# Thin-shell wormhole under non-commutative geometry inspired Einstein-Gauss-
Bonnet gravity.

###### Abstract

Einstein-Gauss-Bonnet gravity is a generalization of the general relativity to
higher dimensions in which the ﬁrst and second-order terms corresponds to
general relativity and Einstein-Gauss-Bonnet gravity respectively. We
construct a new class of ﬁve-dimensional (5D) thin-shell wormholes by the
‘Cut-Paste’ technique from black holes in Einstein-Gauss-Bonnet gravity
inspired by non-commutative geometry starting with a static spherically
symmetric, Gaussian mass distribution as a source and for this structural form
of the thin shell wormhole we have explored several salient features of the
solution, viz., pressure-density profile, equation of state, the nature of
wormhole, total amount of exotic matter content at the shell. We have also
analyzed the linearized stability of the constructed wormhole. From our study
we can assert that our model is found to be plausible with reference to the
other model of thin-shell wormhole available in literature.

###### keywords:

Thin-shell Wormhole, Noncommutative Geometry, Einstein-Gauss-Bonnet Gravity

N. Rahman M. Kalam A. Das S. Islam F. Rahaman M. Murshid

N. Rahman
Department of Physics, Aliah University,
IIA/27, New Town, Kolkata 700160, India
Email<EMAIL_ADDRESS>

Dr. M. Kalam
Department of Physics, Aliah University,
IIA/27, New Town, Kolkata 700160, India
Email<EMAIL_ADDRESS>

Dr. A. Das
Department of Physics, Ashoknagar Vidyasagar Bani Bhaban
High School (H.S.), North 24 Parganas, West Bengal - 743222, India
Email<EMAIL_ADDRESS>

Dr. S. Islam
Department of Mathematics, Amity University Kolkata, Kolkata - 700135, India
Email<EMAIL_ADDRESS>

Dr. S. Islam
Department of Mathematics, Sister Nivedita University, Kolkata - 700156, India
Email<EMAIL_ADDRESS>

Dr. F. Rahaman
Department of Mathematics, Jadavpur University, Kolkata - 700032, India
Email<EMAIL_ADDRESS>

M. Murshid
Department of Physics, Aliah University,
IIA/27, New Town, Kolkata 700160, India
Email<EMAIL_ADDRESS>

##  1 Intoduction

In 1935, Albert Einstein and Nathan Rosen [1] tried to produce a ﬁeld theory
for electrons. Considering a spherically symmetric mass distribution which had
already been used for black holes, Einstein and Rosen implemented a co-
ordinate transformation to eliminate the region comprising the curvature
singularity. But, the solution demonstrates the existence of bridges through
space-time. This bridge can act as a tunnel through which one can travel in
space-time to another part of the universe and it is known as Einstein-Rosen
bridge. It is also known as Schwarzschild wormhole. Theoretically, this bridge
is a shortcut path that minimizes the travel time as well as distance.
However, in 1962, John Archibald Wheeler and Robert W. Fuller [2] argued that
Einstein and Rosen bridge structure is unstable if it connects two parts of
the same universe. Therefore, one can argue that though Schwarzschild wormhole
exists theoretically, but it is not traversable in both directions. In 1988,
Michael S. Morris and Kip S. Thorne [3] first ever proposed a wormhole
solution of Einstein field equations which was found to be traversable. The
solution represents a hypothetical shortcuts between two regions (either of
the same Universe or of two separate Universes) connected by a throat. The
throat of the wormholes can be deﬁned as a two dimensional hypersurface of
minimal area and for the existance of such an wormhole the null energy
condition has to be violated. Hence, all traversable wormholes require exotic
matter which can be held responsible for the violation of the null energy
condition. The requirements of exotic matter for the existence of a wormhole
can be made inﬁnitesimally small by a suitable choice of the geometry [4]. M.
Visser [5] has also proposed another way to minimize the usage of exotic
matter to construct a wormhole, and it is known as ‘Cut and Paste’ technique,
in which the exotic matter is concentrated at the wormhole throat. In the ‘Cut
and Paste’ technique, the wormholes can be theoretically constructed by
cutting and pasting two different manifolds to obtain geodesically complete
new manifold with a throat placed in the joining shell [6]. Using the Darmois-
Israel formalism [47, 48], one can determine the surface stresses of the
exotic matter located in thin shell placed at the joining surfaces. This new
construction of wormhole is known as thin shell wormhole and is extremely
useful to perform the stability analysis for the dynamic case. Several authors
have used this surgical technique (i.e., the Cut and Paste technique) to
construct thin wormholes. Poisson and Visser [6] have analyzed the stability
of a thin wormhole constructed by joining two Schwarzschild spacetimes. Eiroa
and Simeone [59] have constructed the wormholes by applying the cut and paste
technique to two metrics corresponding to a charged black hole solution in low
energy bosonic string theory, with vanishing antisymmetric ﬁeld along with the
Maxwell ﬁeld, whereas they have analyzed cylindrically symmetric thin wormhole
geometry associated to gauge cosmic strings [10]. Thibeault et al. [11] have
analysed the stability as well as the energy conditions of ﬁve dimensional
spherically symmetric thin shell wormholes in Einstein–Maxwell theory along
with the Gauss Bonnet term.

Now, the Einstein-Gauss-Bonnet (EGB) gravity which is a generalization of
Einstein’s general relativity, was originally introduced by Lanczos [12], and
it was rediscovered by David Lovelock [13]. Amid the larger class of general
higher-curvature theories, this theory has some special characteristics having
higher derivative terms. Nevertheless, the field equations in this theory are
of second-order like that of general theory of relativity. In this theory the
Gauss-Bonnet term is present in the low energy effective action of heterotic
string theory [14, 15, 16], and can be considered as the dimensionally
extended version of the four-dimensional Euler density. Also, this term
appears in six-dimensional Calabi-Yau compactifications of M-theory [17]. The
EGB gravity enables the researchers to study how higher curvature corrections
to black hole physics significantly change the qualitative features which are
available due to black holes under the background of general relativity.

The Gauss-Bonnet term naturally appears as the second significant term in
heterotic string effective action. Also, the noncommutative geometry appears
from the study of open string theories. One can show that the gravitational
wave signal GW150914 which has been recently detected by LIGO and Virgo
collaborations [18], can be used to place a bound on the scale of quantum
fuzziness of noncommutative space-time [19]. In a paper [19], author has shown
that the leading noncommutative correction to the phase of the gravitational
waves produced by a binary system appears at the second order of the post-
Newtonian expansion . Also, in another paper [20], the plausibility of using
quantum mechanical transitions that is induced by the combined effect of
gravitational waves and noncommutative structure to probe the spatial
noncommutative nature has been explored. Synder [21, 22] first introduced the
noncommutative spacetime to study the divergences in relativistic quantum
field theory. Noncommutative geometry also aims to place General Relativity
and the Standard Model on the same footing in order to describe gravity, the
electro-weak and strong forces as gravitational forces in a unified spacetime
[23]. Noncommutativity substitutes the point-like structures by smeared
objects [24, 25, 26, 27] and is used to eliminate the divergences which
normally appear in general theory of relativity. It is supposed to be
intrinsic property of spacetime and hence, does not depend on any particular
feature such as curvature.

Although, general relativity retains the usual commutative form, the
noncommutative geometry leads to a smearing of matter distributions which can
be considered as due to the intrinsic uncertainty embodied in the coordinate
commutator as

$\left[x_{\alpha},x_{\beta}\right]=i\theta_{\alpha\beta},$ (1)

where $\theta_{\alpha\beta}$ is an anti-symmetric matrix which determines the
fundamental cell discretization of spacetime. The standard way of modelling
the smearing effect is accomplished by using a Gaussian distribution of
minimal length $\sqrt{\theta}$ due to possible uncertainty. Hence, one can
consider a mass $M$, described by a $\delta$-function distribution for a
static, spherically symmetric, Gaussian-smeared matter source, in
$D-$dimensions [45, 39], as

$\rho_{\theta}(r)=\frac{\mu}{(4\pi\theta)^{(D-1)/2}}e^{{-r^{2}}/{(4\theta)}}$
(2)

where $M$ is the total mass of the source, may be due to a diffused
centralized object such as a wormhole [26] and the particle mass $\mu$ is
supposed to be diffused throughout a region of linear size $\sqrt{\theta}$.
Here, $\theta$ is the noncommutative parameter which can be taken to be of a
Planck length. Thus, the above equation plays the role of a matter source and
the mass is smeared around the region $\sqrt{\theta}$ instead of situated at a
particular point.

One can find a number of studies inspired by noncommutative geometry which are
available in the literature either under the background of general relativity
[30, 32, 33] or under modified gravity theories [34, 35, 36] .The model of
self-sustained traversable wormhole has been studied by R. Garattini and
F.S.N. Lobo [30], whereas Rahaman et. al [31, 32] explored two new wormhole
solutions inspired by noncommutative geometry taking two different energy
density profile admitting conformal motion. P.K.F. Kuhfittig and V. D. Gladney
[33] studied charged wormhole inspired by noncommutative geometry with low
tidal forces. In another paper authors [34] have studied the womhole solutions
inspired by noncommutative geometry in the framework of $f(T)$ gravity whereas
Hassan et al. have [35] studied the womhole solutions in teleparallel gravity
inspired by noncommutative geometry. In a work Samir et al.[36] have
investigated the existence of wormholes by introducing noncommutative geometry
in terms of Gaussian and Lorentzian distributions in $f(R,G)$ gravity. The
present paper searches for a solution of ﬁve-dimensional (5D) Einstein-Gauss-
Bonnet equations in the presence of a static, spherically symmetric Gaussian
mass distribution to ﬁnd a thin-shell wormhole solution inspired by the
noncommutative geometry. The statring point is to solve the non-linear
Einstein-Gauss-Bonnet equations for a static and spherically symmetric metric
for a source as mention by Eq.(2).

The paper is organized as follows: In Section 2 we ﬁnd ageneral solution which
leads to the five dimensional (5D) spherically symmetric static Einstein-
Gauss-Bonnet equations for the source as mentioned by Eq.(2), which further
allows us to compute some quantities, viz., the componets of stress-energy
tensors around the thin shell of a noncommutative inspired thin-shell wormhole
in Section 3. In Section 4 we find the Equation of State (EOS) at the throat
of the thin-shell wormhole. We investigate the motion test particle near the
throat to find the nature wormhole in Section 5 whereas Section 6 deals with
the total exotic matter contet at the shell. We analyse the linearized
stability of our model in Section 7 and finally, in Section 8 we pass some
concluding remarks.

##  2 Black holes in non-commutative geometry inspired Einstein-Gauss-Bonnet
gravity

.

