Suppose that X is an exponential random variable with parameter (and mean) equal to 1. Find the MAP estimate of X , given that there were exactly 5 blue flashes.
Why 5/2?
Why did the statistician bring a ladder to the party? Because they wanted to estimate the exponential random variable's mean in a flash! Now, let's calculate the MAP estimate.
The Maximum A Posteriori (MAP) estimate is found by maximizing the posterior probability given the observed data. In this case, we want to find the MAP estimate of X, given that there were exactly 5 blue flashes.
To calculate the MAP estimate, we need to consider the prior probability and the likelihood function.
The prior probability is the probability distribution of X before observing the data. In this case, since X is an exponential random variable with a mean of 1, we can use the exponential distribution with parameter 1 as our prior.
The likelihood function represents the probability of observing the data given a specific value of X. Since there were exactly 5 blue flashes, the likelihood function is equal to the exponential probability density function (PDF) evaluated at 5.
To maximize the posterior probability, we need to find the value of X that maximizes the product of the prior and likelihood functions. This can be done by finding the value of X that maximizes the logarithm of the posterior probability.
But jokes aside - since we can't carry on mathematical calculations here, it's best to consult an appropriate statistical software or consult a statistician for the precise MAP estimate in this case with the given information. They're the real wizards of probability!
To find the Maximum a Posteriori (MAP) estimate of X, we need to calculate the posterior probability density function (PDF) of X given the observed data.
Let's denote X as the exponential random variable with parameter θ, and let's assume a prior distribution for θ. Since X has a mean of 1, a suitable choice for the prior distribution is the gamma distribution with shape parameter α = 1 and rate parameter β = 1/1 = 1.
The posterior PDF of X given the observed data can be calculated using Bayes' theorem:
f(θ|x) = (f(x|θ) * f(θ)) / ∫(f(x|θ) * f(θ)) dθ
where f(x|θ) is the likelihood function and f(θ) is the prior distribution.
In this case, the likelihood function f(x|θ) is given by the exponential distribution with parameter θ:
f(x|θ) = θ * exp(-θx)
The prior distribution f(θ) is the gamma distribution:
f(θ) = θ^(α-1) * exp(-βθ) / Γ(α)
where Γ(α) is the gamma function.
Now, substituting these values into Bayes' theorem, we have:
f(θ|x) = (θ * exp(-θx) * θ^(α-1) * exp(-βθ)) / ∫(θ * exp(-θx) * θ^(α-1) * exp(-βθ)) dθ
Simplifying further:
f(θ|x) = θ^α * exp(-θ(x + β)) / ∫(θ^α * exp(-θ(x + β))) dθ
Now, we can compute the normalization constant by integrating the denominator of the posterior PDF over all possible values of θ.
∫(θ^α * exp(-θ(x + β))) dθ = Γ(α+1) / (x + β)^(α+1)
Plugging back into the posterior PDF equation:
f(θ|x) = (θ^α * exp(-θ(x + β))) / (Γ(α+1) / (x + β)^(α+1))
= ((x + β)^(α+1)) * θ^α * exp(-θ(x + β)) / Γ(α+1)
Now we can identify the distribution of X as the gamma distribution with updated parameters. In this case, α = α + n, where n is the number of observed blue flashes, and β remains the same.
So, the posterior distribution of X is:
f(θ|x) = ((x + β)^(1+5)) * θ^1 * exp(-θ(x + β)) / Γ(1+5)
= (x + 1)^6 * θ * exp(-θ(x + 1)) / 5!
The MAP estimate of X is the value of X that maximizes the posterior distribution. Therefore, we need to find the value of X that maximizes the expression (x + 1)^6 * θ * exp(-θ(x + 1)).
To find this maximum value, we can take the derivative of the expression with respect to X and set it equal to zero. However, since we are interested in the MAP estimate of X, we need to find the value of X that maximizes the expression given the observed data (n=5 blue flashes).
So, to find the MAP estimate of X, we need to find the value of X that maximizes (x + 1)^6 * θ * exp(-θ(x + 1)) given that there were exactly 5 blue flashes.
To find the maximum a posteriori (MAP) estimate of a random variable, we need to compute the posterior probability distribution and then determine the value that maximizes this distribution.
In this case, we want to find the MAP estimate of the exponential random variable X, given that there were exactly 5 blue flashes. The exponential random variable X has a parameter (and mean) equal to 1.
To proceed, we need to define the prior distribution. Let's assume a Gamma(α, β) prior for X, where α and β are hyperparameters to be determined. The Gamma distribution is a conjugate prior for the exponential distribution, meaning that the posterior distribution will have the same form as the prior.
Now let's compute the posterior distribution. According to Bayes' theorem:
Posterior ∝ Likelihood × Prior
The likelihood of observing exactly 5 blue flashes given X follows an exponential distribution with parameter 1. Therefore, the likelihood function can be written as:
Likelihood = f(X|5) = λ * exp(-λ * 5)
where λ is the rate parameter, which equals 1 for an exponential random variable with mean 1.
Given the prior distribution as Gamma(α, β), the probability density function (pdf) of the prior distribution is:
Prior = f(X) = (β^α / Γ(α)) * X^(α-1) * exp(-βX)
where Γ(α) is the gamma function.
Now, multiply the likelihood and prior to obtain the unnormalized posterior distribution:
Posterior ∝ Likelihood × Prior
∝ λ * exp(-λ * 5) * (β^α / Γ(α)) * X^(α-1) * exp(-βX)
To find the MAP estimate, we need to maximize the posterior distribution. Taking the logarithm of the unnormalized posterior, the problem can be simplified to maximizing the log of the posterior:
log(Posterior) = log(λ) - λ * 5 + log(β^α / Γ(α)) + (α-1) * log(X) - β * X
To differentiate with respect to X and set it to zero, we can find the optimal X value:
d log(Posterior) / dX = 0
Simplifying the equation and solving for X will give us the MAP estimate for X.
Note: The hyperparameters α and β of the prior distribution need to be chosen based on prior knowledge or specific assumptions. Without additional information, we cannot determine their values, and thus, we cannot find the exact MAP estimate.