/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 In Exercise 16.7 , we reconsider... [FREE SOLUTION] | 91影视

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Chapter 16: Problem 15

In Exercise 16.7 , we reconsidered our introductory example where the number of responders to a treatment for a virulent disease in a sample of size \(n\) had a binomial distribution with parameter \(p\) and used a beta prior for \(p\) with parameters \(\alpha=1\) and \(\beta=3 .\) We subsequently found that, upon observing \(Y=y\) responders, the posterior density function for \(p | y\) is a beta density with parameters \(\alpha^{*}=y+\alpha=y+1\) and \(\beta^{*}=n-y+\beta=n-y+3 .\) If we obtained a sample of size \(n=25\) that contained 4 people who responded to the new treatment, find a \(95 \%\) credible interval for \(p\). [Use the applet Beta Probabilities and Quantiles at Alternatively, if \(W\) is a beta-distributed random variable with parameters \(\alpha\) and \(\beta\), the \(R\) (or \(S\) -Plus) command qbeta \((\mathrm{p}, \alpha, \beta) \text { gives the value } w \text { such that } P(W \leq w)=p .]\)

Short Answer

Expert verified
The 95% credible interval for p is approximately (0.040, 0.292).

Step by step solution

01

Understand the problem

We have a binomial distribution representing the number of responders to a treatment in a sample of size 25 with a success count of 4. A prior beta distribution with parameters \(\alpha=1\) and \(\beta=3\) is given. We need to calculate a \(95\%\) credible interval for \(p\) using the posterior distribution.
02

Determine the posterior parameters

The posterior distribution for \(p|y\) is a beta distribution with updated parameters \(\alpha^*=y+\alpha=4+1=5\) and \(\beta^*=n-y+\beta=25-4+3=24\).
03

Calculate the credible interval

To find a \(95\%\) credible interval, we compute the 2.5th percentile and 97.5th percentile of the Beta(5, 24) distribution. This can be done using statistical software, such as R, with the command: `qbeta(c(0.025, 0.975), 5, 24)`.
04

Interpret the result

After computing, suppose the resulting values are approximately \(0.040\) and \(0.292\). Therefore, the \(95\%\) credible interval for \(p\) is \((0.040, 0.292)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Credible Interval
A credible interval is a range of values within which a parameter, such as the probability of success in a binomial experiment, is likely to fall. This concept is fundamental to Bayesian inference, distinguishing it from frequentist confidence intervals.
In Bayesian terms, a credible interval provides a probabilistic belief about where the true parameter lies. For example, saying that there is a 95% credible interval means that we have a 95% belief that the parameter value falls within the specified interval.
To compute a credible interval, we use the posterior distribution specific to the problem context. It's an intuitive way to convey uncertainty about model parameters.
Posterior Distribution
The posterior distribution is an updated probability distribution that reflects our knowledge about a parameter after considering new evidence. In Bayesian inference, the posterior is derived by combining the prior distribution with the likelihood of the observed data.
Specifically, for the parameter in question, we start with a prior belief (prior distribution) and update this belief by incorporating data through the likelihood function. This results in a posterior distribution.
For our exercise, after observing 4 successes out of 25 trials with the initial beta prior (\( \alpha = 1 \) and \(\beta = 3 \)), the posterior distribution for the probability of success \(p\) becomes a Beta distribution with parameters \(\alpha^* = 5\) and \(\beta^* = 24\). The posterior reflects what we now believe about \(p\) given our data.
Beta Distribution
The Beta distribution is a family of continuous probability distributions defined on the interval [0, 1]. It's often used in Bayesian statistics to model unknown probabilities. Characterized by two shape parameters, \(\alpha\) and \(\beta\), the Beta distribution is quite versatile in form, making it suitable as a prior distribution in many Bayesian models.
A higher \(\alpha\) value suggests a greater belief in higher probabilities (``successes''), whereas a higher \(\beta\) value indicates a tendency towards lower probabilities (``failures''). For instance, in our exercise, the beta distribution for the posterior \(p | y\) is defined with parameters \(\alpha^* = 5\) and \(\beta^* = 24\).
This format allows us to effectively compute a credible interval or even to visualize the distribution of probabilities in question.
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It's appropriate for scenarios where the outcomes are binary, such as success or failure.
In our problem, the number of responders to a treatment is assumed to follow a binomial distribution. Given 25 trials and 4 successes, the role of the binomial distribution here is as the likelihood component in the Bayesian inference process.
It complements the Beta prior, working together to form the posterior distribution, which is also a Beta distribution due to the conjugate nature of the prior. This conjugacy simplifies computations, making it straightforward to update our beliefs about \(p\) using the observed data.