The action of Einstein-Gauss-Bonnet gravity in D-dimension [37, 38] can be
written as

$\mathcal{S}=\frac{1}{2\kappa}\int
dx^{D}\sqrt{-g}\left[\mathcal{L}+\alpha\mathcal{L}_{GB}\right]+\mathcal{S}_{m},$
(3)

$\mathcal{S}_{m}$ represents the action associated with matter and $\alpha$ is
a coupling constant with dimension of $(\mbox{length})^{2}$ which is positive
in the hetoretic string theory. For the case of general theory of relativity
$\mathcal{L}=R$, the Ricci scalar and the second-order Gauss-Bonnet term
$\mathcal{L}_{GB}$ is given by

$\mathcal{L}_{GB}=R_{\mu\nu\gamma\delta}R^{\mu\nu\gamma\delta}-4R_{\mu\nu}R^{\mu\nu}+R^{2}.$
(4)

Throughout the paper we assume that $8\pi G=c=1$ and the variation of the
above action with respect to the metric $g_{\mu\nu}$ gives the Einstein-Gauss-
Bonnet equations [40, 41, 42] as

$G_{\mu\nu}^{E}+\alpha G_{\mu\nu}^{GB}=T_{\mu\nu},$ (5)

where $G_{\mu\nu}^{E}$ is the Einstein tensor, while $G_{\mu\nu}^{GB}$ is
explicitly given by [43]

$\displaystyle G_{\mu\nu}^{GB}$ $\displaystyle=$ $\displaystyle
2\;\Big{[}-R_{\mu\sigma\kappa\tau}R_{\quad\nu}^{\kappa\tau\sigma}-2R_{\mu\rho\nu\sigma}R^{\rho\sigma}-2R_{\mu\sigma}R_{\
\nu}^{\sigma}+RR_{\mu\nu}\Big{]}-\frac{1}{2}\mathcal{L}_{GB}g_{\mu\nu}.$ (6)

Here, one can note that the divergence of Einstein-Gauss-Bonnet tensor
$G_{\mu\nu}^{GB}$ vanishes and the spherically symmetric vacuum solutions [37,
38] for this theory is given by

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{D}^{2}$ (7)

where $d\Omega_{D}^{2}$ is the line element on the D unit sphere, i.e.,

$d\Omega_{D}^{2}=d{\theta}_{1}^{2}+\sin^{2}{\theta}_{1}d{\theta}_{2}^{2}+......+\prod_{n=1}^{D-1}\sin^{2}{\theta}_{n}d{\theta}_{D}^{2}$
(8)

and volume of the D unit sphere is given by

$\Omega_{D}=2\frac{\pi^{\frac{D+1}{2}}}{\Gamma(\frac{D+1}{2})}$ (9)

As we have mentioned earlier that noncommutativity eliminates point like
structure in favor of smeared objects in the flat spacetime. The effect of
smearing is mathematically implemented with a Gaussian distribution of minimal
width $\sqrt{\theta}$ [24]. Here, we find 5 dimensional (5D) black hole
solution inspired by the noncommutative geometry in the Einstein-Gauss-Bonnet
gravity. For the matter density $\rho$ mentioned before along with the
condition $g_{00}=1/g_{rr}$ two components of the diagonal stress-energy
tensor reads

$\displaystyle T^{0}_{0}$ $\displaystyle=$
$\displaystyle-T^{r}_{r}=\rho_{\theta}(r)=\frac{\mu}{16\pi^{2}\theta^{2}}e^{{-r^{2}}/{(4\theta)}}.$
(10)

The Bianchi identity $T^{ab};b=0$ [39] reads

$0=\partial_{r}T^{r}_{r}+\frac{1}{2}g^{00}\left[T^{r}_{r}-T^{0}_{0}\right]\partial_{r}g_{00}+\frac{1}{2}\sum
g^{ii}\left[T^{r}_{r}-T^{i}_{i}\right]\partial_{r}g_{ii}$ (11)

and due to the condition $g^{ii}\partial_{r}g_{ii}=2/r$, one can obtain the
energy momentum tensor in the five dimensional (5D) spacetime as

$\displaystyle T^{t}_{t}$ $\displaystyle=$ $\displaystyle
T^{r}_{r}=\rho_{\theta}(r),$ $\displaystyle T^{\theta}_{\theta}$
$\displaystyle=$ $\displaystyle
T^{\phi}_{\phi}=T^{\psi}_{\psi}=\rho_{\theta}(r)+\frac{r}{3}\partial_{r}\rho_{\theta}(r),$
(12)

and the energy-momentum tensor is specified by Eq. (2).

Moreover, Eq. (5) for the matter source as given by Eq. (2), leads to a
general solution [45]

$f_{\pm}(r)=1+\frac{r^{2}}{4{\alpha}}\left[1\pm\sqrt{1+\frac{8\alpha\mu}{r^{4}\pi}\gamma\left(2,\frac{r^{2}}{4\theta}\right)}\right],$
(13)

where $\gamma\left(2\ ,r^{2}/4\theta\,\right)$ is the lower incomplete Gamma
function which is defined as

$\gamma\left(2\
,r^{2}/4\theta\,\right)\equiv\int_{0}^{r^{2}/4\theta}\;du\;u\;e^{-u}.$ (14)

To proceed further, one can define the mass-energy $\mu(r)$ as

$\mu(r)=\frac{2\mu}{\pi}\gamma\left(2\ ,r^{2}/4\theta\,\right),$ (15)

and the total mass-energy ($M$) measured by asymptotic observer [46] is given
by

$M=\lim_{r\rightarrow\infty}\mu(r)=\frac{2\mu}{\pi},$ (16)

and the solution as given by Eq. (13) reads

$f_{\pm}(r)=1+\frac{r^{2}}{4{\alpha}}\left[1\pm\sqrt{1+\frac{8\alpha
M}{r^{4}}\gamma\left(2,\frac{r^{2}}{4\theta}\right)}\right],$ (17)

Here, if one consider $M_{1}=\frac{M}{\sqrt{\theta}}$,
$\alpha_{1}=\frac{\alpha}{\sqrt{\theta}}$ and $r_{1}=\frac{r}{\sqrt{\theta}}$,
then the metric function takes the form

$f(r)=1+\frac{r_{1}^{2}\left(1-\sqrt{\frac{8\alpha_{1}M_{1}-2\alpha_{1}M_{1}e^{-\frac{r_{1}^{2}}{4}}\left(r_{1}^{2}+4\right)}{r_{1}^{4}}+1}\right)}{4\alpha_{1}}$
(18)

Figure 1: The variation in the metric function f(r) with respect to r $(km)$ .

In Fig. (1) the function $f(r)$ has been plotted w.r.t. $r$ for different
values of $\frac{\alpha}{\sqrt{\theta}}$ and from the figure it is clear that
the difference between the values of $f(r)$ at the inner horizon with that at
outer horizon shows an increasing profile with the increase in
$\frac{M}{\sqrt{\theta}}$ value. There also exits a critical value for
$\frac{M}{\sqrt{\theta}}$ to form a black hole and from the it is revealed
that there is no significant dependence on $\alpha$ at two horizons.

##  3 Thin-shell wormholes from black holes in non-commutative geometry
inspired Einstein-Gauss-Bonnet gravity

The thin shell wormhole that we are to construct can be done by coping the
solutions of two regular black holes and the region can be written as
(removing from each of the copies)

$\Omega^{\pm}\equiv\\{r\leq a\mid a>r_{h}\\},$ (19)

where $a$ is a constant and $r_{h}$ is the event horizon of the black hole. As
a result of the above operation, the timelike hypersurfaces can be expressed
topologically,

$d\Omega^{\pm}\equiv\\{r=a\mid a>r_{h}\\}$ (20)

The two timelike hypersurfaces located at $d\Omega^{+}=d\Omega^{-}\equiv\
d\Omega$ along with two asymptotically flat regions that are connected by a
wormhole whose throat is at $d\Omega$. The intrinsic metric upon $d\Omega$ can
be written as

$ds^{2}=-d\tau^{2}+a^{2}(\tau)d\Omega^{2}_{D},$ (21)

where $\tau$ is the proper time on the junction shell and $a(\tau)$ is the
radius of the throat. Using Lanczos equation[47, 48, 49, 50, 51, 52] the
intrinsic surface strees energy tensor can be obtained as

$S^{i}_{j}=-\frac{1}{8\pi}(k^{i}_{j}-\delta^{i}_{j}\ k^{l}_{l})$ (22)

where $k^{\pm}_{ij}=K^{+}_{ij}-K^{-}_{ij}$ is the discontinuity in the second
fundamental form. We can write the second fundamental[53, 54, 55, 56, 57, 58]
form as

$K_{ij}^{\pm}=-n_{\nu}[\frac{\partial^{2}X_{\nu}}{\partial\xi^{i}\partial\xi^{j}}+\Gamma_{\alpha\beta}^{\nu^{\pm}}\frac{\partial
X^{\alpha}}{\partial\xi^{i}}\frac{\partial X^{\beta}}{\partial\xi^{j}}]$ (23)

where $n_{\nu}$ is the unit normal vector to $d\Omega$ and $\xi^{i}$
represents the intrinsic co-ordinates. The parametric equation at the
hypersurface $d\Omega$ is $f(x^{\mu}(\xi^{i}))=0$.

Using the above equation, the normal vector can be expressed as

$n_{\mu}^{\pm}=\pm|g^{\alpha\beta}\frac{\partial F}{\partial
X^{\alpha}}\frac{\partial F}{\partial X^{\beta}}|^{-\frac{1}{2}}\frac{\partial
F}{\partial X^{\mu}}$ (24)

with $n^{\mu}n_{\mu}=+1$. $k_{ij}$ can be written as
$k^{i}_{j}=diag(k^{\tau}_{\tau},k^{\theta_{1}}_{\theta_{1}}k^{\theta_{2}}_{\theta_{2}},....,k^{\theta_{D}}_{\theta_{D}})$,
as a result of spherical symmetry.

Therefore, the surface-energy tensor has a form as
$S^{i}_{j}=diag(-\sigma,p_{\theta_{1}},p_{\theta_{2}},.....,p_{\theta_{D}})$,
where $\sigma$ is the surface energy density and $p$ is the surface pressure.
Now from the Lanczos equation, we can get

$\sigma=-\frac{D}{4\pi a}\sqrt{f(a)+\dot{a}^{2}}$ (25)

$p=p_{\theta_{1}}=p_{\theta_{2}}......=p_{\theta_{D}}=\frac{1}{8\pi}\frac{2\ddot{a}+f^{\prime}(a)}{\sqrt{f(a)+\dot{a}^{2}}}-\frac{D-1}{D}\sigma,$
(26)

where the dots and primes denote the differentiation with respect to $\tau$
and $a$ respectively.

The conservation equations can be written as

$\frac{d}{d\tau}(\sigma a^{D})+p\frac{d}{d\tau}(a^{D})=0$ (27)

or

$\dot{\sigma}+D\frac{\dot{a}}{a}(p+\sigma)=0,$ (28)

which are obeyed by $\sigma$ and $p$. For the static configuration having a
radius $a$, we take $\dot{a}=0$ and $\ddot{a}=0$. All the calculations are
done in five dimensions (5D), i.e., by putting $D=3$. Now Eq.( 25) and Eq.(26)
reduces to

$\sigma=-\frac{3}{4\pi a}\sqrt{f(a)}$ (29)

$\sigma(a)=-\frac{3\sqrt{\frac{a^{2}\left(1-\sqrt{\frac{8\alpha
M\left(1-\frac{e^{-\frac{a^{2}}{4\theta}}\left(a^{2}+4\theta\right)}{4\theta}\right)}{a^{4}}+1}\right)}{4\alpha}+1}}{4\pi
a}$ (30)

Now, for $\sigma_{1}=\sqrt{\theta}\sigma$ we have

$\sigma_{1}=-\frac{3\sqrt{4-\frac{a^{2}\left(\sqrt{\frac{a^{4}-2\left(a^{2}+4\right)e^{-\frac{a^{2}}{4}}\alpha
M+8\alpha M}{a^{4}}}-1\right)}{\alpha}}}{8\pi a}$ (31)

where $a_{1}=\frac{a}{\sqrt{\theta}}$ and

$p=\frac{1}{8\pi}\frac{f^{\prime}(a)}{\sqrt{f(a)}}-\frac{2}{3}\sigma$ (32)

$p(a)=\frac{4\alpha
Me^{-\frac{a^{2}}{4\theta}}\left(4\theta+a^{2}\right)+\theta\left(-\left(16\alpha
M+3a^{4}\right)+\left(3a^{2}+8\alpha a^{2}\right)\sqrt{\frac{8\alpha
M-\frac{2\alpha
Me^{-\frac{a^{2}}{4\theta}}\left(4\theta+a^{2}\right)}{\theta}+a^{4}}{a^{4}}}\right)}{8\pi\alpha\theta
a^{3}\left(\sqrt{\frac{8\alpha M-\frac{2\alpha
Me^{-\frac{a^{2}}{4\theta}}\left(4\theta+a^{2}\right)}{\theta}+a^{4}}{a^{4}}}\right)\sqrt{4-\frac{a^{2}\left(\sqrt{\frac{8\alpha
M-\frac{2\alpha
Me^{-\frac{a^{2}}{4\theta}}\left(4\theta+a^{2}\right)}{\theta}+a^{4}}{a^{4}}}-1\right)}{\alpha}}}.$
(33)

Similarly, for $p_{1}=\sqrt{\theta}p$ we have

$\displaystyle
p_{1}=\frac{4e^{-\frac{a_{1}^{2}}{4}}\alpha_{1}M_{1}(a^{2}_{1}+4)-(16\alpha_{1}M_{1}+3a^{4}_{1})+(3a^{4}_{1}+8\alpha_{1}a^{2}_{1})\sqrt{\frac{2\alpha_{1}M_{1}\left(4-e^{-\frac{a_{1}^{2}}{4}}\left(a_{1}^{2}+4\right)\right)}{a_{1}^{4}}+1}}{8\pi\alpha_{1}a_{1}^{3}\left(\sqrt{\frac{2\alpha_{1}M_{1}\left(4-e^{-\frac{a_{1}^{2}}{4}}\left(a_{1}^{2}+4\right)\right)}{a_{1}^{4}}+1}\right)\sqrt{\frac{4\alpha_{1}+a_{1}^{2}\left(1-\sqrt{\frac{2\alpha_{1}M_{1}\left(4-e^{-\frac{a_{1}^{2}}{4}}\left(a_{1}^{2}+4\right)\right)}{a_{1}^{4}}+1}\right)}{\alpha_{1}}}}$
(34)

where the function $f(a)$ is given by Eq.(5). The negative sign in the surface
density itself implies the existence of exotic matter in the shell which
automatically violates the null energy condition as well as the weak energy
condition.

