Fdtxjr

PlotLabels with equations not expressions

Can anyone tell me why this program fails?

How to deal with a cynical class?

How to write cleanly even if my character uses expletive language?

Have researchers managed to "reverse time"? If so, what does that mean for physics?

Can elves maintain concentration in a trance?

Calculus II Professor will not accept my correct integral evaluation that uses a different method, should I bring this up further?

Connecting top and bottom SMD component pads using via

Why would a flight no longer considered airworthy be redirected like this?

Good allowance savings plan?

Does this AnyDice function accurately calculate the number of ogres you make unconcious with three 4th-level castings of Sleep?

Should we release the security issues we found in our product as CVE or we can just update those on weekly release notes?

How do I hide Chekhov's Gun?

Official degrees of earth’s rotation per day

An Accountant Seeks the Help of a Mathematician

Identifying the interval from A♭ to D♯

Where is the 1/8 CR apprentice in Volo's Guide to Monsters?

Is it true that real estate prices mainly go up?

Old race car problem/puzzle

Why are the outputs of printf and std::cout different

Is it possible that AIC = BIC?

Why do passenger jet manufacturers design their planes with stall prevention systems?

Rules about breaking the rules. How do I do it well?

Make a transparent 448*448 image

How is the replicator dynamic a gradient flow of the Fisher information metric?

Does the curvature determine the metric for all surfacesHamiltonian for Geodesic FlowManifolds: A definition of the Gradient an Algebraic Tangent vector over charts, how to show equivalence?A form of chain rule to differentiate the flow of a vector field on a manifoldGradient flow under varying riemannian metricDerivative of local flow of vector field at originHow does the gradient/jacobian relate to the first derivative of one dimensional functions?Fisher information of sampleFlatness of a statistical manifold with Fisher information metricThe tangential gradient of a function $f$ defined on a manifold

0

$begingroup$

I am trying to understand how the replicator dynamic can be derived as a gradient flow of the Fisher information metric (aka Shahshahani metric). I have a question about understanding a particular proof, but if you have a different straightforward derivation/proof, I would love to see it.

I am trying to understand the following proof (I'll ask my particular questions at the end):

Consider the replicator dynamic $dotp_i = p_i(f_i(p) - barf)$, where $f_i(p) = frac
partial Vpartial p_i$ for $i in 1, dots, m$, and $barf = sum_i p_i f_i(p)$

Lemma:
$$ dotp_i = (nabla_g V)_i = sum_j g_ij fracpartial Vpartial p_j $$
where $g_ij$ is the inverse of the Fisher information metric.

Proof:
For every $xi = (xi_1, dots, xi_k)$ with $sum_j xi_j = 0$, that is, for every tangent vector to the simplex in which the dynamics of $p$ take place, we have:
beginalign
sum_j g_ij dotp_i xi_j &= sum_j frac1p_j dotp_j xi_j \
&= sum_j (f_j(p) - barf) xi_j \
&= sum_j f_j(p) xi_j \
&= sum_j fracpartial Vpartial p_j xi_j \
&textwhich implies the lemma
endalign

My questions are:

• I am not sure why they start the proof how they do. My understanding is they start the proof with the expression of the dot product between the vector pointing in the direction of change ($dotp$) and an arbitrary other tangent vector to $p$. Why start with that dot product?


• I don't know how the last step of the proof implies tne Lemma.

Any clarification, exposition, intuition, or alternative proof that can help me fully understand how the replicator dynamic is derived as a gradient flow of the Fisher metric would be extremely appreciated!

calculus differential-geometry gradient-flows fisher-information evolutionary-game-theory

share|cite|improve this question

edited Mar 12 at 0:08

DrNesbit

asked Mar 11 at 9:48

DrNesbitDrNesbit

617

$endgroup$

add a comment | 

0

$begingroup$

I am trying to understand how the replicator dynamic can be derived as a gradient flow of the Fisher information metric (aka Shahshahani metric). I have a question about understanding a particular proof, but if you have a different straightforward derivation/proof, I would love to see it.

