The process by which certain pulses are emitted in a human body is modeled as a Poisson process. In other words, the times Ym between adjacent pulses are i.i.d. random variables which are exponentially distributed with the arrival rate λ. A healthy patient is characterized by the rate λ0, while an unhealthy one is characterized by λ1

Question

The process by which certain pulses are emitted in a human body is modeled as a Poisson process. In other words, the times Ym between adjacent pulses are i.i.d. random variables which are exponentially distributed with the arrival rate λ. A healthy patient is characterized by the rate λ0, while an unhealthy one is characterized by λ1<λ0. M measurements of inter-pulse times are taken from a patient, and a healthy/unhealthy decision has to be made. However, the values of λ0 and λ1 are not known as they are in fact random. Specifically, each is assumed to be exponentially distributed, λ0 with mean value 
ˉ
λ
0, and λ1 with mean value 
ˉ
λ
1. With this model, the likelihood ratio is replaced by the Bayesian factor. The decision threshold is set to η=
P0

P1
Cfa

Cmd
=1. M=10 observations are made on a patient, yielding the measured values y1,…yM. If 
ˉ
λ
0=1 and 
ˉ
λ
1=
ˉ
λ
0/4, which of the following represents the Bayesian decision rule (numbers are rounded):

BlackTom AI · Accepted Answer

The question asks for the Bayesian decision rule under a Poisson process model for inter-pulse times, with λ0 and λ1 as random (exponential priors) and M = 10 observations y1,…,yM. It also provides specific means for the priors: ˉλ0 = 1 and ˉλ1 = ˉλ0/4 = 0.25, which imply the hyperparameters α0 = 1/ˉλ0 = 1 and α1 = 1/ˉλ1 = 4 for the exponential priors on λ0 and λ1, respectively. The data model for each observation is Yi | λ ∼ Exponential(λ), so the joint likelihood for M observations given λ is L(λ) = λ^M exp(-λ S), where S = ∑_{m=1}^M y_m and M = 10.

From there, the marginal likelihoods under the two hypotheses (healthy H0 with λ0 and unhealthy H1 with λ1) are obtained by integrating the product of the likelihood and the prior over λ:
- p(y|H0) = ∫_0^∞ (λ^M e^{-λ S}) α0 e^{-α0 λ} dλ with α0 = 1.
- p(y|H1) = ∫_0^∞ (λ^M e^{-λ S}) α1 e^{-α1 λ} dλ with α1 = 4.

These integrals yield closed-form expressions (involving Gamma functions and the combined rate S + αk) and define the Bayes factor BF = p(y|H1) / p(y|H0). The decision rule is typically based on comparing BF to a threshold that involves the prior odds P0/P1 (or its reciprocal, depending on the convention), i.e., posterior odds = (P0/P1) × BF, so the decision is made by whether this posterior odds exceed 1 (or equivalently whether BF exceeds the threshold P1/P0, given the chosen convention). In short, the Bayesian rule should be something like: decide H1 if BF > (P1/P0) (or equivalently decide H0 if BF < (P1/P0)), with the exact numerical threshold depending on the stated convention in the problem.

Now, regarding the provided option:
- The option shows something like a sum over m of y_m multiplied by H1 on one side and H0 on the other, with a comparison like ≷, and a numeric 21.3. This form is not consistent with the correct Bayesian marginalization step. The Bayes factor is a ratio of marginal likelihoods p(y|H1) / p(y|H0), each of which is an integral over λ with the exponential priors, not a simple sum of observed times weighted by indicators H1 or H0.
- Specifically, the correct rule does not entail summing y_m and equating to a fixed numeric value like 21.3 directly. Instead, it requires computing the marginal likelihoods (or equivalently the Bayes factor BF), which depend on S = ∑ y_m and on the hyperparameters α0, α1, and M, and then comparing BF to a threshold derived from P0/P1. The displayed expression appears to mix the data sum S with a decision boundary in a way that does not align with the standard integral form for p(y|Hk) and the Bayes factor.
- Without the exact expressions for p(y|H0) and p(y|H1) and without a clear statement of whether the threshold is BF > P1/P0 or BF > P0/P1 (i.e., which convention is adopted), the given line is unlikely to be the correct Bayesian rule. The correct rule should be framed in terms of the Bayes factor and priors, not a unilateral sum like ∑ y_m H1, ∑ y_m H0 with a fixed cutoff.

In summary, the plausible reasoning is that the shown option does not match the standard Bayesian formulation for this problem. The Bayes factor should be computed as the ratio of the two marginal likelihoods p(y|H1) and p(y|H0) obtained by integrating over the exponential priors for λ0 and λ1, and the decision should compare BF to a threshold determined by the prior odds, not a simple sum of observations equated to a fixed numeric threshold such as 21.3. The absence of explicit marginal-likelihood expressions or a clean BF-based inequality in the option indicates a mismatch with the proper Bayesian decision rule.

If more options were provided, the next step would be to explain, for each option, whether it correctly implements: (1) the marginalization over λ under each hypothesis, (2) the Bayes factor BF = p(y|H1)/p(y|H0), and (3) the decision criterion using the given threshold η = P0/P1 (or the conventional BF threshold derived from prior odds). In this case, with only one option available and lacking the proper margin-integral form, we conclude that the presented option does not align with the standard Bayesian decision framework for this problem.

EECE5612.MERGED.202530 midterm part 1

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