statistics – Fisher information of joint distribution of transformed Normal distribution

Suppose $X_1=theta+epsilon_1$ and $$X_i=sqrt{gamma}X_{i-1}+sqrt{1-gamma}epsilon_i+theta(1-sqrt{gamma})$$
Where $gamma in (0,1)$ and $theta$ is the parameter of the model. Also $epsilon_1,epsilon_2,…epsilon_n$ are iid $N(0,1)$.

What is the Fisher information of this model and for what values of $gamma$ does it tensorise. I’ve tried using the Jacobian to find the joint distribution but I’m not sure, especially when determining for which values we have tensorisation. Any help would be much appreaciated.

dnd 5e – If a Cave Fisher receives healing, does it keep its fire vulnerability?

RAW, the vulnerability never goes away

The feature states:

If the cave fisher drops to half its hit points or fewer, it gains vulnerability to fire damage.

There is no listed time or condition that ends this vulnerability, so strictly reading, it does not end, ever.


Ruling as above is a very strict adherence to the rules that need not be used

Such a ruling, however, makes approximately zero sense and clearly wasn’t intended. At my own tables, I would rule that the fire vulnerability only exists while the Cave Fisher is at half its hit points or less.

There are a few ways to explain this narratively, given that their blood is flammable. I would probably just say they gain vulnerability to fire damage because their injuries have exposed their blood and healing can then undo this exposure.

Similarly, you could say that their blood is exposed and they are coated in it and that healing doesn’t undo the exposure but that something like a bucket of water does despite them still being at less than half health. Note that the physical effects of healing and damage, are more or less completely undefined aspects of the game and are part of the narration.

There’s plenty room for narrative to be added and rulings to be made.

linear algebra – Find a way to apply the MLE on Fisher or Covariance matrix to make cross-correlations

I have 2 Fisher matrixes which represent information for the same variables (I mean columns/rows represent the same parameters in the 2 matrixes).

Now I would like to make the cross-correlations synthesis of these 2 matrixes by applying for each parameter the well known formula (coming from Maximum Likelihood Estimator method) :

$$dfrac{1}{sigma_{hat{tau}}^{2}}=dfrac{1}{sigma_1^2}+dfrac{1}{sigma_2^2}quad(1)$$

$sigma_{hat{tau}}$ represents the best estimator representing the combination of a sample1 ($sigma_1$) and a sample2 ($sigma_2$).

Now, I would like to do the same but for my 2 Fisher matrixes, i.e from a matricial point of view.

  1. Firstly, for this, I tried to diagonalize in a simultaneous way each of these 2 Fisher matrix, i.e by finding a common eigenvectors basis for both. Then, I would add the 2 diagonal matrices and I have so a global diagonal Fisher matrix, and after come back in the space of start.

But this method gives the same constraints (by inversing the Fisher matrix) are the same than a classical synthesis between 2 matrices since :

$V^{-1} A V = D_1$

$V^{-1} B V = D_2$ ,

then what I wanted to do by summing the 2 diagonal matrices $D_1$ and $D_2$ is the same thing than doing $A+B$ :

$A+B = V (D_1+ D_2) V^{-1}$

Finally, I had to give up this track

  1. Secondly, I tried to work directly in the space of Covariance matrix. I diagonalized “simultaneously” each of the 2 matrices. (Then, I have no more covariances terms).

and I build another covariance matrix by applying the MLE like putting on the diagonal :

$$sigma_{hat{tau}}^{2}=dfrac{1}{dfrac{1}{sigma_1^2}+dfrac{1}{sigma_2^2}}quad(2)$$

Unfortunately, it doesn’t increase the Figure of Merit (equal to $dfrac{1}{text{det(block(2 parameters))}}$ ), I mean that constraints are not better : as a conclusion, I can’t manage to do cross-correlations sine I have no gain on constraints.

Moreover, with the first method 1) above, after the building of Fisher matrix and its inversion, I can marginalize over the nuisance parameters and re-invert to get a Fisher matrix where nuisance parameters estimations are encoded into it.