Figure 2: The variation in $\sigma$ $(km^{-2})$ with respect to a $(km)$

In Fig. (2) we have plotted the surface energy density with respect to throat
radius $a$ for different values of $\frac{M}{\theta}$ and from the figure it
is revealed that as we are increasing the throat radius, the negative values
of $\sigma$ increases rapidly and reaches a maximum value and after that it
slightly decreases and tends to a saturated value. Fig. (3) shows the
variation of surface pressure with respect to throat radius $a$ and it is
clear from the figure that as we increase the throat radius, the value of $p$
is decreasing rapidly but after a certain point it has no dependence on
$\frac{M}{\theta}$ value. Fig. (4) represents the violation of the null energy
condition which is a necessary for the construction of wormhole.

Figure 3: Pressure p $(km^{-2})$ is plotted with respect to a $(km)$

Figure 4: $\sigma+p$ $(km^{-2})$ is plotted with respect to a $(km)$

##  4 Equation of state

The Equation of State (EOS) is given by

$p(a)=\omega~{}\sigma(a).$ (35)

Figure 5: The variation of Equation of State (EOS) parameter with respect to a
$(km)$

Therefore, by virtue of Eqs. (31) and (34) the equation of state parameter
$\omega$ can be explicitely written as

$\omega=\frac{e^{-\frac{a^{2}}{4}}\left(4\left(a^{2}+4\right)\alpha
M+e^{\frac{a^{2}}{4}}\left(3a^{4}\left(\sqrt{\frac{a^{4}-2\left(a^{2}+4\right)e^{-\frac{a^{2}}{4}}\alpha
M+8\alpha M}{a^{4}}}-1\right)+8\alpha
a^{2}\sqrt{\frac{a^{4}-2\left(a^{2}+4\right)e^{-\frac{a^{2}}{4}}\alpha
M+8\alpha M}{a^{4}}}-16\alpha
M\right)\right)}{3a^{2}\sqrt{\frac{a^{4}-2\left(a^{2}+4\right)e^{-\frac{a^{2}}{4}}\alpha
M+8\alpha
M}{a^{4}}}\left(a^{2}\left(\sqrt{\frac{a^{4}-2\left(a^{2}+4\right)e^{-\frac{a^{2}}{4}}\alpha
M+8\alpha M}{a^{4}}}-1\right)-4\alpha\right)}$ (36)

Fig. (5) shows the variation of equation of state parameter with respect to
throat radius for different values of $\frac{\alpha}{\theta}$ and it reveals
that the negative values of $\omega$ decreases as we increase the throat
radius. Also, there is no significant dependence on $\frac{\alpha}{\theta}$
values.

##  5 The gravitational field

To study the nature of the gravitational field of our wormhole we calculate
the observer’s four acceleration which is given by

$A^{\mu}=u^{\mu}_{;\nu}u^{\nu},$ (37)

where

$u^{\nu}=\frac{\partial
x^{\nu}}{\partial\tau}=(\frac{1}{\sqrt{f(r)}},0,0,0,0).$ (38)

Hence, the only non-zero component can be written as

$A^{r}=\frac{f^{\prime}(r)}{2}$ (39)

or

$A^{r}=\frac{e^{-\frac{a^{2}}{4\theta}}a\left(4\theta^{2}e^{\frac{a^{2}}{4\theta}}\left(\sqrt{\frac{8\alpha
M\left(1-\frac{e^{-\frac{a^{2}}{4\theta}}\left(4\theta+a^{2}\right)}{4\theta}\right)}{a^{4}}+1}-1\right)-\alpha
M\right)}{16\alpha\theta^{2}\sqrt{\frac{2\alpha
M\left(4-\frac{e^{-\frac{a^{2}}{4\theta}}\left(4\theta+a^{2}\right)}{\theta}\right)}{a^{4}}+1}}$
(40)

Now $A^{r}_{1}=\sqrt{\theta}A^{r}$, therefore,

$A^{r}_{1}=\frac{e^{-\frac{a_{1}^{2}}{4}}a_{1}\left(4e^{\frac{a_{1}^{2}}{4}}\left(\sqrt{\frac{2\alpha_{1}M_{1}\left(4-e^{-\frac{a_{1}^{2}}{4}}\left(a_{1}^{2}+4\right)\right)}{a_{1}^{4}}+1}-1\right)-\alpha_{1}M_{1}\right)}{16\alpha_{1}\sqrt{\frac{2\alpha_{1}M_{1}\left(4-e^{-\frac{a_{1}^{2}}{4}}\left(a_{1}^{2}+4\right)\right)}{a_{1}^{4}}+1}}$
(41)

For a test particle moving towards the radial direction from rest the geodesic
equation can be written as

$\frac{d{{}^{2}}r}{d\tau^{2}}=-\Gamma^{r}_{tt}\left(\frac{dt}{d\tau}\right)^{2}=-a^{r}$
(42)

The nature of the wormhole is attractive if $A^{r}>0$ and repulsive if
$A^{r}<0$ depending on the parameters of equation. We have plotted $A^{r}$
with respect to throat radius in Fig. (5) and it shows that the model of
wormhole constructed is attractive in nature under the noncommutative geometry
inspired Einstein-Gauss-Bonnet gravity. Also, the attractive nature is
decreasing with the increase of throat radius.

Figure 6: Accelaration $a^{r}$ $(km)^{-1}$ of the test particle with respect
to a $(km)$

##  6 The total amount of exotic matter

We can determine the total amount of exotic matter in a thin shell wormhole by
using the integral

$\Omega_{\sigma}=\int[\sigma+p]\sqrt{-g}d^{D+1}x$ (43)

A new radial co-ordinate $R=\pm(r-a)$ has been introduced by Eiroa and Simeone
[59] as

$\Omega_{\sigma}=\int_{0}^{2\pi}\int_{0}^{\pi}.....\int_{0}^{\pi}\int_{-\infty}^{\infty}[\rho+p]\sqrt{-g}dRd{\theta_{1}}d{\theta_{2}}.....d{\theta_{D}}$
(44)

Since, the radius of the shell is constant it does not exert any radial
pressure, so that we have $\rho=\delta(R)\sigma(a)$ and one can write

$\Omega_{\sigma}=\int_{0}^{2\pi}\int_{0}^{\pi}.....\int_{0}^{\pi}\rho\sqrt{-g}]|_{r=a}d{\theta_{1}}d{\theta_{2}}.....d{\theta_{D}}=2a^{D}\sigma(a)\frac{\pi^{\frac{D+1}{2}}}{\Gamma(\frac{D+1}{2})}$
(45)

Figure 7: The variation of the total amount of exotic matter $\Omega$ at the
shell with respect to a $(km)$.

$\Omega=-\frac{3}{2}\pi a^{2}\sqrt{\frac{a^{2}\left(1-\sqrt{\frac{8\alpha
M\left(1-\frac{e^{-\frac{a^{2}}{4\theta}}\left(4\theta+a^{2}\right)}{4\theta}\right)}{a^{4}}+1}\right)}{4\alpha}+1}$
(46)

Now $\Omega_{1}=\frac{\Omega}{\theta}$ where

$\Omega_{1}=-\frac{3}{4}\pi
a_{1}^{2}\sqrt{4-\frac{a_{1}^{2}\left(\sqrt{\frac{8\alpha_{1}M_{1}-2\alpha_{1}M_{1}e^{-\frac{a_{1}^{2}}{4}}\left(a_{1}^{2}+4\right)+a_{1}^{4}}{a_{1}^{4}}}-1\right)}{\alpha_{1}}}$
(47)

Fig.(7) shows the variation of total exotic matter content with respect to
throat radius and from figure we can assert that as we increase the value of
throat radius the amount of exotic matter at the shell increases rapidly. So,
to minimize the amount of exotic matter content we have to consider the throat
of the wormhole near the event horizon.

Figure 8: Three dimensional plot of total exotic matter $\Omega$ with respect
to total mass $M$ as well as $\alpha$.

##  7 Linearized stability

To obtain the stability of the thin shell wormhole one has to perform a small
perturbation around a static solution at $a=a_{0}$. One can rearrange the Eq.
(25) to obtain the equation of motion at the thin shell region as

$\dot{a}^{2}+V(a)=0$ (48)

where the potential can be defined as

$V(a)=f(a)-\frac{16\pi^{2}a^{2}\sigma^{2}(a)}{D^{2}}.$ (49)

Now, expanding the potential $V(a)$ into a Taylor series around $a_{0}$ upto
second order we have

$V=V(a_{0})+V^{\prime}(a_{0})(a-a_{0})+\frac{1}{2}V^{\prime\prime}(a_{0})(a-a_{0})^{2}+\mathcal{O}[(a-a_{0})^{3}]$
(50)

where the prime $(^{\prime})$ represents the derivative with respect to $a$.

Figure 9: Plots of $\beta^{2}$ as function of $a_{0}$ for $\alpha=0.1$

As we are linearizing around a static solution at $a=a_{0}$, we have
$V(a_{0})=0$ and $V^{\prime}(a_{0})=0$ and the stable equilibrium
configurations correspond to the condition $V^{\prime\prime}(a_{0})>0$. Here,
one can introduce a the parameter $\beta^{2}(\sigma)$ as the speed of sound
which is given by

$\beta^{2}(\sigma)=\frac{\partial
p}{\partial\sigma}=\frac{p^{\prime}}{\sigma^{\prime}}$ (51)

The first and second order derivatives of $V(a)$ is found to be

$V^{\prime}(a)=f^{\prime}(a)-\frac{32\pi^{2}a\sigma}{D^{2}}\left[\sigma-D(\sigma+p)\right]$
(52)

and

$V^{\prime\prime}(a)=f^{\prime\prime}(a)-\frac{32\pi^{2}}{D^{2}}\left[\\{(D-1)\sigma+Dp\\}^{2}-D\sigma(\sigma+p)(D-1+D\beta^{2})\right]$
(53)

Here, $V(a_{0})=0$ and $V^{\prime}(a_{0})=0$ and for a stable wormhole
solution one must have $V^{\prime\prime}(a_{0})>0$, i.e.,

$\beta^{2}<\frac{D^{2}f^{\prime\prime}\left(a_{0}\right)-32\pi^{2}\left(D^{2}(p+\sigma)(p+2\sigma)-3D\sigma(p+\sigma)+\sigma^{2}\right)}{32\pi^{2}D^{2}\sigma(p+\sigma)}$
(54)

Figure 10: Plots of $\beta^{2}$ as function of $a_{0}$ for $\alpha=0.2$

Using Eqs. (30) and (33) the above inequality takes the form

$\beta^{2}<-\frac{a^{2}_{0}(f_{0}^{\prime})^{2}-2a_{0}f\left(a_{0}f_{0}^{\prime\prime}+(D-1)f_{0}^{\prime}\right)+4(D-1)f^{2}_{0}}{2Df_{0}\left(2f_{0}-a_{0}f_{0}^{\prime}\right)},$
(55)

where $f_{0}=f(a_{0})$ , $f_{0}^{\prime}=f^{\prime}(a_{0})$ and
$f^{\prime\prime}_{0}=f^{\prime\prime}(a_{0})$. Figs. (9) , (10), and (11)
shows the stability regions of the wormhole solutions. From all these figures
one can infer that as one increases the $\frac{M}{\theta}$ value the stability
region is shifted from the higher value of throat radius to the lower value of
throat radius $a$. Also, there is no significant change in the nature of the
stability region as one varies the value of $\alpha$ which is evident from the
figures.