I am trying to understand the following proof (I'll ask my particular questions at the end):

Consider the replicator dynamic $dotp_i = p_i(f_i(p) - barf)$, where $f_i(p) = frac
partial Vpartial p_i$ for $i in 1, dots, m$, and $barf = sum_i p_i f_i(p)$

Lemma:
$$ dotp_i = (nabla_g V)_i = sum_j g_ij fracpartial Vpartial p_j $$
where $g_ij$ is the inverse of the Fisher information metric.

Proof:
For every $xi = (xi_1, dots, xi_k)$ with $sum_j xi_j = 0$, that is, for every tangent vector to the simplex in which the dynamics of $p$ take place, we have:
beginalign
sum_j g_ij dotp_i xi_j &= sum_j frac1p_j dotp_j xi_j \
&= sum_j (f_j(p) - barf) xi_j \
&= sum_j f_j(p) xi_j \
&= sum_j fracpartial Vpartial p_j xi_j \
&textwhich implies the lemma
endalign

My questions are:

• I am not sure why they start the proof how they do. My understanding is they start the proof with the expression of the dot product between the vector pointing in the direction of change ($dotp$) and an arbitrary other tangent vector to $p$. Why start with that dot product?


• I don't know how the last step of the proof implies tne Lemma.

Any clarification, exposition, intuition, or alternative proof that can help me fully understand how the replicator dynamic is derived as a gradient flow of the Fisher metric would be extremely appreciated!

calculus differential-geometry gradient-flows fisher-information evolutionary-game-theory

share|cite|improve this question

edited Mar 12 at 0:08

DrNesbit

asked Mar 11 at 9:48

DrNesbitDrNesbit

617

$endgroup$

add a comment | 

$begingroup$

I am trying to understand how the replicator dynamic can be derived as a gradient flow of the Fisher information metric (aka Shahshahani metric). I have a question about understanding a particular proof, but if you have a different straightforward derivation/proof, I would love to see it.

I am trying to understand the following proof (I'll ask my particular questions at the end):

Consider the replicator dynamic $dotp_i = p_i(f_i(p) - barf)$, where $f_i(p) = frac
partial Vpartial p_i$ for $i in 1, dots, m$, and $barf = sum_i p_i f_i(p)$

Lemma:
$$ dotp_i = (nabla_g V)_i = sum_j g_ij fracpartial Vpartial p_j $$
where $g_ij$ is the inverse of the Fisher information metric.

Proof:
For every $xi = (xi_1, dots, xi_k)$ with $sum_j xi_j = 0$, that is, for every tangent vector to the simplex in which the dynamics of $p$ take place, we have:
beginalign
sum_j g_ij dotp_i xi_j &= sum_j frac1p_j dotp_j xi_j \
&= sum_j (f_j(p) - barf) xi_j \
&= sum_j f_j(p) xi_j \
&= sum_j fracpartial Vpartial p_j xi_j \
&textwhich implies the lemma
endalign

My questions are:

• I am not sure why they start the proof how they do. My understanding is they start the proof with the expression of the dot product between the vector pointing in the direction of change ($dotp$) and an arbitrary other tangent vector to $p$. Why start with that dot product?


• I don't know how the last step of the proof implies tne Lemma.

Any clarification, exposition, intuition, or alternative proof that can help me fully understand how the replicator dynamic is derived as a gradient flow of the Fisher metric would be extremely appreciated!

calculus differential-geometry gradient-flows fisher-information evolutionary-game-theory

share|cite|improve this question

edited Mar 12 at 0:08

DrNesbit

asked Mar 11 at 9:48

DrNesbitDrNesbit

617

$endgroup$

share|cite|improve this question

edited Mar 12 at 0:08

DrNesbit

asked Mar 11 at 9:48

DrNesbitDrNesbit

617

share|cite|improve this question

edited Mar 12 at 0:08

DrNesbit

asked Mar 11 at 9:48

DrNesbitDrNesbit

617

asked Mar 11 at 9:48

DrNesbitDrNesbit

617

asked Mar 11 at 9:48

DrNesbitDrNesbit

617