in other words, P(B) is non-zero, we can compute the probability of A given B. From the conditional probability formula, we have that the probability of A given B is equal to the probability of B and A, divided by the probability of B. The probability of B and A is the same as the probability of A and B. Now, using the chain rule we can re-write the probability of A and B as the probability of A multiplied by the probability of B given A. Now we have a formula to compute posterior probabilities. So, given the prior probabilities: the probability of A and the probability of B, and given the conditional probability of B given A, we can compute the posterior probability: the probability of A given B. Here is another example. Consider the stock price per share of Hooli. The probability of it going up is 90%. Given that the stock price went up, the market was good 75% of the time, fair 20% of the time, and bad 5% of the time. When the stock price went down, those numbers were 60%, 30%, and 10% respectively. Use this information to find the probability of the stock price going up given a fair market. From the given information we have that the probability of a good market given that the stock price went up is 75%, the probability of a good market and given that the stock price went down is 60%, and so on. Using the chain rule, we can multiply the prior probabilities by the conditional probabilities to give us the joint probabilities as so. For example, the probability of a good market and the stock price going up is 0.9 times 0.75 which equals to 0.675. If we add up the columns, we should find that they sum up to the probability of the stock price going up and down. This is a good way to check your work. Adding the two should give you 1. If we add up the rows, we will get the probability of each market condition. So the probability of a good market is 73.5%. The probability of a fair market is 21%, and the probability of a bad market is 5.5%. These should also sum up to 1. Now we can use the formula to calculate the posterior probability. The probability of the stock price going up given a fair market is: the probability of the stock price going up times the probability of a fair market given that the stock price went up, divided by the probability of a fair market. Plugging in the values, we have 0.9 times 0.2, divided by 0.21. Therefore, the posterior probability is 85.7%. The rest of the posterior probabilities can be calculated in the following exercise. Remember to pause the video before checking your solutions. Good luck!

## Definition

The posterior probability is the probability of the parameters $\theta$ given the evidence $X$ : $p(\theta |X)$ .

It contrasts with the likelihood function, which is the probability of the evidence given the parameters: $p(X|\theta )$ .

The two are related as follows:

Let us have a prior belief that the probability distribution function is $p(\theta )$ and observations $x$ with the likelihood $p(x|\theta )$ , then the posterior probability is defined as

$p(\theta |x)={\frac {p(x|\theta )}{p(x)}}p(\theta ).$ The posterior probability can be written in the memorable form as

${\text{Posterior probability}}\propto {\text{Likelihood}}\times {\text{Prior probability}}$ .

## Example

Suppose there is a school having 60% boys and 40% girls as students. The girls wear trousers or skirts in equal numbers; all boys wear trousers. An observer sees a (random) student from a distance; all the observer can see is that this student is wearing trousers. What is the probability this student is a girl? The correct answer can be computed using Bayes' theorem.

The event $G$ is that the student observed is a girl, and the event $T$ is that the student observed is wearing trousers. To compute the posterior probability $P(G|T)$ , we first need to know:

• $P(G)$ , or the probability that the student is a girl regardless of any other information. Since the observer sees a random student, meaning that all students have the same probability of being observed, and the percentage of girls among the students is 40%, this probability equals 0.4.
• $P(B)$ , or the probability that the student is not a girl (i.e. a boy) regardless of any other information ($B$ is the complementary event to $G$ ). This is 60%, or 0.6.
• $P(T|G)$ , or the probability of the student wearing trousers given that the student is a girl. As they are as likely to wear skirts as trousers, this is 0.5.
• $P(T|B)$ , or the probability of the student wearing trousers given that the student is a boy. This is given as 1.
• $P(T)$ , or the probability of a (randomly selected) student wearing trousers regardless of any other information. Since $P(T)=P(T|G)P(G)+P(T|B)P(B)$ (via the law of total probability), this is $P(T)=0.5\times 0.4+1\times 0.6=0.8$ .

Given all this information, the posterior probability of the observer having spotted a girl given that the observed student is wearing trousers can be computed by substituting these values in the formula:

$P(G|T)={\frac {P(T|G)P(G)}{P(T)}}={\frac {0.5\times 0.4}{0.8}}=0.25.$ An intuitive way to solve this is to assume the school has N students. Number of boys = 0.6N and number of girls = 0.4N. If N is sufficiently large, total number of trouser wearers = 0.6N+ 50% of 0.4N. And number of girl trouser wearers = 50% of 0.4N. Therefore, in the population of trousers, girls are (50% of 0.4N)/(0.6N+ 50% of 0.4N) = 25%. In other words, if you separated out the group of trouser wearers, a quarter of that group will be girls. Therefore, if you see trousers, the most you can deduce is that you are looking at a single sample from a subset of students where 25% are girls. And by definition, chance of this random student being a girl is 25%. Every Bayes theorem problem can be solved in this way .

## Calculation

The posterior probability distribution of one random variable given the value of another can be calculated with Bayes' theorem by multiplying the prior probability distribution by the likelihood function, and then dividing by the normalizing constant, as follows:

$f_{X\mid Y=y}(x)={f_{X}(x){\mathcal {L}}_{X\mid Y=y}(x) \over {\int _{-\infty }^{\infty }f_{X}(u){\mathcal {L}}_{X\mid Y=y}(u)\,du}}$ gives the posterior probability density function for a random variable $X$ given the data $Y=y$ , where

• $f_{X}(x)$ is the prior density of $X$ ,
• ${\mathcal {L}}_{X\mid Y=y}(x)=f_{Y\mid X=x}(y)$ is the likelihood function as a function of $x$ ,
• $\int _{-\infty }^{\infty }f_{X}(u){\mathcal {L}}_{X\mid Y=y}(u)\,du$ is the normalizing constant, and
• $f_{X\mid Y=y}(x)$ is the posterior density of $X$ given the data $Y=y$ .

## Credible interval

Posterior probability is a conditional probability conditioned on randomly observed data. Hence it is a random variable. For a random variable, it is important to summarize its amount of uncertainty. One way to achieve this goal is to provide a credible interval of the posterior probability.

## Classification

In classification, posterior probabilities reflect the uncertainty of assessing an observation to particular class, see also Class membership probabilities. While statistical classification methods by definition generate posterior probabilities, Machine Learners usually supply membership values which do not induce any probabilistic confidence. It is desirable to transform or re-scale membership values to class membership probabilities, since they are comparable and additionally more easily applicable for post-processing.