Figure 11: Plots of $\beta^{2}$ as function of $a_{0}$for $\alpha=0.4$

##  8 Conclusion

In this paper we construct a new thin-shell wormhole from black holes in non-
commutative geometry inspired Einstein-Gauss-Bonnet gravity. For that we find
out the non-commutative geometry inspired black hole solution in Einstein-
Gauss-Bonnet gravity. Then, the ‘Cut and Paste’ technique which was first
introduced by M. Visser [5], is employed to construct the wormhole from the
corresponding black hole spacetime assumed to be asymptotically flat in
nature.

For the above mentioned structural form of a thin shell wormhole we have noted
down several salient aspects of the solution as can be described below:

(1) Pressure-density profile: the surface energy density with respect to
throat radius $a$ has been plotted in Fig. (2) for different values of
$\frac{M}{\theta}$ and the figure shows that the negative values of $\sigma$
increases rapidly with the increase of throat radius and reaches a maximum
value and after that it slightly decreases and tends to attain a saturated
value. Fig. (3) shows the variation of surface pressure with respect to throat
radius $a$ and reveals that as we increase the throat radius, the value of $p$
is decreasing rapidly but after a certain point it has no dependence on
$\frac{M}{\theta}$ value. Also, Fig. (4) represents the violation of the null
energy condition which is a necessary for the construction of wormhole.

(2) Equation of state: Fig. (5) which shows the variation of equation of state
parameter with respect to throat radius for different values of
$\frac{\alpha}{\theta}$ reveals that the negative values of $\omega$ decreases
as we increase the throat radius. Also, there is no significant dependence on
$\frac{\alpha}{\theta}$ values.

(3) The gravitational field: The acceleration of test particle $a^{r}$ has
been plotted with respect to throat radius$a$ in Fig. (5) which shows that the
model of wormhole constructed is attractive in nature under the noncommutative
geometry inspired Einstein-Gauss-Bonnet gravity as $a^{r}>0$ through out the
entire range of $a$. Moreover, the attractive nature is decreasing with the
increase of throat radius.

(4) The total amount of exotic matter: Fig.(7) shows the variation of total
exotic matter content with respect to throat radius $a$ and one can assert
that as one increases the value of throat radius the amount of exotic matter
at the shell increases rapidly. So, to minimize the amount of exotic matter
content one has to consider the throat of the wormhole near the event horizon.

(5) Linearized stability: In Figs. (9) , (10), and (11) we have depicted the
stability regions of the wormhole solutions for different values of $\alpha$.
From all these figures one can infer that as one increases the
$\frac{M}{\theta}$ value the stability region is shifted from the higher value
of throat radius to the lower value of throat radius $a$ and there is no
significant change in the nature of the stability region as one varies the
value of $\alpha$.

Finally, we can conclude that our model of thin-shell wormhole which is
constructed from the ‘Cut and Paste’ technique under the noncommutative
geometry inspired Einstein-Gauss-Bonnet Gravity is found to be plausible with
reference to the other model of thin-shell wormhole available in literature.

Acknowledgment

NR is thankful to UGC MANF(MANF-2018-19-WES-96213) for providing financial
support.FR & MK would like to thank the authorities of the Inter-University
Centre for Astronomy and Astrophysics, Pune, India for providing research
facilities.

## References

  * [1] A. Einstein and N. Rosen, Phys. Rev. 48, 73 (1935).
  * [2] R. W. Fuller and J. A. Wheeler, 128, 919 (1962).
  * [3] M. S. Morris,K. S. Throne, Am. J. Phys. 56, 39 (1988).
  * [4] M. Visser, S. Kar , N. Dadhich, Phys. Rev. Lett. 90, 201102 (2003).
  * [5] M. Visser, Nucl. Phys. B 328, 203 (1989).
  * [6] E. Poisson, M. Visser, Phys. Rev. D 52, 7318 (1995).
  * [7] W. Israel, Nuovo Cimemto 44, 1 (1966); erratum : 48, 463 (1967).
  * [8] G. Darmois, “Mémorial des sciences mathématiques XXV”, Fasticule XXV, (Gauthier-Villars, Paris, France, 1927), chap. V.
  * [9] E.F. Eiroa and C. Simeone, Phys. Rev. D 71, 127501 (2005).
  * [10] E.F. Eiroa and C. Simeone, Phys. Rev. D 70, 044008 (2004).
  * [11] M. Thibeault, C. Simeone, and E.F. Eiroa, Gen Relativ Gravit 38, 1593 (2006).
  * [12] C. Lanczos, Ann. Math. 39, 842 (1938).
  * [13] D. Lovelock, J. Math. Phys. 12, 498 (1971).
  * [14] C. Callan, I. Klebanov and M. Perry, Nucl. Phys. B 278, 78 (1986).
  * [15] P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B 258, 46 (1985).
  * [16] D. Gross and J. Sloan, Nucl. Phys. B 291, 41 (1987).
  * [17] M. Guica, L. Huang, W. Li and A. Strominger, J. High Energy Phys. 0610, 036 (2006).
  * [18] B. P. Abbott et al. (Virgo, LIGO Scientific Collaboration), Phys. Rev. Lett. 116, 061102 (2016).
  * [19] A. Kobakhidze, C. Lagger and A. Manning, Phys. Rev. D 94, 064033 (2016)
  * [20] A. Saha and S. Gangopadhyay, Class. Quant. Grav. 33, 205006 (2016)
  * [21] H. S. Snyder, Phys. Rev. 71 , 38 ( 1947).
  * [22] H. S. Snyder, Phys. Rev. 72, 68 (1947).
  * [23] C. A. Stephan .: arXiv:1305.3066 (2013)
  * [24] P. Nicolini, A Smailagic, and E. Spalluci, Phys. Lett. B 632, 547 (2006).
  * [25] P. Nicolini, and E. Spalluci, Class. Quantum Gravit. 27, 015010 (2010).
  * [26] F.S.N. Lobo, and R. Garattini, J. High Energ. Phys. 2013, 65 (2013).
  * [27] F Rahaman et al., Phys. Rev. D 86, 106010 (2012).
  * [28] E. Spallucci, A. Smailagic and P. Nicolini, Phys. Lett. B 670, 449 (2009).
  * [29] T. G. Rizzo, J. High Energ. Phys. 0609, 021 (2006).
  * [30] R. Garattini and F.S.N. Lobo, Phys. Lett. B 671, 146 (2009).
  * [31] F. Rahaman, Saibal Ray, G. S. Khadekar, P. K. F. Kuhfittig, and I. Karar, Int. J. Theor. Phys, 54, 699 (2015).
  * [32] F. Rahaman, S. Karmakar, I. Karar, and S. Ray, Phys. Lett. B 746, 73 (2015).
  * [33] P.K.F. Kuhfittig and V. D. Gladney, J. appl. math. phys 5 , 574 (2017)
  * [34] M. Sharif and Shamaila Rani, Phys. Rev. D 88, 123501 (2013)
  * [35] Z. Hassan, G. Mustafa, P.K. Sahoo, Symmetry 13, 1260 (2021).
  * [36] M. F. Shamir, G. Mustafa, and S. Waseem, Int. J. Geom. Methods Mod. 2020, 17, 2050129.
  * [37] D. G. Boulware, S. Deser, Phys. Rev. Lett. 1985, 55, 2656.
  * [38] J. T. Wheeler, Nucl. Phys. B268, 737 (1986); B273, 732 (1986).
  * [39] T. G. Rizzo, JHEP 0609, 021 (2006).
  * [40] R.G. Cai, Phys. Lett. B 582, 237 (2004).
  * [41] C. Sahabandu, P. Suranyi, C. Vaz and L.C.R. Wijewardhana, Phys. Rev. D 73, 044009 (2006).
  * [42] S. G. Ghosh and D. W. Deshkar, Phys. Rev. D 77, 047504 (2008)
  * [43] D. Kastor and R. B. Mann, J. High Energy Phys. 0604, 048 (2006).
  * [44] E. Herscovich and M. G. Richarte, Phys. Lett. B 689, 192 (2010).
  * [45] E. Spallucci, A. Smailagic and P. Nicolini, Phys. Lett. B 670, 449 (2009).
  * [46] S. Ansoldi, P. Nicolini, A. Smailagic and E. Spallucci, Phys. Lett. B 645, 261 (2007).
  * [47] W. Israel, Nuovo Cimemto 44, 1 (1966); 48, 463(E) (1967).
  * [48] G. Darmois, “Mémorial des sciences mathématiques XXV”, Fasticule XXV, (Gauthier-Villars, Paris, France, 1927), chap. V.
  * [49] K. Lanczos, Ann. Phys. (Berlin) 379, 518 (1924).
  * [50] N. Sen, Ann. Phys. (Berlin) 378, 365 (1924).
  * [51] G. P. Perry and R. B. Mann, Gen. Relativ. Gravit. 24, 305 (1992).
  * [52] P. Musgrave and K. Lake, Classical Quantum Gravity 13, 1885 (1996).
  * [53] F. Rahaman, M. Kalam, and S. Chakraborty, Gen. Relativ. Gravit. 38, 1687 (2006).
  * [54] F. Rahaman, M. Kalam, and K. A. Rahman, Acta Phys. Pol. B 40, 1575 (2009).
  * [55] A. A. Usmani, Z. Hasan, F. Rahaman, Sk. A. Rakib, S. Ray, and P. K. F. Kuhfittig, Gen. Relativ. Gravit. 42, 2901 (2010).
  * [56] F. Rahaman, K.A. Rahman, Sk. A. Rakib, and P. K. F. Kuhfittig, Int. J. Theor. Phys. 49, 2364 (2010).
  * [57] G. A. S. Dias and J. P. S. Lemos, Phys. Rev. D 82, 084023 (2010).
  * [58] F. Rahaman, P. K. F. Kuhfittig, M. Kalam, A. A. Usmani, and S. Ray, Classical Quantum Gravity 28, 155021 (2011).
  * [59] E. F. Eiroa and C. Simeone, Phys. Rev. D71, 127501 (2005)


# A Categorical Foundation for Bayesian Probability

Jared Culbertson and Kirk Sturtz

###### Abstract.

Given two measurable spaces $H$ and $D$ with countably generated
$\sigma$-algebras, a perfect prior probability measure $P_{H}$ on $H$ and a
sampling distribution $\mathcal{S}\colon H\rightarrow D$, there is a
corresponding inference map $\mathcal{I}\colon D\rightarrow H$ which is unique
up to a set of measure zero. Thus, given a data measurement $\mu\colon
1\rightarrow D$, a posterior probability $\widehat{P_{H}}=\mathcal{I}\circ\mu$
can be computed. This procedure is iterative: with each updated probability
$P_{H}$, we obtain a new joint distribution which in turn yields a new
inference map $\mathcal{I}$ and the process repeats with each additional
measurement. The main result uses an existence theorem for regular conditional
probabilities by Faden, which holds in more generality than the setting of
Polish spaces. This less stringent setting then allows for non-trivial
decision rules (Eilenberg–Moore algebras) on finite (as well as non finite)
spaces, and also provides for a common framework for decision theory and
Bayesian probability.

NB: This paper has been published in Applied Categorical Structures,
http://link.springer.com/article/10.1007/s10485-013-9324-9, please contact the
authors at<EMAIL_ADDRESS>for the correct reference.