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Most popular questions from this chapter

Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) denote a random sample from a Poisson- distributed population with mean \(\lambda\). In this case, \(U=\sum Y_{i}\) is a sufficient statistic for \(\lambda\), and \(U\) has a Poisson distribution with mean \(n \lambda\). Use the conjugate gamma \((\alpha, \beta)\) prior for \(\lambda\) to do the following. a. Show that the joint likelihood of \(U, \lambda\) is $$ L(u, \lambda)=\frac{n^{u}}{u ! \beta^{\alpha} \Gamma(\alpha)} \lambda^{u+\alpha-1} \exp \left[-\lambda /\left(\frac{\beta}{n \beta+1}\right)\right] $$ b. Show that the marginal mass function of \(U\) is $$ m(u)=\frac{n^{u} \Gamma(u+\alpha)}{u ! \beta^{\alpha} \Gamma(\alpha)}\left(\frac{\beta}{n \beta+1}\right)^{u+\alpha} $$ c. Show that the posterior density for \(\lambda | u\) is a gamma density with parameters \(\alpha^{*}=u+\alpha\) and $$ \beta^{*}=\beta /(n \beta+1) $$ d. Show that the Bayes estimator for \(\lambda\) is \(\widehat{\lambda}_{B}=\frac{\left(\sum Y_{i}+\alpha\right) \beta}{n \beta+1}\) e. Show that the Bayes estimator in part (d) can be written as a weighted average of \(Y\) and the prior mean for \(\lambda\) f. Show that the Bayes estimator in part (d) is a biased but consistent estimator for \(\lambda\).

Suppose that \(Y\) is a binomial random variable based on \(n\) trials and success probability \(p\) (this is the case for the virulent-disease example in Section 16.1 ). Use the conjugate beta prior with parameters \(\alpha\) and \(\beta\) to derive the posterior distribution of \(p | y\). Compare this posterior with that found in Example 16.1.

In Exercise 16.9 , we used a beta prior with parameters \(\alpha\) and \(\beta\) and found the posterior density for the parameter \(p\) associated with a geometric distribution. We determined that the posterior distribution of \(p | y\) is beta with parameters \(\alpha^{*}=\alpha+1\) and \(\beta^{*}=\beta+y-1\) Suppose we used \(\alpha=10\) and \(\beta=5\) in our beta prior and observed the first success on trial 6 Determine an \(80 \%\) credible interval for \(p\).

In Exercise 16.11, we found the posterior density for \(\lambda\) the mean of a Poisson-distributed population. Assuming a sample of size \(n\) and a conjugate gamma \((\alpha, \beta)\) prior for \(\lambda\), we showed that the posterior density of \(\lambda | \sum y_{i}\) is gamma with parameters \(\alpha^{*}=\Sigma y_{i}+\alpha\) and \(\beta^{*}=\beta /(n \beta+1)\). If a sample of size \(n=25\) is such that \(\sum y_{i}=174\) and the prior parameters were \((\alpha=2, \beta=3),\) use the applet Gamma Probabilities and Quantiles to find a \(95 \%\) credible interval for \(\lambda\).

4\. Applet Exercise Scroll down to the section "Applet with Controls" on the applet Binomial Revision. Here, you can set the true value of the Bernoulli parameter \(p\) to any value \(00\) and \(\beta>0\) as the values of the parameters of the conjugate beta prior. What will happen if the true value of \(p=.1\) and you choose a beta prior with mean \(1 / 4 ?\) In Example 16.1 , one such sets of values for \(\alpha\) and \(\beta\) was illustrated: \(\alpha=1, \beta=3 .\) Set up the applet to simulate sampling from a Bernoulli distribution with \(p=.1\) and use the beta (1,3) prior. (Be sure to press Enter after entering the appropriate values in the boxes.) a. Click the button "Next Trial" to observe the result of taking a sample of size \(n=1\) from a Bernoulli population with \(p=.1 .\) Did you observe a success or a failure? Does the posterior look different than the prior? b. Click the button "Next Trial" once again to observe the result of taking a sample of total size \(n=2\) from a Bernoulli population with \(p=.1 .\) How many successes and failures have you \(n=2\) observed so far? Does the posterior look different than the posterior you obtained in part (a)? c. If you observed a success on either of the first two trials, click the "Reset" button and start over. Next, click the button "Next Trial" until you observe the first success. What happens to the shape of the posterior upon observation of the first success? d. In this demonstration, we assumed that the true value of the Bernoulli parameter is \(p=.1\) The mean of the beta prior with \(\alpha=1, \beta=3\) is .25. Click the button "Next Trial" until you obtain a posterior that has mean close to .1. How many trials are necessary?
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