Stats posterior probability gamma conjugate family

In summary, the question asks for the posterior probability of both observations y4 and y5 being zero, when the prior distribution is a gamma with parameters (a,b) and the sample is of size 3 taken from a Poisson distribution with parameter V. The solution so far shows that the posterior distribution is also a gamma with parameters a*=a+t and b*=b+n, where t is the sum of the previous observations and n is the sample size. To find the probability of both observations being zero, one approach is to compute P(y=0 | D) and then square it. Another approach is to compute P(y4=0 and y5=0) which is proportional to e^-2V. It is unclear which
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binbagsss
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Question

Find the posterior probability that the next two observations y4 and y5 will both be zero? Where the prior distribution is a gamma with parameters (a,b) and the sample is of size of 3 taking from a poisson disribution with parameter V.

So far I have shown that the posterior distribution is also gamma with parameters a*=a+t, where t=y1+y2+...yn, and b*=b+n, n the sample size.

Attempt at the solution

In my notes, a different question but all i have on my notes on this- is that the probability of exactly the next observation being 1 is given as:
##P(y=1 | D)= \int^{\infty}_{0} P(y=1|V) P(V|D) dV,## So wheree P(V|D) is the updated posterior gamma distribution,

where D is the data, the observations y1+...+yn.

This is for one sample and I'm unsure how to approach for 2 samples. The ideas I have are:
1) Compute ##P(y=0 | D)= \int^{\infty}_{0} P(y=1|V) P(V|D) dV## , and then since we want consecutvie 0 to just square this.
2) To compute ##P(y4=0 an y5=0)=\int^{\infty}_{0} P(y4=1 and y5=1|V) P(V|D) dV## , so i.e. is proportional to ##e^{-2V}## ,

Are any of these methods correct? Could someone please explain which is right or wrong, or if both or wrong and what I should be doing,

Thanks for any help, really appreciated.
 
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FAQ: Stats posterior probability gamma conjugate family

1. What is a posterior probability in statistics?

A posterior probability is a conditional probability that takes into account new evidence or information in order to update the probability of a hypothesis being true. It is calculated using Bayes' theorem, which combines the prior probability (initial belief about the hypothesis) with the likelihood of the new evidence given the hypothesis.

2. What is a gamma distribution in statistics?

A gamma distribution is a continuous probability distribution that is commonly used to model variables that are strictly positive, such as reaction times, waiting times, and income. It has two parameters, shape and scale, and its shape can vary from exponential (when shape=1) to normal (when shape is large).

3. How is the gamma distribution related to the conjugate family?

The gamma distribution is a member of the conjugate family of distributions for the exponential family of distributions. This means that when the prior distribution for a parameter in the exponential family is a gamma distribution, the posterior distribution will also be a gamma distribution. This property is useful in Bayesian statistics as it allows for simpler calculations and updates of posterior probabilities.

4. What are the advantages of using the gamma conjugate family in Bayesian statistics?

Using the gamma conjugate family in Bayesian statistics has several advantages. Firstly, it allows for simpler and more efficient calculations of posterior probabilities. Secondly, it can handle a wide range of data types and is flexible in modeling different types of variables. Lastly, it allows for the incorporation of prior beliefs and updating of those beliefs when new evidence is obtained.

5. Can the gamma conjugate family be used for any type of data?

No, the gamma conjugate family is most suitable for modeling data that is strictly positive. It is not appropriate for data that can take on negative values or for binary data. In these cases, other distributions from the conjugate family, such as the normal or beta distribution, may be more appropriate.

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