##  1\. Introduction

Bayesian probability is a subject that has proven very successful in
prediction, inference and model selection [3, 11, 12]. Čencov [21] gives a
categorical foundation for non-Bayesian statistical inference, but as far as
the authors are aware, a categorical framework for Bayesian probability has
not been fully developed. Lawvere took the first steps in this direction by
defining the category of probabilistic mappings in the unpublished manuscript
[15]. Following this, Lawvere and Huber [16] gave a seminar in Zurich on
_Bayesian Sections_ , further developing this category as a basis for Bayesian
probability. The first appearance in the literature was an expansion of these
ideas by Giry [10], who showed that the endofunctor on the category of
measurable spaces $T\colon\mathcal{M}\to\mathcal{M}$ associated to the
probability adjunction given by Lawvere forms a monad, and that Lawvere’s
category of probabilistic mappings is the Kleisli category of that monad.

Subsequently, Meng [17], examined the category of convex sets and affine
linear maps, which can be shown to be equivalent to the category of
Eilenberg–Moore algebras of the Giry monad. This category can be thought of as
the category of “decision rules” since the objects of that category are
certain measurable functions $TX\to X$ whose fibers partition the space of
probability measures on a given space $X$ into positive convex measurable
sets.111Doberkat [6] refers to this process of making decision rules as
_derandomization_ , which in certain applications like probabilistic semantics
may be a more appropriate terminology. Based on the work by Giry restricting
the monad to Polish spaces, Doberkat[7, 5] has since characterized the
Eilenberg–Moore $T$-algebras for these topological spaces. That work, however,
was based upon giving the space of probability measures the $\sigma$-algebra
generated by the weak topology (as used for the Polish space monad) which
results in (nontrivial) finite spaces having no $T$-algebras. In the final
section, we show that this negative result can be circumvented by avoiding
topological conditions and using the initial $\sigma$-algebra generated by the
evaluation maps. Others, including Wendt [22], van Breugel [4] and
Abramsky–Blute–Panangaden [1] have also studied similar constructions.

Central to the study of Bayesian probability is the existence of regular
conditional probabilities. Many textbooks restrict to Polish spaces in order
to prove this existence (e.g., see [8]), though this is not a strictly
necessary condition. Several more general characterizations of conditions
which guarantee the existence of regular conditional probabilities have been
found, either restricting the spaces involved or the joint distributions which
are allowed. In [18], Pachl does not require even countably generated
$\sigma$-algebras, but relies instead on a certain notion of compactness. We
will prefer to follow [9], where Faden gives a necessary and sufficient
condition when we restrict to countably generated spaces. Namely, the
marginals of a joint distribution must give perfect measure spaces. The class
of perfect measures is broad and includes, for example, all Radon measures.
For this reason, we will only consider perfect measures and begin by showing
that the Giry monad restricted to perfect probability measures is still a
monad (this is straightforward based on Theorem 2.2). Note that everything
prior to Section 3.2 holds without restricting to perfect probability measures
(see [10]).

The main theorem (Theorem 4.1) states that inference maps are uniquely
determined by a prior probability and a sampling distribution. This result
follows from the existence of regular conditional probabilities, and we
restate an existence theorem (Theorem 3.1) of Faden [9]. Using our
characterization of Bayesian probability and the well-known fact that the
Kleisli category embeds into the category of $T$-algebras, we can then see
that the category of decision rules provides a common framework for both
decision theory and Bayesian probability. Some of these ideas are similar in
spirit to the general notion of distributions based on commutative monads
found in the recent paper of Kock [13].

##  2\. The Category of Perfect Probabilistic Mappings

We begin with an overview of the category of perfect probabilistic mappings,
which is a slight modification of Lawvere’s category (see [15]) of
probabilistic mappings. The restrictions to have countably generated spaces as
objects and restrict to perfect probability measures in the definition of the
morphisms are required to ensure the existence of regular conditional
probabilities, but are otherwise unnecessary. We include these restrictions
throughout the paper to avoid overcomplicating the statements of the main
theorems. See the introduction for a discussion of alternative definitions.
Most of the following fundamental results are widely known without the
restriction to perfect measures, and we indicate where the usual arguments
must be modified due to this additional constraint. We also review the
definition and foundational properties of perfect probability measures.

Let us fix the following notation. We will denote the $\sigma$-algebra of a
measurable space $X$ by $\Sigma_{X}$, and the category of countably generated
measurable spaces and measurable functions by $\mathcal{M}_{cg}$. For an
object $(X,\Sigma_{X})$ in $\mathcal{M}_{cg}$ we will often drop the
associated $\sigma$-algebra from the notation and denote it simply by $X$ when
the $\sigma$-algebra is obvious or inconsequential. We will use
$(1,\Sigma_{1})$ and $(2,\Sigma_{2})$ for the one-element and two-element
measurable sets with the discrete $\sigma$-algebras, but we will similarly
just write “$1$” or “$2$” when these are used as objects in some category.

######  Definition 2.1.

A measure space $(X,\Sigma,\mu)$ is called perfect if for any measurable
function $f\colon X\to\mathbb{R}$, there exists a Borel set $E\subset f(X)$
such that $\mu(f^{-1}(E))=\mu(X)$. A family $\\{\mu_{i}\\}_{i\in I}$ of
measures on $X$ is equiperfect if given $f$ as before, there exists a single
Borel set $E\subset f(X)$ with $\mu_{i}(f^{-1}(E))=\mu_{i}(X)$ for all $i\in
I$.

The following theorem collects many basic results about perfect probability
measures. We refer the reader to [19] and [9] and the references therein for
proofs and more details on perfect measures.

######  Theorem 2.2.

Let $(X,\Sigma_{X})$ and $(Y,\Sigma_{Y})$ be measurable spaces. Then

  1. (a)

if $P$ is a {0,1}-valued probability measure, then $P$ is perfect

  2. (b)

if $P$ and $Q$ are probability measures on $X$ such that $Q\ll P$, then $Q$ is
perfect,

  3. (c)

if $\Sigma_{X}^{\prime}\subset\Sigma_{X}$ and a probability measure $P$ on $X$
is perfect, then $P$ restricted to $\Sigma_{X}^{\prime}$ is perfect,

  4. (d)

if $f\colon X\to Y$ is a measurable function and a probability measure $P$ on
$X$ is perfect, then $f_{\ast}P$ is a perfect probability measure on $Y$,

  5. (e)

if $J$ is a probability measure on $(X\times Y,\Sigma_{X}\otimes\Sigma_{Y})$
with marginals $P$ and $Q$, then $J$ is perfect if and only if $P$ and $Q$ are
perfect,

  6. (f)

if $f\colon X\times\Sigma_{Y}\to[0,1]$ is measurable for each fixed
$B\in\Sigma_{Y}$ and $P$ is a perfect probability measure on $X$, then the
probability measure on $Y$ defined by $Q(B)=\int_{X}Q(x,B)\,dP$ is perfect if
and only if $\\{f(x,\cdot)\\}_{x\in X}$ is an equiperfect family of
probability measures on $Y$ except on a $P$-null set.

######  Definition 2.3.

The category of perfect probabilistic mappings $\mathcal{P}$ has countably
generated measurable spaces $(X,\Sigma_{X})$ as objects and an arrow between
two such objects $f\colon(X,\Sigma_{X})\to(Y,\Sigma_{Y})$ consists of a
function $f\colon X\times\Sigma_{Y}\to[0,1]$ such that

  1. (i)

for all $B\in\Sigma_{Y}$, the function $f(\cdot,B)\colon X\rightarrow[0,1]$ is
measurable,

  2. (ii)

the collection $\\{f(x,\cdot)\colon\Sigma_{Y}\rightarrow[0,1]\\}_{x\in X}$ is
an equiperfect family of probability measures on $Y$.

That is, morphisms in $\mathcal{P}$ can be thought of as parametrized families
of perfect probability measures that vary measurably. For an arrow
$f\colon(X,\Sigma_{X})\rightarrow(Y,\Sigma_{Y})$ we will often denote the
function $f(\cdot,B)\colon X\to[0,1]$ by $f_{B}$ and the function
$f(x,\cdot)\colon\Sigma_{Y}\to[0,1]$ by $f_{x}$.

Given two arrows

(1) $(X,\Sigma_{X})\xrightarrow{f}(Y,\Sigma_{Y})\xrightarrow{g}(Z,\Sigma_{Z})$

the composition $g\circ f\colon X\times\Sigma_{Z}\rightarrow[0,1]$ is defined
by

(2) $(g\circ f)(x,C)=\int_{y\in Y}g_{C}(y)df_{x}.$

This composition is well-defined due to Theorem 2.2 (f), and associativity
follows easily from the monotone convergence theorem. An important fact is
that every measurable function $f\colon X\to Y$ may be regarded as a
$\mathcal{P}$-morphism $\delta_{f}\colon X\to Y$, where the Dirac (or one
point) measure

(3) $\delta_{f}(x,B)=\begin{cases}1&\text{if }f(x)\in B\\\ 0&\text{if
}f(x)\notin B\end{cases}$

assigns to each $x\in X$ the Dirac measure on $Y$ which is concentrated at
$f(x)$. Taking the measurable function $f$ to be the identity map on a
particular measurable space $X$ gives the arrow
$\delta_{Id_{X}}\colon(X,\Sigma_{X})\to(X,\Sigma_{X})$, i.e., the identity
arrow for $X$ in $\mathcal{P}$. In fact, it is easy to check that the
association $f\mapsto\delta_{f}$ determines a functor
$\delta\colon\mathcal{M}_{cg}\to\mathcal{P}$ taking a measurable space to
itself. Note that this functor is not faithful, however, and so we do not get
an embedding of $\mathcal{M}_{cg}$ into $\mathcal{P}$. We will call a
$\mathcal{P}$ arrow $P\colon X\to Y$ deterministic if for every
$B\in\Sigma_{Y}$ the measurable functions $P_{B}\colon X\to[0,1]$ assume only
the values $0$ or $1$. In fact, every deterministic $\mathcal{P}$-arrow
$P\colon X\to Y$ is of the form $P=\delta_{f}$ for some measurable function
$f$, provided that $X$ has cardinality below the cardinality of the set of all
measurable sets.

The following lemma gives two useful properties which follow easily from
standard exercises in measure theory and the definition of composition in
$\mathcal{P}$.

######  Lemma 2.4.

If $p\colon X\to Y$ is a measurable function and $f\colon Y\to Z$ a
$\mathcal{P}$-morphism, then the composition

$X$$Y$$Z$$\delta_{p}$$f$

is given by $(f\circ\delta_{p})(x,C)=f_{p(x)}(C)$. On the other hand, if
$q\colon Y\to Z$ is a measurable function and $g\colon X\to Y$ a
$\mathcal{P}$-morphism, then the composition

$X$$Y$$Z$$\delta_{q}$$g$

is given by $(\delta_{q}\circ g)(x,C)=g_{x}(q^{-1}C)$.

There are several distinguished objects in $\mathcal{P}$ that play an
important role in many constructions. Any set $X$ with the indiscrete
$\sigma$-algebra $\Sigma_{X}=\\{X,\emptyset\\}$ is a terminal object since any
arrow $P\colon Y\rightarrow X$ is completely determined by the fact that
$P_{y}$ must be a probability measure on $X$. We denote the canonical terminal
object by $1$ since it is isomorphic to the one-element set. Notice that an
arrow $P\colon 1\to X$ is precisely a perfect probability measure on $X$ and
that $1$ is a separator for $\mathcal{P}$. In addition to having a separator,
the category $\mathcal{P}$ also has a coseparator, the two-element set
$2=\\{\top,\bot\\}$ with $\Sigma_{2}$ the discrete algebra on $2$. Moreover,
there is a set bijection
$\hom_{\mathcal{P}}(X,2)\simeq\hom_{\mathcal{M}_{cg}}(X,[0,1]).$

We briefly show how the Giry monad factors through $\mathcal{P}$. Let
$\mathscr{P}X$ denote the set of perfect probability measures on $X$, endowed
with the coarsest $\sigma$-algebra such that the evaluation maps
$ev_{B}\colon\mathscr{P}X\to[0,1]$ given by $ev_{B}(P)=P(B)$ are measurable.
Then we can define a functor $\mathscr{P}\colon\mathcal{P}\to\mathcal{M}_{cg}$
which sends a measurable space $X$ to the space $\mathscr{P}X$ of probability
measures on $X$. On arrows, $\mathscr{P}$ sends the $\mathcal{P}$-arrow
$f\colon X\to Y$ to the measurable function
$\mathscr{P}f\colon\mathscr{P}X\to\mathscr{P}Y$ defined pointwise on
$\Sigma_{Y}$ by

(4) $\mathscr{P}f(P)(B)=\int_{X}f_{B}\,dP.$

That is, $\mathscr{P}f(P)$ gives the probability measure on $Y$ defined by the
composition

(5) $1$$X$$Y$$P$$f$$f\circ P$

in $\mathcal{P}$. Since $\mathscr{P}X=\hom_{\mathcal{P}}(1,X)$ as sets,
another common notation for $\mathscr{P}X$ is $X^{1}$, but we will use the
functor notation for clarity.

