Branching process and posterior distribution.
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I have the following branching process:
(I added the $Z$'s myself for the number of individuals in each gen, hope it's correct.)
Assume the offspring distribution is Poisson with expectation
$lambda$ and also assume that we use an improper prior
$pi(lambda)propto_{lambda}1/lambda$ for $lambda$. Using the
information in the figure, compute the posterior distribution for
$lambda.$
Using Baye's theorem with an observed count $k$, I get
$$pi(lambda|k)propto_{lambda}pi(k|lambda)pi(lambda)propto_{lambda}e^{-lambda}cdotlambda^{k}cdotfrac{1}{lambda}=e^{-lambda}lambda^{k-1}.$$
So we see that this posterior is $pi(lambda|k)simGamma(lambda;k,1).$
However, how do I know what $k$ is based on the figure?
statistics stochastic-processes markov-process
asked Dec 3 at 13:16
Parseval
2,7311718
|
show 1 more comment
up vote
1
down vote
favorite
I have the following branching process:
(I added the $Z$'s myself for the number of individuals in each gen, hope it's correct.)
Assume the offspring distribution is Poisson with expectation
$lambda$ and also assume that we use an improper prior
$pi(lambda)propto_{lambda}1/lambda$ for $lambda$. Using the
information in the figure, compute the posterior distribution for
$lambda.$
Using Baye's theorem with an observed count $k$, I get
$$pi(lambda|k)propto_{lambda}pi(k|lambda)pi(lambda)propto_{lambda}e^{-lambda}cdotlambda^{k}cdotfrac{1}{lambda}=e^{-lambda}lambda^{k-1}.$$
So we see that this posterior is $pi(lambda|k)simGamma(lambda;k,1).$
However, how do I know what $k$ is based on the figure?
statistics stochastic-processes markov-process
asked Dec 3 at 13:16
Parseval
2,7311718
Assuming the next generations of the tree are not represented in the figure (not that they are empty), the figure tells you that the numbers of offspring for this tree are, starting from the root, $$1, 2, 3, 2, 0, 0, 1, 4, 2$$ hence the posterior knowing this tree is $$pi(lambdamid k)proptopi(lambda)prod_kpi(kmidlambda)$$ that is, $$pi(lambdamid k)proptolambda^{-1}(e^{-lambda}lambda)(e^{-lambda}lambda^2)(e^{-lambda}lambda^3)(e^{-lambda}lambda^2)(e^{-lambda}lambda^0)(e^{-lambda}lambda^0)(e^{-lambda}lambda^1)(e^{-lambda}lambda^4)(e^{-lambda}lambda^2)$$
– Did
Dec 3 at 14:20
...or, more simply, $$pi(lambdamid k)propto e^{-9lambda}lambda^{14}$$ Assuming the next generations are empty, one arrives at $$pi(lambdamid k)propto e^{-16lambda}lambda^{14}$$
– Did
Dec 3 at 14:20
@Did Thank you for the answer. Could you elaborate on what you mean by empty and non empty generation? I also don't see how you obtained the number of offsprings? How did you find $1,2,3,2,0,0,1,4,2$?
– Parseval
Dec 3 at 15:50
I looked at each node and counted its descendants.
– Did
Dec 3 at 16:43
@ Did Okay I understnad that now. So can one then say that $$pi(lambda|k)simGamma(lambda;15,9)?$$
– Parseval
Dec 3 at 18:03
|
show 1 more comment
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have the following branching process:
(I added the $Z$'s myself for the number of individuals in each gen, hope it's correct.)
Assume the offspring distribution is Poisson with expectation
$lambda$ and also assume that we use an improper prior
$pi(lambda)propto_{lambda}1/lambda$ for $lambda$. Using the
information in the figure, compute the posterior distribution for
$lambda.$
Using Baye's theorem with an observed count $k$, I get
$$pi(lambda|k)propto_{lambda}pi(k|lambda)pi(lambda)propto_{lambda}e^{-lambda}cdotlambda^{k}cdotfrac{1}{lambda}=e^{-lambda}lambda^{k-1}.$$
So we see that this posterior is $pi(lambda|k)simGamma(lambda;k,1).$
However, how do I know what $k$ is based on the figure?
statistics stochastic-processes markov-process
asked Dec 3 at 13:16
Parseval
2,7311718
I have the following branching process:
(I added the $Z$'s myself for the number of individuals in each gen, hope it's correct.)
