M = {p(x, θ)|θϵΘ} is called an SM [6] if x is a random variable in sample space X and p(x, θ) is the probability density function, which satisfies certain regular conditions.

The most straightforward probabilistic queueing model that can be studied analytically is a single-channel model with exponential inter-arrival durations, service times, and FIFO queue discipline. This is called the M/M/1 queue in queueing theory [10].

Based on the introduction of the function, Parthasarathy [10] suggested a straightforward method for the transient solution of the M/M/1 system to read as

−(λ+μ)t n (λ+μ)t

pn(t) = ρp0(t)e [∑m=1 rm (t) + e ], n = 0,1,2, …, (5)

where stands for the mean rate per unit time at which arrival instants occur, is the mean rate of service

λ(t)

time, ρ = defines the traffic intensity or utilization factor.

μ(t)

n−a1 n−1−a1

rn(t) = ( μβ (1 − δ0a )[In−a (β1t) − In+a (β1t)] + λβ (In+1−a (β1t) −

2 1 1 1 2 1

In−1−a (β1t))).

With β1 = 2√λμ , β2 = 2√ , In(x) Eq. (19) is the modified Bessel function and

−(λ+μ)y

p0(t) = ∫ r1 e dy + δ0a .

Theorem 1. The underlying queue of Eq. (5) has:

I. FIM reads as a b c

[gij] = [b 0 h]. (7)

c h l With

a = − , (8)

ρ2 (1−ρ)2

b = t, (9)

ρ ρ . .

c = + + tμ + μρ , (10)

ρ2 (1−ρ)2

h = (1 + ρ) + tρ , (11)

.2 .. .2 ..

ρ − ρρ ρ + (1 − ρ)ρ

.. . . . . .. (12)

l = ( + + ((1 + ρ)tμ + μ + 2ρ μ + 2μρ + 2ρμ + μtρ , 2 ( )2 ρ 1 − ρ

Δ = det([gij]) = (a(−h ) − b(bl − ch) + c(bh)). (13)

Provided that, . refers to the temporal derivative .

dt ij

II. [g ] reads as:

A B L

adj[gij] ij

[g ] = = [B E F], (14)

L F H where (−h )

A = , (15)

(ch − bl)

B = , (16)

(bh)

L = , (17)

(al − c )

E = , (18)

(bc − ah)

F = , (19)

(−b )

H = . (20)

III. Following (5), we have

−(λ+μ)t −(λ+μ)t n

ℒ(x; θ) = l n(pn(x; θ)) = (l n(ρp0(t)e ) + l n(ρp0(t)e [∑m=1 rm (t) +

(λ+μ)t (21)

e ] ),

θ = (θ1, θ2, θ3) = (ρ, μ, t). (22)

We have

Ψ(θ) = −l n(ρ) − l n(1 − ρ) + μ(1 + ρ)t, p0(t) = 1 − ρ. (23)

Thus,

∂Ψ 1 1 ∂Ψ ∂Ψ ρ ρ

∂1 = = − + + μt, ∂2 = = (1 + ρ)t , ∂3 = = − + + (1 +

∂ρ ρ (1−ρ) ∂μ ∂t ρ (1−ρ)

(24)

ρ)tμ + μ(1 + ρ) + μtρ ,

∂1 ∂1 = − , (25)

2 ( )2 ρ 1 − ρ

∂1 ∂2 = ∂2 ∂1 = t, (26) 2 2 = 0, (27)

ρ ρ . .

∂1 ∂3 = + + tμ + μρ = ∂3 ∂1,

ρ2 (1−ρ)2 (28)

∂2 ∂3 = (1 + ρ) + tρ = ∂3 ∂2, (29)

.2 .. .2 ..

ρ − ρρ ρ + (1 − ρ)ρ

.. . . . . ..

∂3 ∂3 = ( + + (1 + ρ)tμ + μ + 2ρ μ + 2μρ + 2ρμ μtρ ). (30)

2 ( )2 ρ 1 − ρ

Therefore, the fisher information matrix, FIM, is obtained (Eq. (7)).

ij

[g ] reads as:

a b c

adj[gij] Transpose(Cov[gij]) 1 ij −1

[g ] = [gij] = = = transpose(Cov [b 0 h])

∆ ∆ ∆

c h l

2 (31)

(−h ) (ch − bl) (bh) = [(ch − bl) (al − c ) (bc − ah)].

(bh) (bc − ah) (−b )

Thus,

(−h ) (ch − bl) (bh) A B L

ij 1 2

[g ] = transpose(Cov[(ch − bl) (al − c ) (bc − ah)]) = [B E F].

