The number of defects per 10 meters of cloth produced by a...

The number of defects per 10 meters of cloth produced by a weaving machine has the Poisson distribution with mean $\mu$. You examine 100 meters of cloth produced by the machine and observe 71 defects. (a) Your prior belief about $\mu$ is that it has mean 6 and standard deviation 2 . Find a gamma $(r, v)$ prior that matches your prior belief. (b) Find the posterior distribution of $\mu$ given that you observed 71 defects in 100 meters of cloth. (c) Calculate a $95 \%$ Bayesian credible interval for $\mu$.

The number of defects per 10 meters of cloth produced by a weaving machine has the Poisson distribution with mean $\mu$. You examine 100 meters of cloth produced by the machine and observe 71 defects.
(a) Your prior belief about $\mu$ is that it has mean 6 and standard deviation 2 . Find a gamma $(r, v)$ prior that matches your prior belief.
(b) Find the posterior distribution of $\mu$ given that you observed 71 defects in 100 meters of cloth.
(c) Calculate a $95 \%$ Bayesian credible interval for $\mu$.

Key Concepts

Bayesian Credible Interval

A Bayesian credible interval is an interval derived from the posterior distribution within which the parameter is believed to lie with a certain probability, such as 95%. Unlike frequentist confidence intervals, credible intervals are interpreted directly in terms of the probability of the parameter falling within the interval, providing a straightforward probabilistic statement about the parameter based on the observed data and prior beliefs.

Bayesian Inference

Bayesian inference is a framework for statistical inference in which prior beliefs about the parameters are updated in the light of new evidence, by applying Bayes’ theorem to combine the prior distribution with the likelihood derived from observed data. This approach results in a posterior distribution that encapsulates both the prior knowledge and the information provided by the new data.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, provided these events occur with a known constant mean rate and independently of the time since the last event. In Bayesian analysis, it is often used as the likelihood function when modeling count data, serving as a natural choice when dealing with rare events over a continuous interval.

Gamma Distribution

The Gamma distribution is a continuous probability distribution that is commonly used as a prior in Bayesian analysis for positive continuous parameters, such as rates in Poisson processes. It is characterized by its shape and rate (or scale) parameters, and its flexibility makes it suitable for representing a variety of beliefs about the parameter’s possible values. Its moments (mean and variance) are easily expressed in terms of these parameters, which allows for straightforward matching of prior beliefs.

Conjugate Prior

A conjugate prior is a prior distribution that, when combined with a given likelihood through Bayes' theorem, results in a posterior distribution that is in the same family as the prior. This property greatly simplifies Bayesian updating because it allows for analytic solutions and easy interpretation of the updated parameters. For Poisson likelihoods, the Gamma distribution serves as a conjugate prior for the rate parameter.

Transcript

00:01 In this exercise, we have the number of customers arriving in a one -hour period, modeled by a poisson distribution with mean mu.

00:13 Now, it's believed that the mean is 15.

00:17 So for part a, we're asked to define a prior distribution for mean from the gamma family of distributions, such that the mean is 15, and the standard deviation is 5.

00:41 Now for a gamma distribution, the mean is equal to alpha times beta.

00:49 Remember the gamma distribution has this form.

01:17 And so returning to the question, then the standard deviation for a gamma distribution is equal to the square root of alpha times beta.

01:27 So we can solve for alpha and beta here if we take the mean divided by the standard deviation.

01:35 It gives the square root of alpha is equal to 3, which means that we can solve for alpha in beta here.

01:40 Means that alpha is equal to 9.

01:46 And then using either equation, we can solve for beta.

01:57 That'll be 15 over 9.

02:05 So that means our prior distribution, function of mu, is we just have to give it the gamma form, gamma distribution form, using our solved alpha and beta.

02:43 So that answer is part a.

02:46 Now for part b, we have the number of customers that have arrived in 10 randomly one hour, randomly selected one hour intervals.

02:56 And based on this new information, we're asked to find the posterior distribution for mu.

03:03 So remember the arrivals are modeled by a pluson distribution...

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