But I can’t do the same for method 2). I can fix parameters (remove directly lines/columns) in Fisher matrix but this produces too high FoM (and so too small constraints) since I have less error parameters to estimate.

So, I am looking for a way to apply the MLE on Fisher or Covariance matrix to make cross-correlations.

Any suggestion/track/clue is welcome.

ap.analysis of pdes – Positivity of solution for Fisher Equation

How can we prove that if $y=y(t,x)$ is the solution of the problem:

$$begin{cases} dfrac{partial y}{partial t}(t,x)-dDelta y(t,x)=r(x)y(t,x)-rho(x) y^2(t,x), (t,x)in (0,T)times Omega \ dfrac{partial y}{partialnu}(t,x)=0, (t,x)in (0,T)timespartialOmega \ y(0,x)=y_0(x)geq 0, xin Omegaend{cases}$$

then $y(t,x)geq 0$ on $(0,T)timesOmega$?

I cannot apply the maximum principle for parabolic equation because of the logistic nonlinear term. All the derivatives are in the distributional sense and the solution $y$ is mild, i.e. $yin C((0,T),L^2(Omega))$.

We assume that $r,rhoin L^{infty}(Omega)$ are positive. I read in some article that the maximum principle can be applied but I cannot see how…

I have the same question for a nonlocal logistic term of the form: $$y(t,x)int_{Omega} varrho(x,x’)y(t,x’) dx’$$

where $varrhoin L^{infty}(OmegatimesOmega)$. How can we prove that if the above problem (with this nonlocal logistic term ) has a solution, then it must be positive?

eigenvalues – How to make a cross-correlation between 2 Fisher matrices from a pure mathematical point of view?

Firstly, I want to give you a maximum of informations and precisions about my issue. If I can’t manage to get the expected results, I will launch a bounty, maybe some experts or symply people who have been already faced to a similar problem would be able to help me.

1)

I have 2 covariance matrices known $Cov_1$ and $Cov_2$ that I want to cross-correlate. (Covariance matrix is the inverse of Fisher matrix).

I describe my approach to cross-correlate the 2 covariance matrices (the constraints are expected to be better than the constraints infered from a “simple sum” (elements by elements) of the 2 Fisher matrices).

  • For this, I have performed a diagonalisation of each Fisher matrix $F_1$ and $F_2$ associated of Covariance matrices $Cov_1$ and $Cov_2$.

  • So, I have 2 different linear combinations of random variablethat are uncorraleted, i.e just related by eigen values ($1/sigma_i^2$) as respect of their combination.

These eigen values of diagonalising are contained into diagonal matrices $D_1$ and $D_2$.

2) I can’t build a “global” Fisher matrix directly by summing the 2 diagonal matrices since the linear combination of random variables is different between the 2 Fisher matrices.

I have eigen vectors represented by $P_1$ and $P_2$ matrices.

That’s why I think that I could perform a “global” combination of eigen vectors where I can respect the MLE (Maximum Likelihood Estimator) as each eigen value :

$$dfrac{1}{sigma_{hat{tau}}^{2}}=dfrac{1}{sigma_1^2}+dfrac{1}{sigma_2^2}quad(1)$$

because $sigma_{hat{tau}}$ corresponds to the best estimator from MLE method.

So, I thought a convenient linear combination of each eigen vectors $P_1$ and $P_2$ that could allow to achieve it would be under a new matrix P whose each column represents a new eigein global vector like this :

$$P = aP_1 + bP_2$$

3) PROBLEM: : But there too, I can’t sum eigen values under the form $D_1 + D_2$ since the new matrix $P= a.P_1 + b.P_2$ can’t have in the same time the eigen values $D_1$ and also $D_2$ eigen_values, can it ?

