Abstract
In this paper, we present some inequalities for Csiszár f-divergence between two probability measures on time scale. These results extend some known results in the literature and offer new results in h-discrete calculus and quantum calculus. We also present several inequalities for divergence measures.
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1 Introduction
In many applications of probability theory the essential problem is determining an appropriate measure of distance (or divergence) among two probability distributions. Consequently, many different divergence measures were introduced and extensively studied by various authors, for instance, the Csiszár f-divergence (Kullback–Leibler divergence, Hellinger distance, and total-variation distance), Rényi divergence, and Jensen–Shannon divergence; see [9, 13, 18, 20].
Csiszár [6] introduced the following:
Definition 1
Let \(f: \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}\) be a convex function. Let \(\tilde{\mathbf{r}} = (r_{1}, r_{2}, \ldots , r_{n})\) and \(\tilde{\mathbf{s}} = (s_{1}, s_{2}, \ldots , s_{n})\) be such that \(\sum_{\nu =1}^{n}r_{\nu }=1\) and \(\sum_{\nu =1}^{n}s_{\nu }=1\). Then the f-divergence functional is defined as
where f satisfies the following conditions:
Dragomir [7, 8] has done a plenty of work giving different types of bounds on the distance and divergence measures. Jensen’s inequality plays a vital role to get inequalities for divergences between probability distributions. Horvath et al. [11] introduced a new functional based on the f-divergence functional and obtained some estimates for the new functional, the f-divergence and Rényi divergence by utilizing cyclic refinement of Jensen’s inequality. Recently, Adil et al. [12] obtained some inequalities for convex functions and their applications to Csiszár divergence.
The main objective behind the theory of time scales is unifying continuous and discrete analysis introduced by Stefan Hilger in 1988 and established in the comprehensive books [4, 5]. Various dynamic derivatives on time scales not just give a helpful route in useful applications, but also demonstrate their extraordinary appearance in approximations. It may be beneficial to examine if such useful features can be kept up or even improved in a specific way while different dynamic derivatives are utilized in the same application simultaneously.
Guseinov [10] examined the process of Riemann and Lebesgue integration on time scales. Many authors established time scale version of linear and nonlinear integral inequalities [1, 17, 19]. The time scale integral inequalities have been used to study the boundedness, uniqueness, and so on of the solutions of different dynamic equations [14, 16]. Ansari et al. [2] introduced the differential entropy of a continuous random variable on time scales and established some Shannon-type inequalities on arbitrary time scales. It was shown that the obtained inequalities are used to estimate the bounds of differential entropy for some particular distributions. Some classical inequalities and their converses for multiple integration on time scales were investigated in [3].
The setup of this paper is as follows. Section 2 is confined to the basic definitions and preliminary results of time scales calculus. Our aim in Sect. 3 is deriving some new inequalities for Csiszár f-divergence on arbitrary time scales and finding some inequalities for Csiszár divergence in h-discrete calculus and quantum calculus. To the best of the author’s knowledge, no contribution is available in the literature for Csiszár divergence inequalities in quantum calculus. Section 4 is concerned to the study of some divergence measures on time scales including the bounds of the Kullback–Leibler distance, triangular discrimination, Hellinger discrimination, Jeffreys distance, Bhattacharyya distance, and harmonic distance in terms of some special means such as identric, logarithmic, arithmetic, and geometric means. The upper bounds of these divergence results in quantum calculus are also part of discussion.
2 Preliminaries
In this paper, we assume that a time scale \(\mathbb{T}\) is an arbitrary nonempty closed subset of the real line. The following definitions and results are extracted from [4].
Definition 2
Consider a time scale \(\mathbb{T}\) that is a closed and bounded subset of real numbers and \(\omega \in \mathbb{T}\). Then the mappings \(\sigma : \mathbb{T} \rightarrow \mathbb{T}\) and \(\rho : \mathbb{T} \rightarrow \mathbb{T}\) satisfying
are known as forward and backward jump operators on \(\mathbb{T}\), respectively.