In fact, $\mathscr{P}X$ is an important object in $\mathcal{P}$ as well, and
based on the definition of the $\sigma$-algebra on $\mathscr{P}X$, we can
define the evaluation morphism $\varepsilon_{X}\colon\mathscr{P}X\to X$ by
$\varepsilon_{X}(P,A)=P(A)$. With this, we are able to characterize the
relationship between $\mathcal{M}_{cg}$ and $\mathcal{P}$, first proved in
[10] for general measurable spaces and probability measures. To see that this
proof goes through when restricting to countably generated spaces and perfect
measures, we only need to observe that the $\sigma$-algebra defined above for
$\mathscr{P}X$ is countably generated when $X$ is countably generated and that
pushforwards take perfect measures to perfect measures.

######  Theorem 2.5.

The functors $\delta\colon\mathcal{M}_{cg}\to\mathcal{P}$ and
$\mathscr{P}\colon\mathcal{P}\to\mathcal{M}_{cg}$ form an adjunction

$\mathcal{M}_{cg}$$\mathcal{P}$$\delta$$\mathscr{P}$$\delta\dashv\mathscr{P}$

with the unit of the adjunction $\eta_{X}(x)=\delta_{\\{x\\}}$ and the counit
$\varepsilon_{X}\colon\mathscr{P}X\to X$.

Thus, we can realize the Giry monad as the composition
$T=\mathscr{P}\,\circ\,\delta$, and moreover, $\mathcal{P}$ is equivalent to
the smallest category through which $T$ factors—i.e., it is equivalent to the
Kleisli category $K(T)$ of the Giry monad. Hence every $\mathcal{P}$ arrow
$P\colon X\to Y$ corresponds uniquely to a measurable arrow $X\to TY$.

##  3\. Joint Distributions and Conditionals

Given a family of objects $\\{X_{i}\\}_{i\in I}$ we can form the cartesian
product $\prod_{i\in I}X_{i}$ and endow this set with the product
$\sigma$-algebra generated by all the projection maps $\prod_{i\in
I}X_{i}\stackrel{{\scriptstyle\pi_{j}}}{{\longrightarrow}}X_{j}$, one for each
index $j\in I$. It is easy to see that

(6) $\left(\left(\prod_{i\in I}X_{i},\bigotimes_{i\in
I}\Sigma_{X_{i}}\right),\\{\delta_{\pi_{i}}\\}_{i\in I}\right)$

does not give a categorical product. In fact, only weak products and
equalizers exist in $\mathcal{P}$, as the uniqueness condition fails for both
constructions. We use the terminology “product space” to denote the set
product of any family $\\{(X_{i},\Sigma_{X_{i}})\\}_{i\in I}$ of objects with
the product $\sigma$-algebra and not to imply that that object

$\left(\prod_{i\in I}X_{i},\bigotimes_{i\in I}\Sigma_{X_{i}}\right)$

with projections satisfies any universality condition. We will call a
probability measure $J\colon 1\to(\prod_{i\in I}X_{i},\otimes_{i\in
I}\Sigma_{X_{i}})$ on a product space a joint distribution. We do not mean to
imply that these are distributions of a random variable, but rather indicate a
measure on a product space which is not necessarily a product measure. These
joint distributions are the main objects of study in Bayesian probability.

Given any joint distribution $J\colon 1\to\prod_{i\in I}X_{i}$, for each $j\in
I$ we have the diagram

(7) $1$$X_{j}$$\prod_{i\in I}X_{i}$$J$$\delta_{\pi_{j}}$$\delta_{\pi_{j}}\circ
J$

where the composite $\delta_{\pi_{j}}\circ J$ is called the _marginal_
(distribution) of $J$ on component $X_{j}$ and is given by
$(\delta_{\pi_{j}}\circ J)(A_{j})=J(\pi_{j}^{-1}A_{j})$ by Lemma 2.4.

Given only the probability measures on the components, $\\{P_{i}\\}_{i\in I}$,
there are many joint distributions on the product space whose marginals are
the given family $\\{P_{i}\\}_{i\in I}$. By using _relationships_ in the form
of conditionals between the components, we bring into play additional
knowledge that permits the determination of the appropriate joint
distribution. If the uncertainty of component $X_{j}$, as expressed by a
probability measure $P_{j}$ on component $X_{j}$, depends conditionally on a
parameter which varies over component $X_{i}$ then we have the $\mathcal{P}$
arrow $h\colon X_{i}\to X_{j}$. These conditionals—which are the morphisms in
$\mathcal{P}$—are the key to determining a unique joint distribution. The
relationship between the components $X_{i}$ and $X_{j}$ is mediated by the
conditional $h$ and expresses the relationship $P_{j}=h\circ P_{i}$.

###  3.1. Constructing a Joint Distribution Given Conditionals

We now show how marginals and conditionals can be used to determine joint
distributions in $\mathcal{P}$. This development follows that of [1] where we
first learned of this approach (the category $\mathcal{S}toch$ of stochastic
kernels in that paper is nearly what we call $\mathcal{P}$, although our
restriction to countably generated spaces and perfect measures is replaced in
the paper by restricting to Polish spaces). Given a $\mathcal{P}$-arrow
$h\colon X\to Y$ and a perfect probability measure $P_{X}\colon 1\to X$ on
$X$, consider the diagram

(8) $1$$X$$Y$$X\times Y$$P_{X}$$\delta_{\pi_{X}}$$\delta_{\pi_{Y}}$$J_{h}$$h$

where $J_{h}$ is the uniquely determined joint distribution on the product
space $X\times Y$ defined on the rectangles of the $\sigma$-algebra
$\Sigma_{X}\otimes\Sigma_{Y}$ by

(9) $J_{h}(A\times B)=\int_{A}h_{B}\,dP_{X}.$

Then it follows from Theorem 2.2 that $J_{h}$ is perfect. The marginal of
$J_{h}$ with respect to $Y$ then satisfies $\delta_{\pi_{Y}}\circ J_{h}=h\circ
P_{X}$ and the marginal of $J_{h}$ with respect to $X$ is $P_{X}$, both of
which are also perfect. By a symmetric argument, if we are given a probability
measure $P_{Y}$ and conditional probability $k\colon Y\to X$ then we obtain a
unique perfect joint distribution $J_{k}$ on the product space $X\times Y$
given on the rectangles by

(10) $J_{k}(A\times B)=\int_{B}k_{A}\,dQ.$

However, if we are given $P_{X},P_{Y},h,k$ as indicated in the diagram

(11) $1$$X$$Y$$X\times
Y$$P_{Y}$$P_{X}$$\delta_{\pi_{X}}$$\delta_{\pi_{Y}}$$J_{k}$$J_{h}$$h$$k$

then we have that $J_{h}=J_{k}$ if and only if the compatibility conditions

(12) $\begin{array}[]{lcl}P_{X}&=&k\circ P_{Y}\\\ P_{Y}&=&h\circ
P_{X}\end{array}$

are satisfied. Thus if the compatibility conditions are satisfied, then we can
realize the product rule of probability in $\mathcal{P}$ as

(13) $\int_{A}h_{B}\,dP_{X}=J(A\times B)=\int_{B}k_{A}\,dP_{Y}.$

In the extreme case, suppose we have a $\mathcal{P}$-arrow $h\colon X\to Y$
which factors through the terminal object $1$ as

(14) $X$$Y$$1$$h$$!$$Q$

where $!$ represents the unique arrow from $X\to 1$. If we are also given a
perfect probability measure $P\colon 1\to X$, then we can calculate the joint
distribution determined by $P$ and $h=Q\circ!$ as

(15) $\begin{array}[]{lcl}J(A\times B)&=&\int_{A}(Q\circ!)_{B}\,dP\\\
&=&P(A)\cdot Q(B)\end{array}$

so that $J=P\otimes Q$. This is precisely the situation where we say that the
marginals $P$ and $Q$ are _independent_. Thus in $\mathcal{P}$ independence
corresponds to a special instance of a conditional—one that factors through
the terminal object.

###  3.2. Constructing Regular Conditionals given a Joint Distribution

The following result is the basis from which the inference maps in Bayesian
probability theory are subsequently constructed. The proof of the following
theorem can be found in [9], where several equivalent conditions are
identified.

######  Theorem 3.1.

If $J:1\to X\times Y$ is a $\mathcal{P}$-morphism with marginals $P_{X}$ on
$X$ and $P_{Y}$ on $Y$, then there exists a $\mathcal{P}$-arrow $f$ that makes
the diagram

(16) $1$$Y$$X$$P_{Y}$$P_{X}$$f$

commute and satisfies

(17) $J(A\times B)=\int_{B}f_{A}\,dP_{Y}.$

Moreover, the morphism $f$ is the unique $\mathcal{P}$-morphism with these
properties, up to a set of $P_{Y}$-measure zero.

Interestingly, we can use Theorem 3.1 to obtain a seemingly stronger
statement, i.e., that the regular conditional probability factors through the
product. Though this is not difficult to prove, we will prefer this stronger
statement in the sequel.

######  Theorem 3.2.

If $J:1\to X\times Y$ is a $\mathcal{P}$-morphism with marginal distributions
$P_{X}$ and $P_{Y}$ on $X$ and $Y$, then there exist $\mathcal{P}$ arrows $f$
and $g$ such that the diagram

(18) $1$$Y$$X$$X\times
Y$$P_{Y}$$P_{X}$$J$$\delta_{\pi_{Y}}$$\delta_{\pi_{X}}$$\delta_{\pi_{X}}\circ
f$$f$$g$$\delta_{\pi_{Y}}\circ g$

commutes and

(19) $\int_{A}(\delta_{\pi_{Y}}\circ g)_{B}\,dP_{X}=J(A\times
B)=\int_{B}(\delta_{\pi_{X}}\circ f)_{A}\,dP_{Y}.$

###### Proof.

We can apply Theorem 3.1 to see that there is a $\mathcal{P}$-arrow $f\colon
Y\to X\times Y$ satisfying $J=f\circ P_{Y}$ such that

(20) $\int_{C}f_{A\times B}\,dP_{Y}=J\left(A\times(B\cap C)\right).$

Then from Lemma 2.4, we know that $(\delta_{\pi_{X}}\circ f)(y,A)=f(y,A\times
Y)$ and so

(21) $\displaystyle\int_{B}(\delta_{\pi_{X}}\circ f)_{A}\,dP_{Y}$
$\displaystyle=\int_{B}f_{A\times Y}\,dP_{Y}$ (22)
$\displaystyle=J\left(A\times(Y\cap B)\right)$ (23) $\displaystyle=J(A\times
B)$

Similarly we obtain a $\mathcal{P}$-arrow $g\colon X\to X\times Y$ satisfying
$J=g\circ P_{X}$ and

(24) $\int_{A}(\delta_{\pi_{Y}}\circ g)_{B}\,dP_{X}=J(A\times B).$

With these facts, it is a simple exercise to check that the diagram commutes.
∎

Note that if the joint distribution $J$ is obtained by a probability measure
$P_{X}$ and a conditional $h\colon X\rightarrow Y$ using the method described
by Diagram 8, then using the above result and notation it follows $P_{X}$-a.s.
that $h=\delta_{\pi_{Y}}\circ g$.