2 L F H (bh) (bc − ah) (−b )

As requested, It is notable that FIM should satisfy the symmetry requirement.

In what follows, the components of α (or ∇ )- connection are obtained. These calculated expressions are needed to obtain the corresponding GEs of the parametric coordinates of M/M/1 QM.

From Eq. (3), the reader can check that.

(α) 1 1

Γ = −(1 − α)( + ),

11,1 (1−ρ)3 ρ3 (32)

(α) (α) (α) (α) (α) (α) (α) (α) (α) (α)

0 = Γ22,2 = Γ12,2 = Γ22,1 = Γ21,2 = Γ11,2 = Γ12,1 = Γ21,1 = Γ22,3 = Γ = Γ32,2, (33)

23,2 (α) . 1 1

Γ = (1 − α)ρ ( − ),

31,1 (1−ρ)3 ρ3 (34)

(α) (α) 1 1

Γ = Γ = ( 1 − α)ρ ( − ), (35)

13,1 11,3 3 3

(1 − ρ) ρ

(α) (α) (1 − α)

Γ = Γ = , (36)

13,2 12,3 .. .. .2

( ) (1 − α) ρ ρ 2ρ

α .. .

Γ13,3 = [ + + + tμ + 2μ ], (37)

2 ( )2 ( )3 2 ρ 1 − ρ 1 − ρ

( ) ( ) (1 − α)ρ

Γ23,1 = Γ21,3 = , (38)

. ..

( ) (1 − α)(1 + 2ρ + tρ )

Γ23,3 = , (39)

( ) ( ) (1 − α) Γ31,2 = Γ32,1 = . (40) Engaging the same procedure, the remaining components can be determined.

I. The IMEs of the server utilization coordinate, ρ of the transient M/M/1QM.

The IMEs corresponding to the arrival rate coordinate, ρ are

2 1 i j

d θ 1(0) dθ dθ

+ Γ ( ) ( ) = 0,i, j = 1,2,3.

dt2 ij dt dt

Now, we are in a situation of trying to find the path of motion of family of families of IMEs corresponding to the server utilization coordinate, ρ.

. 1 1

2Aρ ( − ) +

2 . (1−ρ)3 ρ3 d ρ L Lρ dρ dμ 1

[ + [ + ]( )( ) +[ ( .. .. .2 ) +

dt2 2 2 dt dt 2 ρ ρ 2ρ .. .

B + L [ + + + tμ + 2μ ]

ρ2 (1−ρ)2 (1−ρ)3

.. .2 .2 ..

1 . 1 1 ρρ −2ρ 2ρ +(1−ρ)ρ

(2Aρ ( − ) + B + ( + +

2 (1−ρ)3 ρ3 ρ3 (1−ρ)3 .. . . . .. dρ dt . 1 1 dρ 2

tμ + ρ μ + μ + μρ ) L))]( )( )+[ρ L( − )]( ) +

dt dt (1−ρ)3 ρ3 dt 1 . . .. 1 . .. dμ dt

[ (Aρ + B(1 + 2ρ + tρ )) + (A +(ρ + tρ )L)]( )( )+ (41)

2 2 dt dt

.. .2 .2 ..

1 ρρ −2ρ 2ρ +(1−ρ)ρ .. . . . .. .

[ (( + + tμ + ρ μ + μ + μρ ) A + (ρ +

2 ρ3 (1−ρ)3

. .. … . .2 .. . .. … . .2 ..

.. (ρ ρ −ρρ )ρ−2ρ (ρ −ρρ ) (ρ ρ +(1−ρ)ρ )(1−ρ)+2ρ (ρ +(1−ρ)ρ )

tρ )B +[ + +

ρ3 (1−ρ)3 ... .. . .. . .. . ..

(1 + ρ)tμ + (1 + ρ)μ + 3ρ μ + μ + 2ρ μ + 2μρ +

. . .. ... .. . .. dt

4μ ρ + 2ρμ + μtρ + μρ + μ tρ ]L)] ( ) ] = 0.

dt

It can be verified that one of the closed form solutions of Eq. (41) is determined by the paths of motion:

ρ(t) = constant, μ(t) = constant , such that ρ ∈ (0,1), μ > 1. (42)

II. The IMEs of the Mean service Rate coordinate, μ of the transient M/M/1 QM.

The IMEs of are

2 2 i j

d θ 1(0) dθ dθ

+ Γ ( ) ( ) = 0,i, j = 1,2,3.

dt2 ij dt dt

Now, we are in a situation of trying to find the path of motion of family of families of IMEs corresponding to the mean service rate, .

2 . .. .. .2

d μ F Fρ dρ dμ 1 . 1 1 ρ ρ 2ρ ..