A function \(z : \mathbb{T} \rightarrow \mathbb{R}\) is right-dense continuous or rd-continuous if it is continuous at right-dense points in \(\mathbb{T}\) and its left-sided limits exist (finite) at left-dense points in \(\mathbb{T}\). The set of all rd-continuous functions is denoted by \(C_{\mathrm{rd}}\).
Here we define \(\mathbb{T}^{k}\) as follows:
Definition 3
Let \(z : \mathbb{T} \rightarrow \mathbb{R}\) and \(\omega \in \mathbb{T}^{k}\). Then we define the delta derivative \(z^{\Delta }(\omega )\) as the number (provided it exists) such that for each \(\epsilon > 0\), there exists a neighborhood U of ω such that
for all \(\lambda \in U\). We say that z is delta differentiable at ω.
If \(\mathbb{T}= \mathbb{R}\), then \(z^{\Delta }\) is the the usual derivative \(z^{\prime }\), whereas \(z^{\Delta }\) becomes the forward difference operator \(\Delta z(\omega ) = z(\omega +1) - z(\omega )\) for \(\mathbb{T} = \mathbb{Z}\). If \(\mathbb{T} = \overline{q^{\mathbb{Z}}} = \{q^{n}: n \in \mathbb{Z} \} \cup \{0\}\) with \(q > 1\), then \(z^{\Delta }\) is the so-called q-difference operator
Theorem 1
(Existence of antiderivatives)
Every rd-continuous function has an antiderivative. In particular if \(x_{0} \in \mathbb{T}\), then F is defined by
which is an antiderivative of f.
For \(\mathbb{T} = \mathbb{R}\), we get \(\int _{a}^{b}z(\omega )\Delta \omega = \int _{a}^{b}z(\omega ) \,d\omega \), and if \(\mathbb{T} = \mathbb{N}\), then \(\int _{a}^{b}z(\omega )\Delta \omega = \sum_{\omega =a}^{b-1}z( \omega )\), where \(a, b \in \mathbb{T}\) with \(a\leq b\).
3 Main results
Let \(\mathbb{T}\) be a time scale and consider the set of all probability density functions on \(\mathbb{T}\),
In this paper, we assume that \(\tilde{r}, \tilde{s} \in \Omega \).
Definition 4
The Csiszár f-divergence on time scales is defined as
where f is convex on \((0,\infty )\).
By suitable substitutions to f in Definition 4 we can obtain several divergences on time scales. For instance, if we choose \(f(x) = x^{2} - 1\), then we find the Pearson \(\chi ^{2}\)-divergence on time scales denoted by \(D_{\chi ^{2}}\) and defined as
We begin with the following result.
Theorem 2
Let \(\psi : [0, \infty ) \rightarrow \mathbb{R}\) be a mapping convex on the interval \([\zeta _{1}, \zeta _{2}] \subset [0, \infty )\), where \(\zeta _{1}\leq 1 \leq \zeta _{2}\). If
then
Proof
As ψ is convex on \([\zeta _{1}, \zeta _{2}]\), we can write
Choose \(t = \frac{\zeta _{2}-x}{\zeta _{2}-\zeta _{1}}\), \(x \in [\zeta _{1}, \zeta _{2}]\). Then \(1-t = \frac{x-\zeta _{1}}{\zeta _{2}-\zeta _{1}}\), and from (2) we get
Using \(x = \frac{\tilde{r}(y)}{\tilde{s}(y)}\), \(y \in \mathbb{T}\), in (3), we get
Multiplying (4) by \(\tilde{s}(y) > 0\), integrating over \(\mathbb{T}\), and using the equalities \(\int _{a}^{b} \tilde{r}(y) \Delta y = \int _{a}^{b} \tilde{s}(y) \Delta y = 1\), we obtain
which is the stated result. □
Example 1
For \(\mathbb{T} = \mathbb{R}\), Theorem 2 becomes [7, Theorem 1 on p. 2].
Example 2
Choosing \(\mathbb{T} = h\mathbb{Z}\), \(h > 0\), in Theorem 2, we get
Remark 1
Inequality (5) in h-discrete calculus is an extension of specific upper bound for the Csiszár divergence obtained by Lovričević et al. [15, Corollary 4.1].
