Question

3 MLE and MAP [15 pts] Consider the Poisson distribution, which models the count of events occurring within a fixed interval. If X ~ Poisson(λ), the probability mass function is: $$P(X = x | \lambda) = \frac{\lambda^x e^{-\lambda}}{x!}$$ where E[X] = λ. Assume we have n i.i.d. samples D = {$x_1, ..., x_n$} from Poisson(λ). Question 3.1 [4 points] Write down the likelihood function for D given λ and find the Maximum Likelihood Estimate (MLE) of λ. Question 3.2 [6 points] Now, assume λ has a Gamma prior distribution with parameters (α, β), where: $$P(\lambda | \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} \lambda^{\alpha - 1} e^{-\beta\lambda}$$ where Γ(α) = (α - 1)! (here we assume α is a positive integer). Compute the posterior distribution for λ given D and determine if the Gamma distribution is a conjugate prior for the Poisson distribution. Question 3.3 [5 points] Derive an expression for the Maximum A Posteriori (MAP) estimate of λ under the Gamma(α, β) prior.

Name: 3 mle and map 15 pts consider the poisson distribution which models the count of events occurring within a fixed interval if x poisson the probability mass function is px x lambda fraclambda 46359
Uploaded: 2024-10-09T23:56:49-08:00
Duration: 6 min 20 s
Channel: Jon Southam
Description: 3 mle and map 15 pts consider the poisson distribution which models the count of events occurring within a fixed interval if x poisson the probability mass function is px x lambda fraclambda 46359

          3 MLE and MAP [15 pts]
Consider the Poisson distribution, which models the count of events occurring within a fixed interval. If
X ~ Poisson(λ), the probability mass function is:
$$P(X = x | \lambda) = \frac{\lambda^x e^{-\lambda}}{x!}$$
where E[X] = λ. Assume we have n i.i.d. samples D = {$x_1, ..., x_n$} from Poisson(λ).
Question 3.1 [4 points] Write down the likelihood function for D given λ and find the Maximum Likelihood Estimate (MLE) of λ.
Question 3.2 [6 points] Now, assume λ has a Gamma prior distribution with parameters (α, β), where:
$$P(\lambda | \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} \lambda^{\alpha - 1} e^{-\beta\lambda}$$
where Γ(α) = (α - 1)! (here we assume α is a positive integer). Compute the posterior distribution for λ given D and determine if the Gamma distribution is a conjugate prior for the Poisson distribution.
Question 3.3 [5 points] Derive an expression for the Maximum A Posteriori (MAP) estimate of λ under the Gamma(α, β) prior.

$3 mle and map 15 pts consider the poisson distribution which models the count of events occurring within a fixed interval if x poisson the probability mass function is px x lambda fraclambda 46359$

Added by Nerea Q.

Question

Please give Ace some feedback