######  Remark 3.3.

(Tonneli’s Theorem) Given a $\mathcal{P}$-morphism $J:1\to X\times Y$, with
marginals $P_{X}$ and $P_{Y}$ let $\gamma\colon X\rightarrow X\times Y$ and
$\varphi\colon Y\rightarrow X\times Y$ be the $\mathcal{P}$ arrows satisfying
$\gamma\circ P_{X}=J$ and $\varphi\circ P_{Y}=J$ whose existence is guaranteed
by Theorem 3.2. Given any measurable function $F\colon X\times
Y\rightarrow[0,1]$ we have the diagram

(25) $1$$Y$$X$$X\times
Y$$2$$P_{Y}$$P_{X}$$\delta_{\pi_{Y}}$$\delta_{\pi_{X}}$$\phi$$\gamma$$J$$\overline{F}$$\overline{f}=\overline{F}\circ\gamma$$\overline{g}=\overline{F}\circ\varphi$

where the top two triangles commute. Thus we can define
$\overline{f}=\overline{F}\circ\gamma$ and
$\overline{g}=\overline{F}\circ\varphi$ so that the entire diagram commutes.
From this, it follows that

(26) $\int_{X}f\,dP_{X}=\int_{X\times Y}F\,dJ=\int_{Y}g\,dP_{Y}$

and we can realize Tonneli’s Theorem as the special case with $J=P_{X}\otimes
P_{Y}$.

This formulation provides the context for the following optimal transportation
problem: given marginals $P_{X}$ and $P_{Y}$ that model the supply and demand
constraints, and a cost function $F$ (defined up to a scalar constant)
representing the unit cost to transport a product from $x\in X$ to $y\in Y$,
what joint distribution $J$ on $X\times Y$ with marginals $P_{X}$ and $P_{Y}$
minimizes the objective function $\int_{X\times Y}F\,dJ$? The optimal
assignment is then the conditional probability $X\rightarrow Y$ determined by
the optimal joint distribution $J$. For example, this problem is investigated
and a unique solution is given in certain cases in [14].

##  4\. Bayesian probability in $\mathcal{P}$

If we replace $X$ and $Y$ in Diagram 18 by $D$(ata) and $H$(ypotheses), and
the composites $\delta_{\pi_{Y}}\circ g$ by $\mathcal{S}$(ampling
distribution) and $\delta_{\pi_{X}}\circ f$ by $\mathcal{I}$(nference), then
we can define $P_{D}=\mathcal{S}\circ P_{H}$ to obtain

(27) $\leavevmode\hbox to129.32pt{\vbox
to89.47pt{\pgfpicture\makeatletter\hbox{\hskip
64.80101pt\lower-26.00789pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{53.6833pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-61.468pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$H$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{52.62704pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$D$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{-6.033pt}{50.87657pt}\pgfsys@lineto{-49.63464pt}{7.27492pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.7071}{-0.7071}{0.7071}{-0.7071}{-49.63464pt}{7.27492pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-44.44846pt}{34.35945pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{H}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{6.033pt}{50.87657pt}\pgfsys@lineto{49.63464pt}{7.27492pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.7071}{-0.7071}{0.7071}{0.7071}{49.63464pt}{7.27492pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{31.52945pt}{34.35945pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$P_{D}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{-48.81003pt}{0.0pt}\pgfsys@lineto{48.63405pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{48.63405pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.63579pt}{3.533pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mathcal{S}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}
{{}{{}{}}{}}{{}{{}{}}{}}{{}{}}{{}}
{{}{{}{}}{}}{{{}}{{}}}{{}}{{}{{}{}}{}}{{{}}{{}}}{ {}{}{}}{}{{}}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{}{}{}}{}{}{}{}{{}}{}{}{}{}{{}}\pgfsys@moveto{52.89369pt}{-6.94966pt}\pgfsys@curveto{48.76767pt}{-14.09488pt}{-48.76767pt}{-14.09488pt}{-52.66367pt}{-7.348pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.50008}{0.866}{-0.866}{-0.50008}{-52.66365pt}{-7.348pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.80556pt}{-22.67488pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mathcal{I}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
In the context of Bayesian probability the probability measure $P_{H}$ is
often called a _prior probability_.

In this notation, the product rule in $\mathcal{P}$ given in Equation 13
becomes

(28)
$\int_{\mathcal{D}}\mathcal{I}_{\mathcal{H}}\,dP_{D}=J(\mathcal{H}\times\mathcal{D})=\int_{\mathcal{H}}\mathcal{S}_{\mathcal{D}}\,dP_{H}$

where $\mathcal{H}\in\Sigma_{H}$ and $\mathcal{D}\in\Sigma_{D}$. We will spend
the remainder of this sections showing how this interpretation of these spaces
in $\mathcal{P}$ provides a categorical foundation for Bayesian probability.
First, we briefly review the fundamental concepts in Bayesian probability
theory which can be found in [12] and then proceed to show how $\mathcal{P}$
is the appropriate category for this theory. Generally, a Bayesian model is
comprised of a number of items including

  1. (i)

two measurable spaces $H$ and $D$ representing hypotheses and data,
respectively,

  2. (ii)

a probability measure $P_{H}$ on the $H$ space called the prior probability,

  3. (iii)

a $\mathcal{P}$ arrow $\mathcal{S}\colon H\rightarrow D$ called the sampling
distribution,

  4. (iv)

a $\mathcal{P}$ arrow $\mathcal{I}\colon D\rightarrow H$ called the inference
map,

Note that the data space $D$ can also be thought of as the event space for
some experiment, and the $\sigma$-algebra on $D$ is determined by
distinguishable data. The prior probability measures are updated via the
inference map as one takes measurements, which correspond to probability
measures $\mu$ on $D$. These updated probability measures are then called
posterior probabilities and are given by
$\widehat{P}_{H}=\mathcal{I}\circ\mu$. The posterior $\widehat{P}_{H}$ then
becomes the prior probability for the next step and the process continues as
more measurements are taken. At each step in a Bayesian process, the posterior
probability is a representation of knowledge about the hypotheses based on all
of the data that has been accumulated up to that point.

Using the sampling distribution $\mathcal{S}$ and the prior probability
$P_{H}$ on $H$, we can define a joint distribution $J$ on the product space
$H\times D$ as in Section 3.1 by defining it on the rectangles as $J(A\times
B)=\int_{A}\mathcal{S}_{B}\,dP_{H}$. The $D$-marginal (prior probability on
data) is then $P_{D}=\mathcal{S}\circ P_{H}=\delta_{\pi_{D}}\circ J$. Using
Theorem 3.2, we have the following theorem.

######  Theorem 4.1.

Given $\mathcal{P}$ arrows representing a prior probability $P_{H}\colon 1\to
H$ and a sampling distribution $\mathcal{S}\colon H\to D$, the inference map
$\mathcal{I}\colon D\to H$ is determined uniquely up to a set of
$P_{D}$-measure zero.

###### Proof.

Let $\mathcal{I}\colon D\rightarrow H$ be the composition
$\delta_{\pi_{H}}\circ f$, where $\delta_{\pi_{H}}$ is the projection $H\times
D\to H$ and $f\colon D\rightarrow H\times D$ is the $\mathcal{P}$-arrow
satisfying

(29) $J(U\times V)=\int_{D}f_{U\times V}\,dP_{D}$

whose existence is given by Theorem 3.2. Thus

(30) $\int_{A}\mathcal{S}_{B}\,dP_{H}=J(A\times
B)=\int_{B}\mathcal{I}_{A}\,dP_{D}$

and this inference arrow $\mathcal{I}$ is unique in that if
$\mathcal{I}^{\prime}$ also satisfies equation 30 then the set $\\{y\in
Y\mid\mathcal{I}_{y}\neq\mathcal{I}^{\prime}_{y}\\}$ has $P_{D}$-measure zero.
∎

Thus the complete process works in the following way. A prior probability
$P_{H}$ and sampling distribution $\mathcal{S}$ are specified, from which one
determines the inference map $\mathcal{I}$. Once measurements $\mu\colon 1\to
D$ are taken, we then calculate the posterior probability by
$\mathcal{I}\circ\mu$. This updating procedure can be characterized by the
diagram

(31) $1$$H$$D$$P_{H}$$\mu$$\mathcal{S}$$\mathcal{I}$$\mathcal{I}\circ\mu$

where the solid lines indicate arrows given a priori, the dotted line
indicates the arrow determined using Theorem 3.2, and the dashed lines
indicate the updating after a measurement. Note that if there is no
uncertainty in the measurement, then $\mu=\delta_{\\{x\\}}$ for some $x\in D$,
but in practice there is usually some uncertainty in the measurements
themselves.

Following the calculation of the posterior probability, the sampling
distribution is then updated, if required. The process can then repeat: using
the posterior probability and the updated sampling distribution the updated
joint probability distribution on the product space is determined and the
corresponding (updated) inference map determined. We can then continue to
iterate as long as new measurements are received. For some problems (such as
with the standard urn problem with replacement of balls) the sampling
distribution does not change from iterate to iterate, but the inference map is
updated since the posterior probability on the hypothesis space changes with
each measurement. The model selection problem (either once at the beginning of
this process, or iteratively throughout) can also be modeled as a meta-
Bayesian process, where the hypothesis space is the space of potential models
and the data constitutes some information that would inform on the suitability
of a given model.

######  Remark 4.2.

We know from Theorem 4.1 that the inference map $\mathcal{I}$ is uniquely
determined by $P_{H}$ and $\mathcal{S}$ up to a set of $P_{D}$-measure zero.
However, there is no reason a priori that a measurement $\mu\colon 1\to D$ is
required to be absolutely continuous with respect to $P_{D}$. In $\mu$ is not
absolutely continuous with respect to $P_{D}$, then a different choice of
inference map $\mathcal{I}^{\prime}$ could yield a different posterior
probability—i.e., we could have
$\mathcal{I}\circ\mu\neq\mathcal{I}^{\prime}\circ\mu$. Thus we make the
assumption that measurement probabilities on $D$ are absolutely continuous
with respect to the prior probability $P_{D}$ on $D$. This is a reasonable
assumption, however, since if a data event is impossible (has $P_{D}$-measure
zero) under a certain model, then the model should not be expected to make an
meaningful inference when presented with that data. On the other hand, it is
easy to see that if a measurement $\mu\ll P_{D}$, then $\mathcal{I}\circ\mu\ll
P_{H}$, as expected.

We emphasize that this procedure can be employed for any perfect prior
probability and any regular conditional probability. For example, given a
perfect prior $P\colon 1\to\mathscr{P}X$ and the conditional
$\varepsilon_{X}\colon\mathscr{P}X\to X$ there corresponds a unique inference
map $\mathcal{I}\colon X\to\mathscr{P}X$ satisfying, for all $A\in\Sigma_{X}$
and for all $\mathcal{B}\in\Sigma_{\mathscr{P}X}$,

(32)
$\int_{\mathscr{P}X}{\varepsilon_{X}}_{A}\,dP=\int_{X}\mathcal{I}_{\mathcal{B}}\,d(\varepsilon_{X}\circ
P).$

In the case where $X=2=\\{\top,\bot\\}$ (the two element set), these “higher
order distributions” $P\colon 1\to\mathscr{P}2$ can be used to explicate the
concept of $A_{p}$ distributions as characterized by Jaynes [12, Chapter 18].
Using our notation, a proposition $A$, which is a morphism $A\colon 1\to 2$ in
the category $\mathcal{S}et$ of sets, has an associated probability of truth,
say $Pr(A)=p$. Hence $A$ determines a $\mathcal{P}$-morphism
$\overline{A}\colon 1\to 2$, with $\overline{A}(\\{\top\\})=p$. The
information supplied by the arrow $\overline{A}$ consists only of the single
value $p$ and fails to indicate how sensitive this proposition is to
additional data. The confidence that one has in the value $p$ can be supplied
by the higher order distributions which are probability measures on the space
$\mathscr{P}2$ of probability measures on $2$. Since $\mathscr{P}2$ consists
precisely of the Bernoulli distributions
$B_{\theta}=\theta\delta_{\top}+(1-\theta)\delta_{\bot}$, where
$\theta\in[0,1]$, it follows that $\mathscr{P}2\simeq[0,1]$. Consequently, any
distribution

(33) $1\xrightarrow{A_{p}}\mathscr{P}2$

has an expected value which can be calculated using the composition

(34) $1\xrightarrow{A_{p}}\mathscr{P}2\xrightarrow{\varepsilon_{2}}2.$

Thus $E(A_{p})=(\varepsilon_{2}\circ A_{p})(\\{\top\\})=p$ and any such
distribution provides a more informative measure. For example, the two
distributions on $\mathscr{P}2\simeq[0,1]$ specified by
$p=\delta_{\frac{1}{2}}$ and $p^{\prime}$ the uniform (Lebesque) measure both
have expected value $\frac{1}{2}$. Yet clearly, the first is deterministic,
expressing a (complete) confidence in the statement that the expected value of
the proposition

(35) $1\xrightarrow{~{}~{}\overline{A}~{}~{}}2$

where $\overline{A}(\top)=\frac{1}{2}$ is $\frac{1}{2}$. On the other hand,
the distribution $p^{\prime}$ also determines $\overline{A}$, but instead
expresses a maximal ignorance modeled by the uniform distribution.