Example 3
Choosing \(\mathbb{T} = q^{\mathbb{N}_{0}}\) (\(q > 1\)) in Theorem 2, we have
Remark 2
Equation (6) represents the Csiszár divergence in quantum calculus, which is new up to the knowledge of the authors.
Theorem 3
Consider a differentiable convex function \(\psi : [0, \infty ) \rightarrow \mathbb{R}\) on the interval \([\zeta _{1}, \zeta _{2}]\) and \(I_{\psi } \) defined in Theorem 2. Then we have
where \(D_{\chi ^{2}}(\tilde{r}, \tilde{s}) := \int _{a}^{b}\tilde{s}(y) [ (\frac{\tilde{r}(y)}{\tilde{s}(y)} )^{2} - 1 ] \Delta y\).
Proof
Since ψ is a differentiable convex function, we have
Now assume that \(a_{1}, a_{2} \in [\zeta _{1}, \zeta _{2}]\) and consider \(\alpha _{1}, \alpha _{2}\geq 0\) such that \(\alpha _{1} + \alpha _{2} > 0\). Then using \(u_{1} = \frac{\alpha _{1}a_{1}+\alpha _{2}a_{2}}{\alpha _{1}+\alpha _{2}}\) and \(u_{2} = a_{1}\) in (10), we get
Rewrite (11) with \(u_{2} = a_{2}\):
Multiplying (11) by \(\alpha _{1}\) and (12) by \(\alpha _{2}\) and then adding the resultant inequalities, we get
Dividing (13) by \(-(\alpha _{1}+\alpha _{2})\), we obtain
Now using \(\alpha _{1} = \zeta _{2} - x\), \(\alpha _{2} = x - \zeta _{1}\), \(a_{1} = \zeta _{1}\), \(a_{2} = \zeta _{2}\) in (14), we get
Putting \(x = \frac{\tilde{r}(y)}{\tilde{s}(y)}\) in (15) and multiplying by \(\tilde{s}(y)\), we obtain
for all \(y \in \mathbb{T}\).
By taking Δ-integral on both sides of (16) with \(\int _{a}^{b}\tilde{r}(y)\Delta y = \int _{a}^{b}\tilde{s}(y)\Delta y = 1\) we get
which is inequality (8). Inequality (9) is obvious, since
□
Example 4
Choosing \(\mathbb{T} = \mathbb{R}\) in Theorem 3, we get [7, Theorem 2 on p. 3].
Example 5
Choosing \(\mathbb{T} = h\mathbb{Z}\), \(h > 0\), in Theorem 3, we obtain
Example 6
Choosing \(\mathbb{T} = q^{\mathbb{N}_{0}}\) (\(q > 1\)) in Theorem 3, we get
Remark 3
Equation (17) is new in quantum calculus, which involves the Csiszár divergence and Pearson \(\chi ^{2}\)-divergence.
Theorem 4
Consider a twice differentiable function \(\Psi : [0, \infty ) \rightarrow \mathbb{R}\) on \([\zeta _{1}, \zeta _{2}]\) with
If \(\zeta _{1}\leq \frac{\tilde{r}(y)}{\tilde{s}(y)} \leq \zeta _{2}\) for all \(y \in \mathbb{T}\), then
where \(D_{\chi ^{2}}(\tilde{r}, \tilde{s})\) is defined in Theorem 3.
Proof
Define \(\eta _{m} : [0, \infty ) \rightarrow \mathbb{R}\) by \(\eta _{m}(t) = \Psi (t) - \frac{1}{2}m t^{2}\). Then \(\eta ^{\prime \prime } _{m}(t) = \Psi ^{\prime \prime }(t) - m \geq 0\), \(t \in [\zeta _{1}, \zeta _{2}]\), and this implies that \(\eta _{m}\) is convex on \([\zeta _{1}, \zeta _{2}]\). By using (1) for \(\eta _{m}\) instead of ψ we get
However,
and by (20) we obtain
Simplification of the left-hand side of (21) gives
and (18) is proved. Similarly, inequality (19) can be obtained for the mapping \(\eta _{m}(t) = \frac{1}{2}M t^{2} - \Psi (t)\). □
Example 7
Choosing \(\mathbb{T} = \mathbb{R}\) in Theorem 4, we get [7, Theorem 3 on p. 4].
