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Chapter 3: Problem 29

A drug is assumed to be effective with an unknown probability \(p\). To estimate \(p\) the drug is given to \(n\) patients. It is found to be effective for \(m\) patients. The method of maximum likelihood for estimating \(p\) states that we should choose the value for \(p\) that gives the highest probability of getting what we got on the experiment. Assuming that the experiment can be considered as a Bernoulli trials process with probability \(p\) for success, show that the maximum likelihood estimate for \(p\) is the proportion \(m / n\) of successes.

Short Answer

Expert verified
The MLE of \(p\) is \(\frac{m}{n}\), the proportion of effective outcomes.

Step by step solution

01

Understand Bernoulli Trials

In a Bernoulli trials process, each trial results in a success with probability \(p\) and failure with probability \(1-p\). If there are \(n\) independent trials and \(m\) of them are successes, this constitutes a series of Bernoulli trials.
02

Define the Likelihood Function

For a given \(p\), the probability of observing exactly \(m\) successes in \(n\) trials, assuming a Bernoulli process, is given by the binomial distribution: \[ L(p) = \binom{n}{m} p^m (1-p)^{n-m} \] where \(\binom{n}{m}\) is the binomial coefficient which is constant for fixed \(m\) and \(n\).
03

Set Up the Maximization Problem

We need to maximize the likelihood function \(L(p)\) with respect to \(p\). Since \(\binom{n}{m}\) is constant, maximizing \(L(p)\) is equivalent to maximizing the logarithm of the likelihood function, which simplifies the expression.
04

Calculate the Log-Likelihood Function

The log-likelihood function is given by: \[ \log L(p) = \log \left( \binom{n}{m} \right) + m \log(p) + (n-m) \log(1-p) \] We can ignore the first term as it does not depend on \(p\) and focus on maximizing: \[ m \log(p) + (n-m) \log(1-p) \]
05

Derive the Critical Point

To find the maximum, take the derivative of the log-likelihood with respect to \(p\) and set it to zero: \[ \frac{d}{dp} \left[ m \log(p) + (n-m) \log(1-p) \right] = \frac{m}{p} - \frac{n-m}{1-p} = 0 \]
06

Solve for \(p\)

Setting the derivative equal to zero gives:\[ \frac{m}{p} = \frac{n-m}{1-p} \] Cross-multiply to solve for \(p\): \[ m(1-p) = (n-m)p \]\[ m - mp = np - mp \]\[ m = np \]\[ p = \frac{m}{n} \] Thus, the value of \(p\) that maximizes the likelihood is \(\frac{m}{n}\).
07

Conclusion

Therefore, the maximum likelihood estimate (MLE) of the probability \(p\) of the drug being effective is the proportion of successes, \(\frac{m}{n}\). This is the result we wanted to show.