Assume the offspring distribution is Poisson with expectation
$lambda$ and also assume that we use an improper prior
$pi(lambda)propto_{lambda}1/lambda$ for $lambda$. Using the
information in the figure, compute the posterior distribution for
$lambda.$
Using Baye's theorem with an observed count $k$, I get
$$pi(lambda|k)propto_{lambda}pi(k|lambda)pi(lambda)propto_{lambda}e^{-lambda}cdotlambda^{k}cdotfrac{1}{lambda}=e^{-lambda}lambda^{k-1}.$$
So we see that this posterior is $pi(lambda|k)simGamma(lambda;k,1).$
However, how do I know what $k$ is based on the figure?
statistics stochastic-processes markov-process
statistics stochastic-processes markov-process
asked Dec 3 at 13:16
Parseval
2,7311718
asked Dec 3 at 13:16
Parseval
2,7311718
asked Dec 3 at 13:16
Parseval
2,7311718
asked Dec 3 at 13:16
Parseval
2,7311718
asked Dec 3 at 13:16
Parseval
2,7311718
2,7311718
Assuming the next generations of the tree are not represented in the figure (not that they are empty), the figure tells you that the numbers of offspring for this tree are, starting from the root, $$1, 2, 3, 2, 0, 0, 1, 4, 2$$ hence the posterior knowing this tree is $$pi(lambdamid k)proptopi(lambda)prod_kpi(kmidlambda)$$ that is, $$pi(lambdamid k)proptolambda^{-1}(e^{-lambda}lambda)(e^{-lambda}lambda^2)(e^{-lambda}lambda^3)(e^{-lambda}lambda^2)(e^{-lambda}lambda^0)(e^{-lambda}lambda^0)(e^{-lambda}lambda^1)(e^{-lambda}lambda^4)(e^{-lambda}lambda^2)$$
– Did
Dec 3 at 14:20
...or, more simply, $$pi(lambdamid k)propto e^{-9lambda}lambda^{14}$$ Assuming the next generations are empty, one arrives at $$pi(lambdamid k)propto e^{-16lambda}lambda^{14}$$
– Did
Dec 3 at 14:20
@Did Thank you for the answer. Could you elaborate on what you mean by empty and non empty generation? I also don't see how you obtained the number of offsprings? How did you find $1,2,3,2,0,0,1,4,2$?
– Parseval
Dec 3 at 15:50
I looked at each node and counted its descendants.
– Did
Dec 3 at 16:43
@ Did Okay I understnad that now. So can one then say that $$pi(lambda|k)simGamma(lambda;15,9)?$$
– Parseval
Dec 3 at 18:03
|
show 1 more comment
Assuming the next generations of the tree are not represented in the figure (not that they are empty), the figure tells you that the numbers of offspring for this tree are, starting from the root, $$1, 2, 3, 2, 0, 0, 1, 4, 2$$ hence the posterior knowing this tree is $$pi(lambdamid k)proptopi(lambda)prod_kpi(kmidlambda)$$ that is, $$pi(lambdamid k)proptolambda^{-1}(e^{-lambda}lambda)(e^{-lambda}lambda^2)(e^{-lambda}lambda^3)(e^{-lambda}lambda^2)(e^{-lambda}lambda^0)(e^{-lambda}lambda^0)(e^{-lambda}lambda^1)(e^{-lambda}lambda^4)(e^{-lambda}lambda^2)$$
– Did
Dec 3 at 14:20
...or, more simply, $$pi(lambdamid k)propto e^{-9lambda}lambda^{14}$$ Assuming the next generations are empty, one arrives at $$pi(lambdamid k)propto e^{-16lambda}lambda^{14}$$
– Did
Dec 3 at 14:20
@Did Thank you for the answer. Could you elaborate on what you mean by empty and non empty generation? I also don't see how you obtained the number of offsprings? How did you find $1,2,3,2,0,0,1,4,2$?
– Parseval
Dec 3 at 15:50
I looked at each node and counted its descendants.