Example 8
Choosing \(\mathbb{T} = h\mathbb{Z}\), \(h > 0\), in Theorem 4, we have
Example 9
Choosing \(\mathbb{T} = q^{\mathbb{N}_{0}}\) (\(q > 1\)) in Theorem 4, we have
Remark 4
In Example 9, we get some new inequalities involving the Csiszár divergence for quantum calculus.
Corollary 1
Under the conditions of Theorem 4, if \(m \geq 0\), then
Proof
We just need to show that
which follows from the proof of Theorem 3, since
□
Example 10
Choosing \(\mathbb{T} = \mathbb{R}\) in Corollary 1, we get [7, Corollary 1 on p. 5].
4 Bounds of some divergence measures
First we recall some special means:
and
4.1 Kullback–Leibler divergence on time scales
Let \(\psi : (0, \infty ) \rightarrow \mathbb{R}\) be the convex mapping \(\psi (t) = t\ln t\). Then
where \(D(\tilde{r}, \tilde{s})\) is the Kullback–Leibler distance.
Proposition 1
If
then
where \(G(\cdot , \cdot )\) is the usual geometric mean, \(L(\cdot , \cdot )\) is the logarithmic mean, and \(I(\cdot , \cdot )\) is the identric mean.
Proof
Using Theorem 2 for \(\psi (t) = t\ln t \), we obtain
□
Example 11
Choosing \(\mathbb{T} = \mathbb{R}\) in Proposition 1, we get [7, Proposition 1 on p. 6].
Example 12
Choosing \(\mathbb{T} = h\mathbb{Z}\), \(h > 0\), in Proposition 1, we obtain
Remark 5
Equation (24) is an extension of the specific bound for the Kullback–Leibler divergence obtained by Lovričević et al. [15, Corollary 4.4].
Example 13
Choosing \(\mathbb{T} = q^{\mathbb{N}_{0}}\) (\(q > 1\)) in Proposition 1, we have
Remark 6
Equation (25) shows an upper bound for the Kullback–Leibler divergence, which is new in quantum calculus.
Proposition 2
Under the conditions of Proposition 1, we get
where \(D_{\chi ^{2}}(\tilde{r}, \tilde{s})\) is defined in Theorem 3.
Proof
Apply Theorem 3 for \(\psi (t) = t\ln t\):
□
Example 14
Putting \(\mathbb{T} = \mathbb{R}\) in Proposition 2, we get [7, Proposition 2 on p. 6].
Example 15
Choosing \(\mathbb{T} = q^{\mathbb{N}_{0}}\) (\(q > 1\)), in Proposition 2, we have
By using Theorem 4 we can improve (26) as follows.
Proposition 3
Let r̃, s̃ satisfy (23). Then we have
Proof
Consider \(\psi (t) = t\ln t\) in Theorem 4. In this case, \(\psi ^{\prime \prime }(t) = \frac{1}{t}\), \(t \in [\zeta _{1}, \zeta _{2}]\), and then
which gives the desired result. □
Remark 7
For \(\mathbb{T} = \mathbb{R}\) in Proposition 3, we get [7, Proposition 3 on p. 7].
Example 16
Choosing \(\mathbb{T} = q^{\mathbb{N}_{0}}\) (\(q > 1\)) in Proposition 3, we have
Now consider the convex mapping \(\psi (t) = -\ln t\). We get
By using Theorem 2 we obtain the following result.
Proposition 4
Let r̃, s̃ satisfy (23). Then
Proof
Using (1) for \(\psi (t) = -\ln t\), we get
□
Remark 8
For \(\mathbb{T} = \mathbb{R}\) in Proposition 4, we get [7, Proposition 4 on p. 7].
Proposition 5
Let r̃, s̃ satisfy (23). Then we have
where \(D_{\chi ^{2}}(\tilde{r}, \tilde{s})\) is defined in Theorem 3.
Proof
Apply Theorem 3 for \(\psi (t) = - \ln t\):
□
Remark 9
For \(\mathbb{T} = \mathbb{R}\), Proposition 5 becomes [7, Proposition 5 on p. 7].