[ + [ + ]( )( ) +[ (2Bρ ( − ) + E + F [ + + + tμ +

dt2 2 2 dt dt 2 (1−ρ)3 ρ3 ρ2 (1−ρ)2 (1−ρ)3

.. .2 .2 ..

. 1 . 1 1 ρρ −2ρ 2ρ +(1−ρ)ρ .. . . . ..

2μ ]) + (2Bρ ( − ) + E + ( + + tμ + ρ μ + μ + μρ )

2 (1−ρ)3 ρ3 ρ3 (1−ρ)3 dμ dt . 1 1 dρ 1 . . .. 1 .

F)]( ) ( )+[ρ ( − ) F] ( ) + [ (Bρ + E(1 + 2ρ + tρ )) + (B +(ρ +

dt dt (1−ρ)3 ρ3 dt 2 2

.. .2 .2 ..

.. dρ dt 1 ρρ −2ρ 2ρ +(1−ρ)ρ .. . . . .. .

tρ )F)]( )( )+[ (( + + tμ + ρ μ + μ + μρ ) B + (ρ +

dt dt 2 ρ3 (1−ρ)3

. .. … . .2 .. . .. … . .2 ..

.. (ρ ρ −ρρ )ρ−2ρ (ρ −ρρ ) (ρ ρ +(1−ρ)ρ )(1−ρ)+2ρ (ρ +(1−ρ)ρ )

tρ )E + [ + +

ρ3 (1−ρ)3

Thus, 1 1 .. . . 2

(( − )[ ((1+ρ)tμ +μ+2ρμ ]−(tμ ) )

ρ2 (1−ρ)2 1 1 2

(( − )(−((1+ρ)) )−(t2[((1+ρ)tμ..+μ+2ρμ.]))

2 ρ2 (1−ρ)2

d μ 1

[ +[ 1 1 +

dt2 2 .

(t(tμ )−( − )(1+ρ))

ρ2 (1−ρ)2

.. .

[tμ + 2μ ]

1 1 2 2 .. .

(( − )(−((1+ρ)) )−(t [((1+ρ)tμ +μ+2ρμ ]))

ρ2 (1−ρ)2

. 1 1

(t(tμ )−( − )(1+ρ))

ρ2 (1−ρ)2

+ (43)

1 1 2

(( − )(−((1+ρ)) )−(t2[((1+ρ)tμ..+μ+2ρμ.]))

ρ2 (1−ρ)2

. 1 1

(t(tμ )−( − )(1+ρ))

ρ2 (1−ρ)2

.. . dμ 1 ..

(tμ + μ ) )]( )+[ (( tμ +

1 1 2 dt 2 (( − )(−((1+ρ)) )−(t2[((1+ρ)tμ..+μ+2ρμ.]))

ρ2 (1−ρ)2 . ..

. (tμ (1−ρ)−t[((1+ρ)tμ +μ])

μ ) +] ] = 0.

1 1 2

(( − )(−((1+ρ)) )−(t2[((1+ρ)tμ..+μ+2ρμ.]))

ρ2 (1−ρ)2

It is obvious that for arbitrary constant values of ρ, μ, we have a closed form solution for Eq. (43). As time becomes sufficiently large, i.e., t → ∞, Eq. (42) reduces to

.. .2 . ...

3μ 3μ μ μ

[ − − ] = 0. (44)

2 2(1+ρ) μ..

2γt

We propose the closed form solution, μ = , for some arbitrary non-zero constant . substituting in Eq.

(1+γt)

(44) implies:

18γ + + 6γ = 0. (45) (1 + γt)(1 + ρ)

As t → ∞, γ = 0,0,0, −3. The only accepted value is γ = −3, μ = , which tends to the value μ = 2 as

(3t−1)

t → ∞.

III. The IMEs of the temporal coordinate, t for the underlying QM.

Engaging a similar lengthy mathematical manipulation, it can be verified that the IMEs of the temporal coordinate, t, at infinite time is characterized by the family of families of temporal curves:

μ(t) = ζ1+ζ2t + ζ3t . (46)

for some arbitrary non-zero constants ζ1,ζ2, ζ3, 1 > 1.

IV. Ig Applications To Ml Stochastic gradient descent is a popular optimisation approach in ML that seeks to determine which direction of descent is the steepest within probability distributions’ parameter space. Amari et al. [11] created the natural gradient, which is superior to the ordinary gradient since it more accurately captures this direction. Furthermore, even with gradient descent, FIM can be used to calculate the predicted change in output about parameter changes. Artificial Neural Networks (ANNs) are functions with multiple parameters that map input vectors to output vectors. They are widely applied in regression, computer vision, and speech processing. ANNs can be trained through supervised learning, where a training set of input-output pairs is used, or through unsupervised learning, which does not require a training set. The back-propagation algorithm efficiently computes the gradient descent by determining the contribution of each parameter to the error. Overfitting, a common issue in ANNs, can be mitigated by techniques such as using a validation set, early stopping, data augmentation, and dropout [12].