I mean, I wonder how to build this new diagonal matrix $D’$ such that I could write :

$$P^{-1} cdot F_{1} cdot P + P^{-1} cdot F_{2} cdot P=D’$$

If $a$ and $b$ could be scalars, I could for example to start from taking the relation :

$$P^{-1} cdot F_{1} cdot P = a^2*D_1quad(1)$$

and $$P^{-1} cdot F_{2} cdot P = b^2*D_2quad(2)$$

with $(1)$ and $(2)$ making appear the relation : $$Var(aX+bY) = a^2 Var(X) + b^2 Var(Y) + 2ab Cov(X,Y) = a^2 Var(X) + b^2 Var(Y)$$ since we are in a new basis $P$ that respect $(1)$ and $(2)$.

But the issue is that $a$ and $b$ seems to be matrices and not scalars, so I don’t know how to proceed to compute $D’$.

4) CONCLUSION :

Is this approach correct to build a new basis $P = a.P_1 + b.P_2$ and $D’ = a.a.D_1 + b.b.D_2$ assuming $a$ and $b$ are matrices ?

The key point is : if I can manage to build this new basis, I could return back to the starting space, the one of single parameters (no more combinations of them) by simply doing :

$$F_{text {cross}}=P . D’ cdot P^{-1}$$ and estimate the constraints with covariance matrix : $C_{text{cross}}=F_{text {cross}}^{-1}$.

If my approach seems to be correct, the most difficulty will be to determine $a$ and $b$ parameters (which is under matricial form, at least I think since with scalar form, there are too many equations compared to 2 unknown).

Sorry if there is no code for instant but I wanted to set correctly the problematic of this approach before trying to implement.

Hoping I have been clear enough.

Any help/suggestion/track/clue is welcome to solve this problem, this would be fine to tell it.

probability or statistics – Fisher information for Gaussian

I would like to calculate the Fisher information for
$f(x|alpha)=frac{1}{(2pi delta^2(alpha))^{1/2}} Exp{-frac{(x-mu(alpha))^2}{2delta^2(alpha)}}$

using $FisherInformation({mu, delta(alpha)}, f)$ but shows me an error. Is there a better way?

Functional Analysis: Heat Flow, Fisher Information Decay, and Displacement Convexity $ lambda $

Throughout the post I will work on the flat bull $ mathbb T ^ d = mathbb R ^ d / mathbb Z ^ d $ and $ rho $ will represent any measure of probability $ mathcal P ( mathbb T ^ d) $. This question is related to two of my previous posts, the universal decomposition rate of fisherman's information along the heat flux and the improved regularization of the convex lambda gradient fluxes.

  • Fact 0: the square root of Wasserstein $ W_2 $ induces a (formal) Riemannian structure in the probability measures space, giving meaning to the Wasserstein gradients $ operatorname {grad} _ {W_2} F ( rho) $ of a functional $ F: mathcal P ( mathbb T ^ d) to mathbb R $ at a point $ rho $

  • Fact 1: heat flow $ partial_t rho_t = Delta rho_t $
    is the Wasserstein gradient flow
    $$
    dot rho_t = – operatorname {grad} _ {W_2} H ( rho_t)
    $$

    Boltzmann's entropy
    $$
    H ( rho) = int _ { mathbb T ^ d} rho log rho
    $$

  • Fact 2: Boltzmann's entropy is $ lambda $– (displacement) convex for some $ lambda $.
    Its functional dissipation is Fisher's information,
    $$
    F ( rho): = | operatorname {grad} _ {W_2} H ( rho) | ^ 2 _ { rho} = int _ { mathbb T ^ d} | nabla log rho | ^ 2 rho
    $$

  • Fact 3: for abstract metric gradient streams (in the sense of (AGS)) and $ lambda $– convex functional $ Phi: X to mathbb R cup { infty } $ one expects a smoothing effect for gradient flows $ dot x_t = – operatorname {grad} Phi (x_t) $ in the way
    begin {equation}
    | nabla Phi (x_t) | ^ 2 leq frac {C_ lambda} {t} Big ( Phi (x_0) – inf_X Phi Big)
    tag {R}
    end {equation}

    at least for small moments, where $ C_ lambda $ depends only on $ lambda $ but not in $ x_0 $ see p. (AG, Proposition 3.22 (iii)).