Further improvement of (27) is as follows.
Proposition 6
Under the assumptions of Theorem 4, we have
Proof
Apply Theorem 4, for which \(\psi ^{\prime \prime }(t) = \frac{1}{t^{2}}\) and
for all \(t \in [\zeta _{1}, \zeta _{2}]\). □
Remark 10
Choosing \(\mathbb{T} = \mathbb{R}\) in Proposition 6, we get [7, Proposition 6 on p. 8].
4.2 Triangular discrimination on time scales
Let \(\psi : [0, \infty ) \rightarrow \mathbb{R}\) be the convex mapping \(\psi (t) = \frac{(t - 1)^{2}}{t + 1}\). Then
where \(D_{\Delta }(\tilde{r}, \tilde{s})\) is the triangular discrimination.
Proposition 7
Under the assumptions of Theorem 2, we have
Proof
Using Theorem 2 for \(\psi (t) = \frac{(t - 1)^{2}}{t + 1}\), we obtain
□
The following example gives an upper bound for triangular discrimination, which is new in quantum calculus.
Example 17
Choosing \(\mathbb{T} = q^{\mathbb{N}_{0}}\) (\(q > 1\)) in Proposition 7, we have
Proposition 8
Under the conditions of Theorem 3, we have
Proof
Apply Theorem 3 with \(\psi (t) = \frac{(t - 1)^{2}}{t+1}\), which gives \(\psi ^{\prime }(t) = 1- \frac{4}{(1 + t)^{2}}\) and
□
Proposition 9
Under the assumptions of Theorem 4, we have
Proof
Use Theorem 4 for \(\psi (t) = \frac{(t - 1)^{2}}{t+1}\), which implies that \(\psi ^{\prime \prime }(t) = \frac{8}{(1 + t)^{3}}\) and
□
4.3 Hellinger discrimination on time scales
Let \(\psi : [0, \infty ) \rightarrow \mathbb{R}\) be the convex mapping \(\psi (t) = \frac{1}{2}(\sqrt{t} - 1)^{2}\). Then
where \(h^{2}(\tilde{r}, \tilde{s})\) is the Hellinger discrimination.
Proposition 10
Under the assumptions of Theorem 2, we get
Proof
Use Theorem 2 for \(\psi (t) = \frac{1}{2}(\sqrt{t} - 1)^{2}\) to obtain
□
Example 18
For \(\mathbb{T} = \mathbb{R}\) in Proposition 10, we get [7, Proposition 7 on p. 8].
The following example gives an upper bound for the Hellinger discrimination, which is new in quantum calculus.
Example 19
Choosing \(\mathbb{T} = q^{\mathbb{N}_{0}}\) (\(q > 1\)) in Proposition 10, we have
Proposition 11
Under the assumptions of Theorem 3, we have
where \(A(\cdot ,\cdot )\) is the arithmetic mean.
Proof
Apply Theorem 3 with \(\psi (t) = \frac{1}{2}(\sqrt{t} - 1)^{2}\), which implies that \(\psi ^{\prime }(t) = \frac{1}{2} - \frac{1}{2\sqrt{t}}\) and
□
Remark 11
For \(\mathbb{T} = \mathbb{R}\), Proposition 11 becomes [7, Proposition 8 on p. 9].
Proposition 12
Under the assumptions of Theorem 4, we have
Proof
Use Theorem 4 for \(\psi (t) = \frac{1}{2}(\sqrt{t} - 1)^{2}\), which gives \(\psi ^{\prime \prime }(t) = \frac{1}{4t^{\frac{3}{2}}}\) and, obviously,
□
Remark 12
Choosing \(\mathbb{T} = \mathbb{R}\) in Proposition 12, we get [7, Proposition 9 on p. 9].
4.4 Jeffreys distance on time scales
Let \(\psi : (0, \infty ) \rightarrow \mathbb{R}\) be the convex mapping \(\psi (t) = (t - 1)\ln (t)\). Then
where \(D_{J}(\tilde{r}, \tilde{s})\) is the Jeffreys distance.
