The process by which certain pulses are emitted in a human body is modeled as a Poisson process. In other words, the times between adjacent pulses are i.i.d. random variables which are exponentially distributed with the arrival rate . A healthy patient is characterized by the rate , while an unhealthy one is characterized by . measurements of inter-pulse times are taken from a patient, and a healthy/unhealthy decision has to be made. However, the values of and are not known as they are in fact random. Specifically, each is assumed to be exponentially distributed, with mean value , and with mean value . With this model, the likelihood ratio is replaced by the Bayesian factor. The decision threshold is set to . observations are made on a patient, yielding the measured values . If and , which of the following represents the Bayesian decision rule (numbers are rounded):

Question

The process by which certain pulses are emitted in a human body is modeled as a Poisson process. In other words, the times  between adjacent pulses are i.i.d. random variables which are exponentially distributed with the arrival rate . A healthy patient is characterized by the rate , while an unhealthy one is characterized by .  measurements of inter-pulse times are taken from a patient, and a healthy/unhealthy decision has to be made. However, the values of  and  are not known as they are in fact random. Specifically, each is assumed to be exponentially distributed,  with mean value , and  with mean value . With this model, the likelihood ratio is replaced by the Bayesian factor. The decision threshold is set to .  observations are made on a patient, yielding the measured values . If  and , which of the following represents the Bayesian decision rule (numbers are rounded):

BlackTom AI · Accepted Answer

The provided data include a question about a Bayesian decision rule for classifying a patient as healthy or unhealthy based on inter-pulse times, where the model assumes a Poisson process with exponentially distributed inter-arrival times. However, the input also shows that the answer options are empty, so there is no concrete choice to evaluate. I’ll first restate what is given and then outline the general Bayesian decision framework and what the rule would look like in this setup.

Restating the problem context:
- Observations: y1, y2, ..., yn are inter-pulse times.
- Conditional on the latent rate λ, each yi ~ Exponential(λ), and the joint density is p(y|λ) = λ^n exp(−λ t), where t = sum_{i=1}^n yi.
- The rate λ is unknown and has a prior distribution that differs by class:
  * Under healthy (H0): λ ~ Exponential(mean α) = prior p0(λ) = (1/α) exp(−λ/α).
  * Under unhealthy (H1): λ ~ Exponential(mean β) = prior p1(λ) = (1/β) exp(−λ/β).
- The goal is to decide between H0 and H1 using a Bayesian rule with a given threshold (the exact threshold value isn’t present in the provided options).
- The Bayesian approach replaces the likelihood ratio with the Bayes factor (marginal likelihood ratio) obtained by integrating out λ under each prior.

Key concepts and derivation:
- Marginal likelihood under a class (integrating out λ):
  m(y | H0) = ∫ p(y | λ) p0(λ) dλ = ∫ [λ^n exp(−λ t)] (1/α) exp(−λ/α) dλ
             = (1/α) ∫ λ^n exp[−λ (t + 1/α)] dλ
             = (1/α) Γ(n+1) / [t + 1/α]^{n+1}.
  Similarly,
  m(y | H1) = (1/β) Γ(n+1) / [t + 1/β]^{n+1}.
- Bayes factor (ratio of marginal likelihoods):
  B_{01} = m(y | H0) / m(y | H1) = [(1/α) Γ(n+1) / (t + 1/α)^{n+1}] / [(1/β) Γ(n+1) / (t + 1/β)^{n+1}]
           = (β/α) * [ (t + 1/β)^{n+1} / (t + 1/α)^{n+1} ].
- Bayesian decision rule (with prior odds and a decision threshold):
  The posterior odds in favor of H0 given data y is:
     P(H0 | y) / P(H1 | y) = [P(H0) / P(H1)] * B_{01}.
  The decision rule compares this posterior odds to a threshold η (the problem statement indicates a threshold, but the exact numeric value isn’t provided in the options). Concretely, one would decide H0 if:
     [P(H0) / P(H1)] * B_{01} > η,
  otherwise decide H1.
- If the problem states a simple threshold on the Bayes factor itself (i.e., compare B_{01} to some threshold τ that incorporates the prior odds and misclassification costs), the rule can be written as:
     decide H0 if B_{01} > τ.* (Here τ encodes the prior odds and the loss ratio; without the specific τ provided, we cannot state the exact numeric cutoff.)

What the missing answer options would typically look like:
- They would express a comparison that depends on t = sum yi and n, using a function of t, α, β, and possibly the threshold. The canonical form involves the Bayes factor B_{01} as above, potentially with a threshold like η = (P(H1)/P(H0)) * ratio of costs, or a direct comparison B_{01} ≷ τ.
- A common numeric form would be something like: decide H0 if (β/α) * [ (t + 1/β)^{n+1} / (t + 1/α)^{n+1} ] > η, where η encodes the prior odds and loss ratio. Any alternative should be equivalent after algebraic simplification.

Why options that might appear (had they been present) could be right or wrong in this context:
- A correct option would present a Bayes-factor–based rule that uses t = sum yi and n, with the (t + 1/β) and (t + 1/α) terms raised to the power n+1, plus the (β/α) prefactor. It would also reflect the presence of the prior odds or a threshold η that relates to the specified decision criterion.
- An incorrect option might ignore the marginalization (i.e., pretend to use the simplistic likelihood ratio p(y|H0)/p(y|H1) without integrating λ), or misplace the roles of α and β, or omit the (n+1) exponent on the (t + 1/α) and (t + 1/β) terms, or misstate the direction of the inequality.
- If an option uses a sum over m or a representation like ∑ y_m rather than t = ∑ y_i, that would be implying the same sufficient statistic, but the given standard derivation clearly reduces to a function of t, so any option not recognizing t as the sufficient statistic would be incorrect.

Important note about the current prompt state:
- The input shows answer_options as an empty list, which means there is no available choice to evaluate directly. Consequently, one cannot select a correct option from the provided list. If you provide the actual options, I can map them to the Bayes-factor form described above and explain which is correct and why, with step-by-step reasoning linked to the math of the model.

EECE5612.MERGED.202530 midterm part 1

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