##  5\. The Category of Decision Rules

Recall that the Giry monad $T$ factors through $\mathcal{P}$ via the
adjunction of Theorem 2.5. At the other end of the spectrum of categories
through which the Giry monad factors is the Eilenberg–Moore category
$\mathcal{M}_{cg}^{T}$, consisting of the Eilenberg–Moore algebras of the Giry
monad. From the theory of monads (see [2], for example), we know that
$\mathcal{P}$ then embeds into $\mathcal{M}_{cg}^{T}$, which has additional
structure that is useful for dealing with other aspects of probability theory
and decision making which $\mathcal{P}$ is not equipped for. Let us briefly
recall the definition of a $T$-algebra.

If $(T,\eta,\mu)$ is a monad in a category $\mathcal{C}$, a $T$-algebra
$(X,\alpha)$ is a pair consisting of an object $X$ in $\mathcal{C}$ and a
$\mathcal{C}$ arrow $\alpha\colon TX\to X$ such that the diagrams

(36)
$T^{2}X$$TX$$TX$$X$$T\alpha$$\mu_{X}$$\alpha$$\alpha$$X$$TX$$X$$\alpha$$\eta_{X}$$Id$

commute; the first diagram is called the associative law and the second
diagram the unit law. A morphism of $T$-algebras
$(X,\alpha)\stackrel{{\scriptstyle f}}{{\longrightarrow}}(Y,\beta)$ is an
arrow $f\colon X\to Y$ of $\mathcal{C}$ such that the diagram

(37) $TX$$TY$$X$$Y$$Tf$$\beta$$\alpha$$f$

commutes.

When $T$ is the Giry monad, an algebra
$TX\stackrel{{\scriptstyle\alpha}}{{\longrightarrow}}X$ consists of a
measurable space $X$ with a countably generated $\sigma$-algebra, the space
$TX=\mathscr{P}X$ of probability measures on $X$, and a measurable map
$\alpha$ satisfying the two defining properties of a $T$-algebra. The
measurable map $Tf$ in the definition of a morphism of $T$-algebras is the
pushforward map: given $P\in TX$ the pushforward by $f$ is
$Tf(P)=f_{\ast}P\in\mathscr{P}Y$.

The $T$-algebras $(X,\alpha)$ are often called decision rules since the
measurable map $\alpha$ assigns (decides) a value in $X$ to each probability
measure $P$ on $X$. Alternatively, we can think of a decision rule as
collapsing a probability distribution to a definite value, or derandomizing a
probability distribution as in [6]. For this reason, we often use the
descriptive characterization of Čencov [21] and call the category
$\mathcal{M}_{cg}^{T}$ the category of decision rules.222Čencov did not work
in $\mathcal{M}_{cg}^{T}$, but rather in $\mathcal{P}$, restricting to
measurable spaces $X$ such that $\mathscr{P}X$ has a $\sigma$-algebra
generated by finitely many atoms. The primary difference in the current
approach and that of Čencov is that we take a Bayesian viewpoint, while he is
attempting to describe the standard statistical inference perspective.

Embedding the main consequence of the existence of regular conditional
probabilities for Bayesian probability into $\mathcal{M}_{cg}^{T}$, we have
the following.

######  Theorem 5.1.

Given a measurable function $\mathcal{S}\colon H\rightarrow TD$, there exists
a measurable function $\mathcal{I}\colon D\to TH$ such that
$\hat{\mathcal{I}}=\mu_{H}\circ\mathcal{I}$ is a retraction of
$\hat{\mathcal{S}}=\mu_{D}\circ T\mathcal{S}$ in $\mathcal{M}_{cg}$.

###### Proof.

This follows immediately from Theorem 4.1 and the embedding of $\mathcal{P}$
into $\mathcal{M}_{cg}^{T}$ and can be summarized in the diagram

(38) $\leavevmode\hbox to193.46pt{\vbox
to100.66pt{\pgfpicture\makeatletter\hbox{\hskip
11.51181pt\lower-50.33102pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.1788pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$TH$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{76.06348pt}{38.36026pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T^{2}D$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{162.82176pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$TD$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{75.77946pt}{-46.99802pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T^{2}H$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{11.7118pt}{5.85321pt}\pgfsys@lineto{72.11908pt}{36.06613pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.89436}{0.44733}{-0.44733}{0.89436}{72.11906pt}{36.06613pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.79993pt}{24.59555pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T\mathcal{S}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{98.18608pt}{36.2676pt}\pgfsys@lineto{158.87737pt}{5.91371pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.89438}{-0.44731}{0.44731}{0.89438}{158.87735pt}{5.91373pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{132.27042pt}{26.46521pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mu_{D}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{159.28876pt}{-5.71407pt}\pgfsys@lineto{98.8815pt}{-35.92429pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.8944}{-0.4473}{0.4473}{-0.8944}{98.8815pt}{-35.92429pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{132.41243pt}{-31.28836pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T\mathcal{I}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{72.24646pt}{-36.12865pt}\pgfsys@lineto{12.1232pt}{-6.0556pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.89436}{0.44736}{-0.44736}{-0.89436}{12.12321pt}{-6.0556pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{}}{} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{27.31067pt}{-28.82779pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\mu_{H}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\par{{}{}{}}{{{}}{{}}}{}{{}}{}{ {}{}{}}{{{}}{{}}}{}
{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{11.7118pt}{3.0pt}\pgfsys@lineto{158.82877pt}{3.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{158.82877pt}{3.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{82.72249pt}{6.533pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\hat{\mathcal{S}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {
{}{}{}}{{{}}{{}}}{}{{}}{}{{}{}{}}{{{}}{{}}}{}
{}{}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{159.28876pt}{-3.0pt}\pgfsys@lineto{12.1718pt}{-3.0pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{12.1718pt}{-3.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{82.7225pt}{-13.75523pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\hat{\mathcal{I}}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
∎

One of the primary advantages to using the existence of regular conditional
probabilities guaranteed by Theorem 3.2 is that we only require that
measurable spaces have countably generated $\sigma$-algebras and that the
measures are perfect. In contrast, much of the previous work involving the
Giry monad and regular conditional probabilities requires resorting to
topological arguments and restricting to Polish spaces. For example, in [5],
Doberkat characterizes the $T$-algebras for the Giry monad under the Polish
space assumption, and proves the counter-intuitive result that there are no
non-trivial decision rules for finite spaces in $\mathcal{M}_{cg}^{T}$. In
contrast, we exhibit a finite space having a $T$-algebra when one does not
require topological restrictions.

######  Example 5.2.

An important such case is the decision rule $d\colon T2\rightarrow 2$ given by

(39) $d(P)=\begin{cases}\top&\text{if }P(\\{\top\\})=1\\\ \bot&\text{if
}P(\\{\top\\})<1.\end{cases}$

The function $d$ is measurable since
$d^{-1}(\\{\top\\})=\\{\delta_{\top}\\}\in\Sigma_{T2}$. The associativity
identity

(40)
$T^{2}2$$T2$$T2$$2$$Td$$d$$\mu_{2}$$d$$Q$$Qd^{-1}$$\mu_{2}(Q)$$d(\mu_{2}(Q))=d(Qd^{-1})$$Td$$d$$\mu_{2}$$d$

where $\mu$ is the monad multiplication defined by

(41) $\mu_{2}(Q)(A)=\int_{q\in T(2)}ev_{A}(q)\,dQ,$

is satisfied since both routes map the element $Q\neq\delta_{\delta_{\top}}\in
T^{2}(2)\mapsto\bot$ while $\delta_{\delta_{\top}}\mapsto\top$. The unit law
$Id_{2}=d\circ\eta_{2}$ is trivial to verify.

The decision rule $d\colon T2\rightarrow 2$ partitions the space $T2$ into
$\delta_{\top}$ and all measures on $2$ whose value on $\\{\top\\}$ is of
measure less than one. There are many other decision rules for $2$, and any
other finite or nonfinite space. Characterizing decision rules without the
requirement for continuity is an open problem.

##  6\. Acknowledgements

The authors would like to express gratitude to F.W. Lawvere, P.F. Stiller and
T. Nguyen for many fruitful conversations regarding these ideas. We also thank
the reviewer for the helpful comments provided that have led to a much clearer
exposition. This work was partially supported by the Air Force Office of
Scientific Research, for which the authors are extremely grateful.

## References

  * [1] Abramsky, S., Blute, R., Panangaden, P.: Nuclear and trace ideals in tensored $\ast$-categories.  J. Pure Appl. Algebra 143, 3–47 (1999)
  * [2] Barr, M., Wells, C.: Toposes, triples and theories.  Repr. Theory Appl. Categ. (12) (2005).  Corrected reprint of the 1985 original (Springer-Verlag)
  * [3] Berger, J.: Statistical Decision Theory and Bayesian Analysis, 2nd edn.  Springer Series in Statistics. Springer-Verlag, New York (1985)
  * [4] van Breugel, F.: The metric monad for probabilistic nondeterminism (2005).  Unpublished
  * [5] Doberkat, E.E.: Characterizing the Eilenberg–Moore algebras for a monad of stochastic relations (2004).  Internal Memorandum No. 147
  * [6] Doberkat, E.E.: Derandomizing probabilistic semantics through Eilenberg–moore algebras for the Giry monad (2004).  Internal Memorandum No. 149
  * [7] Doberkat, E.E.: Kleisli morphisms and randomized congruences for the giry monad.  J. Pure Appl. Algebra 211(3), 638–664 (2007)
  * [8] Dudley, R.: Real Analysis and Probability.  No. 74 in Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2002)
  * [9] Faden, A.M.: The existence of regular conditional probabilities: necessary and sufficient conditions.  Ann. Probab. 13(1), 288 – 298 (1985)
  * [10] Giry, M.: A categorical approach to probability theory.  Categorical Aspects of Topology and Analysis 915, 68–85 (1981)
  * [11] Howson, C., Urbach, P.: Scientific Reasoning: The Bayesian Approach, 2nd edn.  Open Court Publishing, Chicago (1993)
  * [12] Jaynes, E.T.: Probability Theory: The Logic of Science.  Cambridge University Press, Cambridge (2003)
  * [13] Kock, A.: Commutative monads as a theory of distributions.  Theory Appl. Categ. 26, 97–131 (2012)
  * [14] Korman, J., McCann, R.: Optimal transportation with capacity constraints (2012).  ArXiv:1201.6404v2
  * [15] Lawvere, W.F.: The category of probabilistic mappings (1962).  Unpublished
  * [16] Lawvere, W.F.: Bayesian sections (2011).  Personal communication
  * [17] Meng, X.: Categories of convex sets and of metric spaces, with applications to stochastic programming and related areas.  Ph.D. thesis, State University of New York at Buffalo (1988)
  * [18] Pachl, J.K.: Disintegration and compact measures.  Math. Scand. 43, 157 –168 (1978)
  * [19] Rodine, R.H.: Perfect probability measures and regular conditional probabilities.  Ann. Math. Statist. 37, 1273 – 1278 (1966)
  * [20] Ryll-Nardzewski, C.: On quasi-compact measures.  Fund. Math. 40, 125–130 (1953)
  * [21] Čencov, N.: Statistical Decision Rules and Optimal Inference, _Translations of Mathematical Monographs_ , vol. 53.  American Mathematical Society (1982)
  * [22] Wendt, M.: The category of disintegrations.  Cahiers Topologie Géom. Différentielle Catég. 35(4), 291–308 (1994)