– Did
Dec 3 at 16:43
@ Did Okay I understnad that now. So can one then say that $$pi(lambda|k)simGamma(lambda;15,9)?$$
– Parseval
Dec 3 at 18:03
Assuming the next generations of the tree are not represented in the figure (not that they are empty), the figure tells you that the numbers of offspring for this tree are, starting from the root, $$1, 2, 3, 2, 0, 0, 1, 4, 2$$ hence the posterior knowing this tree is $$pi(lambdamid k)proptopi(lambda)prod_kpi(kmidlambda)$$ that is, $$pi(lambdamid k)proptolambda^{-1}(e^{-lambda}lambda)(e^{-lambda}lambda^2)(e^{-lambda}lambda^3)(e^{-lambda}lambda^2)(e^{-lambda}lambda^0)(e^{-lambda}lambda^0)(e^{-lambda}lambda^1)(e^{-lambda}lambda^4)(e^{-lambda}lambda^2)$$
– Did
Dec 3 at 14:20
Assuming the next generations of the tree are not represented in the figure (not that they are empty), the figure tells you that the numbers of offspring for this tree are, starting from the root, $$1, 2, 3, 2, 0, 0, 1, 4, 2$$ hence the posterior knowing this tree is $$pi(lambdamid k)proptopi(lambda)prod_kpi(kmidlambda)$$ that is, $$pi(lambdamid k)proptolambda^{-1}(e^{-lambda}lambda)(e^{-lambda}lambda^2)(e^{-lambda}lambda^3)(e^{-lambda}lambda^2)(e^{-lambda}lambda^0)(e^{-lambda}lambda^0)(e^{-lambda}lambda^1)(e^{-lambda}lambda^4)(e^{-lambda}lambda^2)$$
– Did
Dec 3 at 14:20
...or, more simply, $$pi(lambdamid k)propto e^{-9lambda}lambda^{14}$$ Assuming the next generations are empty, one arrives at $$pi(lambdamid k)propto e^{-16lambda}lambda^{14}$$
– Did
Dec 3 at 14:20
...or, more simply, $$pi(lambdamid k)propto e^{-9lambda}lambda^{14}$$ Assuming the next generations are empty, one arrives at $$pi(lambdamid k)propto e^{-16lambda}lambda^{14}$$
– Did
Dec 3 at 14:20
@Did Thank you for the answer. Could you elaborate on what you mean by empty and non empty generation? I also don't see how you obtained the number of offsprings? How did you find $1,2,3,2,0,0,1,4,2$?
– Parseval
Dec 3 at 15:50
@Did Thank you for the answer. Could you elaborate on what you mean by empty and non empty generation? I also don't see how you obtained the number of offsprings? How did you find $1,2,3,2,0,0,1,4,2$?
– Parseval
Dec 3 at 15:50
I looked at each node and counted its descendants.
– Did
Dec 3 at 16:43
I looked at each node and counted its descendants.
– Did
Dec 3 at 16:43
@ Did Okay I understnad that now. So can one then say that $$pi(lambda|k)simGamma(lambda;15,9)?$$
– Parseval
Dec 3 at 18:03
@ Did Okay I understnad that now. So can one then say that $$pi(lambda|k)simGamma(lambda;15,9)?$$
– Parseval
Dec 3 at 18:03
|
show 1 more comment
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Assuming the next generations of the tree are not represented in the figure (not that they are empty), the figure tells you that the numbers of offspring for this tree are, starting from the root, $$1, 2, 3, 2, 0, 0, 1, 4, 2$$ hence the posterior knowing this tree is $$pi(lambdamid k)proptopi(lambda)prod_kpi(kmidlambda)$$ that is, $$pi(lambdamid k)proptolambda^{-1}(e^{-lambda}lambda)(e^{-lambda}lambda^2)(e^{-lambda}lambda^3)(e^{-lambda}lambda^2)(e^{-lambda}lambda^0)(e^{-lambda}lambda^0)(e^{-lambda}lambda^1)(e^{-lambda}lambda^4)(e^{-lambda}lambda^2)$$
– Did
Dec 3 at 14:20
...or, more simply, $$pi(lambdamid k)propto e^{-9lambda}lambda^{14}$$ Assuming the next generations are empty, one arrives at $$pi(lambdamid k)propto e^{-16lambda}lambda^{14}$$
– Did
Dec 3 at 14:20
@Did Thank you for the answer. Could you elaborate on what you mean by empty and non empty generation? I also don't see how you obtained the number of offsprings? How did you find $1,2,3,2,0,0,1,4,2$?
– Parseval
Dec 3 at 15:50
I looked at each node and counted its descendants.
– Did
Dec 3 at 16:43
@ Did Okay I understnad that now. So can one then say that $$pi(lambda|k)simGamma(lambda;15,9)?$$
– Parseval
Dec 3 at 18:03