Proposition 13
Let r̃, s̃ satisfy (23). Then we get
Proof
Use Theorem 2 for \(\psi (t) = (t - 1)\ln t \) to get
□
The following example gives an upper bound for the Jeffreys distance, which is new in quantum calculus.
Example 20
Choosing \(\mathbb{T} = q^{\mathbb{N}_{0}}\) (\(q > 1\)) in Proposition 13, we have
Proposition 14
Under the assumptions of Theorem 3, we get
Proof
Apply Theorem 3 for \(\psi (t) = (t - 1)\ln t\), for which
□
Proposition 15
Under the assumptions of Theorem 4, we have
Proof
Consider \(\psi (t) = (t - 1)\ln t\) in Theorem 4. In this case, \(\psi ^{\prime \prime }(t) = \frac{t + 1}{t^{2}}\), \(t \in [\zeta _{1}, \zeta _{2}]\), and then
which gives the desired result. □
4.5 Bhattacharyya distance on time scales
Let \(\psi : [0, \infty ) \rightarrow \mathbb{R}\) be a convex mapping, \(\psi (t) = -\sqrt{t}\). Then
where \(D_{B}(\tilde{r}, \tilde{s})\) is the Bhattacharyya distance.
Proposition 16
Under the assumptions of Theorem 2, we obtain
Proof
Use Theorem 2 for \(\psi (t) = -\sqrt{t}\) to obtain
□
Example 21
Choosing \(\mathbb{T} = q^{\mathbb{N}_{0}}\) (\(q > 1\)) in Proposition 16, we have
Remark 13
In Example 21, we get an upper bound for the Bhattacharyya distance, which is new in quantum calculus.
Proposition 17
Under the assumptions of Theorem 3, we have
where \(A(\cdot ,\cdot )\) is the arithmetic mean.
Proof
Apply Theorem 3 with \(\psi (t) = -\sqrt{t}\), which implies that \(\psi ^{\prime }(t) = - \frac{1}{2\sqrt{t}}\) and
□
Proposition 18
Under the assumptions of Theorem 4, we have
Proof
Use Theorem 4 for \(\psi (t) = - \sqrt{t}\), which implies that \(\psi ^{\prime \prime }(t) = \frac{1}{4t^{\frac{3}{2}}}\) and, obviously,
□
4.6 Harmonic distance on time scales
Let \(\psi : (0, \infty ) \rightarrow \mathbb{R}\) be the convex mapping \(\psi (t) = \frac{-2t}{1 + t}\). Then
where \(D_{\mathrm{Ha}}(\tilde{r}, \tilde{s})\) is the harmonic distance.
Proposition 19
Under the assumptions of Theorem 2, we have
Proof
Use Theorem 2 for \(\psi (t) = \frac{-2t}{1 + t}\) to get
□
The following example gives an upper bound for the harmonic distance, which is new in quantum calculus.
Example 22
Choosing \(\mathbb{T} = q^{\mathbb{N}_{0}}\) (\(q > 1\)) in Proposition 19, we obtain
Proposition 20
Under the conditions of Theorem 3, we have
Proof
Apply Theorem 3 with \(\psi (t) = \frac{-2t}{1 + t}\), which implies that \(\psi ^{\prime }(t) = \frac{-1}{(1 + t)^{2}}\) and
□
Proposition 21
Under the assumptions of Theorem 4, we get
Proof
Use Theorem 4 for \(\psi (t) = \frac{-2t}{1 + t}\), which implies that \(\psi ^{\prime \prime }(t) = \frac{2}{(1 + t)^{3}}\) and
□
5 Conclusion
In this paper, we introduced the Csiszár f-divergence on time scales and establish inequalities involving the Csiszár f-divergence on time scales. The obtained results are an extension of some known results in the literature and report new results in h-discrete calculus and quantum calculus.
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The research of Josip Pečarić is supported by the Ministry of Education and Science of the Russian Federation (Agreement number 02.a03.21.0008).
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Ansari, I., Khan, K.A., Nosheen, A. et al. Some inequalities for Csiszár divergence via theory of time scales. Adv Differ Equ 2020, 698 (2020). https://doi.org/10.1186/s13662-020-03159-x
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DOI: https://doi.org/10.1186/s13662-020-03159-x