In Reinforcement Learning (RL) [13], when an agent needs to prepare for unknown tasks, one approach is unsupervised skill discovery. These algorithms learn a set of policies without using a reward function and are like representation learning algorithms in supervised learning. While prior work has shown that these methods can accelerate downstream tasks, our analysis reveals that they do not learn skills that are optimal for every possible reward function. However, the distribution over skills can provide an optimal initialization that minimizes regret against adversarially chosen reward functions, assuming a specific adaptation procedure. Notably, understanding which state marginals are achievable is crucial for analyzing the behavior of unsupervised skill learning algorithms [13] and their relevance to state-dependent reward functions. The set of achievable state marginal distributions can be characterized by a set of linear equations or by a complementary description based on a specific property.

This alternative perspective provides insights into the relationship between learned skills and downstream tasks in RL. So, if we have a set of possible state distributions, any combination of those distributions within the convex hull is also possible. This means that there exists a policy that can achieve any state distribution within that convex hull. By taking the convex hull of the state distributions of all deterministic policies, we can obtain the set of all possible state distributions, which is referred to as a polytope. Each vertex of the polytope represents a deterministic policy, and any policy's state distribution can be expressed as a combination of the state distributions at the vertices, as visualized by Fig. 2.

The objective of maximizing the expected return is equivalent to maximizing the dot product between the state marginal distribution and the reward vector. This visualization helps us understand how different reward-maximizing policies are related to the state marginal polytope and provides insights into the relationship between state-dependent reward functions and optimal policies [13].

The analysis of where the skills learned by Mutual Information-based Skill Learning (MISL) are positioned on the state marginal polytope, which has motivated [13] to undertake this analysis by examining the mutual information objective and dissecting it to understand how the learned skills align with the polytope. In the context of skill learning [13], maximizing mutual information is equivalent to minimizing the maximum difference between a prior distribution over states (represented by a green square) and any possible state distribution.

This concept, known as information geometry, helps optimize skill-learning algorithms by assigning higher probabilities to policies with state distributions that deviate further from the average state distribution. The goal is to find skills as distinct as possible from each other and the average state distribution. Most importantly, the opaque circles represent the skills discovered by the Mutual Information Skill Learning (MISL) algorithm. The dashed line represents equidistant state marginals from the green square, indicating an equal divergence from the prior distribution over states. This visualization helps illustrate the relationship between the skills learned by MISL and their distribution in relation to the average state distribution. This is illustrated in Fig. 3. Reliable detection of Out-Of-Distribution (OOD) samples [14] is crucial for ensuring the safety of modern machine learning systems. The work of [14] introduced IGEOOD, an effective method that can detect OOD samples using any pre-trained neural network, regardless of the level of access to the ML model. By leveraging the geodesic distance between data distributions, IGEOOD combines confidence scores from the neural network's logits outputs and learned features to achieve superior performance compared to other state-of-the-art methods across different network architectures and datasets [14].

The authors [14] introduced the concept of IG and derived a metric based on the Fisher-Rao distance to measure the mismatch between probability density functions of a pre-trained neural classifier. Experimental results demonstrate that IGEOOD performs competitively with state-of-the-art methods in different setups, including black-box and grey-box scenarios, and achieves improved performance on various benchmarks. Fig. 4 showcases a comparison between Fisher-Rao and Mahalanobis distances for distinguishing between 1D Gaussian distributions in the context of OOD detection. It highlights the motivation to use the Fisher-Rao metric, which measures the dissimilarity between probability models, as it shows better performance in distinguishing between in-distribution and OOD samples compared to other distance-based approaches.

In the context of the experiment, the authors [14] compared three different settings for OOD detection. They use a unified metric, but with different formulations based on the type of distribution. For the Deep Neural Network (DNN) outputs, they use the softmax posterior probability distribution, while for the intermediate layers, they employ a model based on diagonal Gaussian Probability Density Functions (PDFs). This unified framework combines a single distance measure for both the softmax outputs and the latent features of the neural network, contributing to the separation of in-distribution and OOD samples, as illustrated in Fig. 5. The distributions are shown for three different settings, specifically for a pre-trained DenseNet model on CIFAR-10 dataset, considering both in-distribution and OOD data from TinyImageNet (downsampled). This analysis helps evaluate the effectiveness of the IGEOOD score in distinguishing between in-distribution and OOD samples [13].