Suppose that Y is an exponential random variable with parameter (and mean) equal to 2. Find the MAP estimate of Y, given that there were exactly 10 red flashes.
To find the Maximum A Posteriori (MAP) estimate of the exponential random variable Y, we need to consider the prior distribution of Y and the observed data.
In this case, the prior distribution of Y is exponential with parameter (and mean) equal to 2. Let's denote it as Y ~ Exp(2).
The likelihood function represents the probability of observing the data given the parameter of the random variable, and in this case, it is the probability of observing exactly 10 red flashes given Y. Since Y is an exponential random variable, the probability density function (PDF) of Y is given by:
f(y|λ) = λ * e^(-λy)
where λ is the parameter of the exponential distribution.
The likelihood of observing exactly 10 red flashes given Y can be written as:
L(λ) = f(10|λ) = λ * e^(-10λ)
Next, we need to find the posterior distribution by combining the prior and likelihood. The posterior distribution represents our updated belief about the parameter after considering the observed data. According to Bayes' theorem, the posterior distribution can be calculated as:
P(λ|10) ∝ L(λ) * P(λ)
where P(λ) is the prior distribution of λ.
Since we know that the prior distribution of Y is exponential with parameter (and mean) equal to 2, we can write:
P(λ) = Exp(2) = 2 * e^(-2λ)
Substituting the likelihood function and prior distribution into Bayes' theorem, we get:
P(λ|10) ∝ (λ * e^(-10λ)) * (2 * e^(-2λ))
To find the MAP estimate of Y, we need to find the value of λ that maximizes the posterior distribution. Let's denote this value as λ_map.
To maximize the posterior distribution, we can take the natural logarithm (ln) of both sides, because the logarithm is a monotonically increasing function. This gives:
ln(P(λ|10)) ∝ ln((λ * e^(-10λ)) * (2 * e^(-2λ)))
Expanding the terms and removing constants, we have:
ln(P(λ|10)) ∝ ln(2λ * e^(-2λ) * λ * e^(-10λ))
Simplifying further:
ln(P(λ|10)) ∝ ln(2λ * λ * e^(-2λ) * e^(-10λ))
ln(P(λ|10)) ∝ ln(2) + ln(λ) + ln(λ) + (-2λ) + (-10λ)
Taking the derivative of ln(P(λ|10)) with respect to λ and setting it equal to zero, we can find the value of λ that maximizes the posterior distribution:
d[ln(P(λ|10))]/dλ = 0
Solving this equation will give us the MAP estimate of Y.
To find the maximum a posteriori (MAP) estimate of the exponential random variable Y, given that there were exactly 10 red flashes, we need to calculate the posterior probability distribution and find the value that maximizes it.
The posterior probability distribution can be calculated using Bayes' theorem:
P(Y|X) = (P(X|Y) * P(Y)) / P(X)
Where:
- P(Y|X) is the posterior probability distribution
- P(X|Y) is the likelihood function
- P(Y) is the prior probability distribution
- P(X) is the evidence
In this case, we are given that Y follows an exponential distribution with a parameter (and mean) equal to 2, so we can assign a prior distribution to Y:
P(Y) = Exp(λ), where λ = 1/2
The likelihood function can be calculated using the exponential distribution:
P(X|Y) = (λ * exp(-λX))
Where X is the given data, in this case, X = 10.
To calculate the evidence, we integrate the joint probability distribution over all possible values of Y:
P(X) = ∫[0,∞] (P(X|Y) * P(Y)) dY
Let's calculate the evidence first:
P(X) = ∫[0,∞] (λ * exp(-λX) * exp(-λY)) dY
= λ * exp(-λX) * ∫[0,∞] (exp(-λY)) dY
= λ * exp(-λX) * [(-1/λ) * exp(-λY)] [Limits: 0 to ∞]
= (1/2) * exp(-1/2 * 10)
Now, we can calculate the posterior probability distribution:
P(Y|X) = (P(X|Y) * P(Y)) / P(X)
= (λ * exp(-λX) * exp(-λY)) / P(X)
= (1/2 * exp(-1/2 * 10) * exp(-Y/2)) / ((1/2) * exp(-1/2 * 10))
= exp(-Y/2)
Since we need to find the MAP estimate, we are interested in finding the value of Y that maximizes the posterior probability distribution, which is exp(-Y/2).
As Y approaches infinity, exp(-Y/2) approaches 0, so to maximize the probability, Y should be minimized.
Therefore, the MAP estimate of Y given that there were exactly 10 red flashes is Y = 0.