  • Done 3 & # 39 ;: with the same notation as in fact 3, an alternative regularization can be established as
    begin {equation}
    | nabla Phi (x_t) | ^ 2 leq frac {1} {2e ^ { lambda t} -1} | nabla Phi (y) | ^ 2 + frac {1} {( int_0 ^ te ^ { lambda s} ds) ^ 2} dist ^ 2 (x_0, y),
    qquad
    forall y in X
    tag {R & # 39;}
    end {equation}

  • Fact 4: In the Torus, Fisher's information decays at a universal rate, that is, there is $ C = C_d $ depending on the dimension just such that for everyone $ rho_0 in mathcal P ( mathbb T ^ d) $ and $ t> 0 $, the solution $ rho_t $ of the heat flux emanating from $ rho_0 $ satisfies
    begin {equation}
    F ( rho_t) leq frac {C} {t}
    label{*}
    end {equation}

    This follows from the Li-Yau (LY) inequality, see this post of mine and F. Baudoin's answer.

Question: Is there more to ($ * $) that only the convexity of the Boltzmann functional? If functional driving were out of upper limits $ Phi (x_0) leq C $ (for all $ x_0 in X $) in the regularization estimate (R), we would immediately obtain the universal decomposition $ | nabla Phi (x_t) | leq frac {C} {t} $.
However, in the specific context of Acts 0-2, it is clearly not true that Boltzmann's entropy has upper limits. In fact, there are many measures of probability with infinite entropy, for example any Dirac mass. Since (R) is optimal, I suppose it cannot be deduced simply (*) from general $ lambda $convexity arguments, and there is more than meets the eye. But is there a connection? Note that both the Li-Yau inequality and the displacement convexity of Boltzmann entropy are highly dependent on the non-negative Ricci curvature of the underlying torus.

I desperately tried to use any modified regularization estimates (eg R & # 39; and variants of it instead of R), but so far it has not been possible. I am starting to believe that there is no direct implication, and that Li-Yau's work is really profoundly ad-hoc (don't get me wrong, I just want to say that your results cannot be generalized to abstract gradient flows, and that your result / Proof really takes advantage of the specific structure and configuration of heat flow in Riemannian collectors, not just any gradient flow.) I would greatly appreciate any input or idea!


(AG) Ambrosio, L. and Gigli, N. (2013). A user guide for optimal transportation. In Modeling and optimization of flows in networks (pp. 1-155). Springer, Berlin, Heidelberg.

(AGS) Ambrosio, L., Gigli, N. and Savaré, G. (2008). Gradient flows: in metric spaces and in the probability measures space. Springer Science & Business Media.

(LY) Li, P. and Yau, S. T. (1986). In the parabolic nucleus of the operator Schrödinger. Acta Mathematica, 156, 153-201.

dg. differential geometry: is there any geometric indication for the trace of the Fisher information matrix?

Consider a parametric family $ p_ theta (x) $ of distributions, with parameter $ theta in Theta subseteq mathbb R ^ p $.

If the map $ theta mapsto p_ theta (x) $ it is continuously differentiable in $ theta_0 in Theta $ for $ p _ { theta_0} $-almost every $ x $, then Fisher's information in $ theta_0 $ is the psd matrix given by $ F ( theta_0) = mathbb E_ {x sim p _ { theta_0}} (s _ { theta_0} (x) s _ { theta_0} (x) ^ T) $, where $ s _ { theta_0} (x): = partial_ theta log (p_ theta (x)) | _ { theta = theta_0} in mathbb R ^ p $ is the scoring function in $ x $.

What is the geometric interpretation of the trace of $ F ( theta_0) $ ?

If it is well known that $ KL (p _ { theta_0 + Delta theta} | p _ { theta_0}) = frac {1} {2} Delta theta ^ TF ( theta_0) Delta theta + or ( | Delta theta | ^ 2) $.

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