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Key Concepts

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

Bernoulli Trials
When we consider each application of the drug to a patient as an individual experiment, we are describing what is known as a Bernoulli trial. Bernoulli trials are a fundamental concept in probability and statistics. They are essentially experiments or processes that result in one of two outcomes: success or failure.
For the drug scenario, success is when the drug proves effective, and failure is when it does not work.
  • Each trial (or patient) is independent, which means the outcome of one does not affect another.
  • The probability of success is the same for each trial, denoted by \( p \).
  • The probability of failure is then \( 1-p \).
Collectively, a series of such Bernoulli trials can provide a sample data set, allowing us to estimate the unknown probability \( p \) of success.
Binomial Distribution
The binomial distribution comes into play when we want to model the number of successes in a fixed number of independent Bernoulli trials. For our problem, the binomial distribution helps describe the likelihood of achieving exactly \( m \) successes in \( n \) trials, given the probability of success \( p \).
The formula for the binomial probability is:\[L(p) = \binom{n}{m} p^m (1-p)^{n-m}\]Here, \( \binom{n}{m} \) is the binomial coefficient and represents the number of ways to arrange \( m \) successes in \( n \) trials. Importantly, for our maximum likelihood estimation, we don't need to compute \( \binom{n}{m} \) as it remains constant for fixed \( n \) and \( m \).
  • The exponent \( m \) of \( p \) reflects the number of successful outcomes.
  • The exponent \( n-m \) of \( (1-p) \) accounts for the trials that resulted in failure.
This distribution forms the backbone for calculating the likelihood that our observed number of drug successes could have occurred under various values of \( p \).
Log-Likelihood Function
The log-likelihood function is a transformation we use to make our calculations simpler, particularly when maximizing the likelihood. By taking the logarithm of the likelihood function, we convert products into sums, which are much easier to differentiate.
The log-likelihood formula for our binomial setup becomes:\[\log L(p) = \log \left( \binom{n}{m} \right) + m \log(p) + (n-m) \log(1-p)\]Because \( \log \left( \binom{n}{m} \right) \) is constant for our scenario, it doesn't influence the maximization with respect to \( p \). We focus only on:\[m \log(p) + (n-m) \log(1-p)\]To find the \( p \) that maximizes this expression, we derive it with respect to \( p \) and set the derivative to zero. The solution shows us that the maximum likelihood estimate (MLE) for \( p \) is the ratio \( \frac{m}{n} \), indicating the observed proportion of effectiveness amongst the trials. This transformation and simplification allow for a more straightforward mathematical optimization process, providing clear insights into how we infer the probability \( p \) from the experimental data.
  • Taking derivatives simplifies the equation further for solving.
  • Setting derivatives to zero finds the critical points for maxima.

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Most popular questions from this chapter

Charles claims that he can distinguish between beer and ale 75 percent of the time. Ruth bets that he cannot and, in fact, just guesses. To settle this, a bet is made: Charles is to be given ten small glasses, each having been filled with beer or ale, chosen by tossing a fair coin. He wins the bet if he gets seven or more correct. Find the probability that Charles wins if he has the ability that he claims. Find the probability that Ruth wins if Charles is guessing.

If a set has \(2 n\) elements, show that it has more subsets with \(n\) elements than with any other number of elements.

Find a formula for the probability that among a set of \(n\) people, at least two have their birthdays in the same month of the year (assuming the months are equally likely for birthdays).

Barbara Smith is interviewing candidates to be her secretary. As she interviews the candidates, she can determine the relative rank of the candidates but not the true rank. Thus, if there are six candidates and their true rank is \(6,1,4,2,3,5,\) (where 1 is best) then after she had interviewed the first three candidates she would rank them \(3,1,2 .\) As she interviews each candidate, she must either accept or reject the candidate. If she does not accept the candidate after the interview, the candidate is lost to her. She wants to decide on a strategy for deciding when to stop and accept a candidate that will maximize the probability of getting the best candidate. Assume that there are \(n\) candidates and they arrive in a random rank order. (a) What is the probability that Barbara gets the best candidate if she interviews all of the candidates? What is it if she chooses the first candidate? (b) Assume that Barbara decides to interview the first half of the candidates and then continue interviewing until getting a candidate better than any candidate seen so far. Show that she has a better than 25 percent chance of ending up with the best candidate.

Let \(n(\bmod m)\) denote the remainder when the integer \(n\) is divided by the integer \(m\). Write a computer program to compute the numbers \(\left(\begin{array}{c}n \\ j\end{array}\right)(\bmod m)\) where \(\left(\begin{array}{l}n \\ j\end{array}\right)\) is a binomial coefficient and \(m\) is an integer. You can do this by using the recursion relations for generating binomial coefficients, doing all the arithmetic using the basic function \(\bmod (n, m)\). Try to write your program to make as large a table as possible. Run your program for the cases \(m=2\) to 7 . Do you see any patterns? In particular, for the case \(m=2\) and \(n\) a power of 2 , verify that all the entries in the \((n-1)\) st row are 1 . (The corresponding binomial numbers are odd.) Use your pictures to explain why this is true.
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