In Bayesian statistics, we consider parameters in models as random variables and its probability distributions, and we calculate the posterior distribution by using Bayes’ theorem.
In Hamiltonian dynamical system, any time evolution is defined by Hamiltonian equations and expressed by canonical transformations (or symplectic diffeomorphisms) on phase spaces. Phase spaces and equations of motion are abstract symplectic manifolds and Hamiltonian vector fields respectively. Under time evolution for Hamiltonian dynamical system, Hamiltonian functions and the phase volume are preserved. These are direct consequences of skew-symmetricity of symplectic structures. In the case where the dimension of a phase space is greater than or equal to 4, there are other conserved quantities that are called symplectic capacities. Symplectic capacities are far from trivial and are deep result in symplectic geometry. For detail, see     .
In this paper we prove that Bayesian updating for multivariate normal population mean vector can be expressed by an affine symplectic diffeomorphism (affine canonical transformation). The main result is the following.
Theorem 1. There exists a linear symplectic diffeomorphism on such that the first component of the composition maps a prior distribution to the posterior, where is the parallel translation on .
In this theorem to reformulate Bayesian updating from symplectic geometric viewpoint, we consider the cotangent space of which its base space contains population mean . The reason why we use the cotangent space is as follows. If we assume is a point in , to express Bayes’ theorem by an affine symplectic transformation on , in the case where variance is known, we have to find an element in such that . (Note that are all .) However usually in Bayesian updating, we fix a type of the posterior which is a section of density function, and then we normalize it:
It is well know that any canonical transformations are volume-preserving. Hence we cannot expect the existence of desired transformation. Moreover, in this case the population mean is in , so we can only treat even dimensional case. The key to getting rid of this drawback is to use of Lagrangian submanifolds. Consider a symplectic manifold which has a Lagrangian submanifold containing , and construct desired canonical transformation on the total space. Canonical transformations on the total space may change a measure on Lagrangian submanifolds.
There is another approach to Bayesian inference from symplectic-contact geometric viewpoint due to Mori   . Mori considers the square of the parameter space of normal distributions and its Lagrangian submanifold to describe Bayes’ theorem by Hamiltonian follows, and he simultaneously gives Bayesian updating for mean and variance in univariate case. Taking account of Mori’s considerations, we should use the Poincaré type symplectic form to express a Bayesian updating for covariant matrices while we use the canonical symplectic structure on for mean vectors. For information geometry and a relation between information geometry and symplectic geomerty, see    .
2. Bayesian Updating
In this section we review Bayes’ theorem for multivariate normal distributions. For detail, see e.g.,  .
Consider a posterior distribution of mean vector for a multivariate normal distribution with covariant matrix. Fix a positive definite symmetric matrix . First we treat the case of covariance matrix is known. Let a prior distribution of is distributed :
The posterier distribution with sample y is and
Next we consider the case of the variance is unknown. If we denote a priori distribution of by , , then the posterior is , , where
3. Symplectic Group and Affine Canonical Transformation
In this section we review properties of the symplectic group and Hamiltonian flows.
Denote the set of all linear symplectic transformations on by
and call the symplectic group, where .
Let z be a vector in and be the canonical symplectic structure on 2n dimensional vector space , then a necessary and sufficient condition for is . For any we have , where denotes the determinant of matrix S. We also have for . In general is a connected Lie group of dimension , and the Lie algebra is given by
. If we write in terms of block matrices by , then
Hence the inverse matrix of S is given by . For details, see Abraham-Marsden  and de Gosson  .
If we consider time evolutions of by a Hamiltonian flow, the resulting function is distributed multivariate normal.
Lemma 1. If we evolve a density function such that by a linear Hamiltonian system with transition matrix , then we have , where .
For any Hamiltonian equation
the transition matrix is given by and satisfies for any t.
Lemma 1 shows that if we evolve a density function distributed a normal distribution by Hamiltonian equations, then the result is also distributed a normal distribution whose variance is obtained to original variance by multiplying the transition matrix from left and its transpose from right. The proof is straightforward as follows. By and , we have
4. Proof of the Theorem
To prove the theorem, we explicitly construct affine symplectic diffeomorphisms.
First we consider known variance case. Let be the canonical symplectic structure on . We consider is in the first factor of , and take the matrix which corresponds to apriori distribution. Let
then we have and
Hence the Bayesian updating can be expressed as
where denotes the parallel translation on , and denotes the composition of maps.
In the case where unknown variance, we take a matrix as apriori distribution and set
then the desired transformation is given by
In this paper we show that Bayesian updating can be expressed by an affine symplectic diffeomorphism on whose base space contains a population mean vector. Bayesian updating is widely used in several areas, and recently it is usual to use computers to determine the posterior, implicitly. However our theorem expresses the posterior explicitly and concretely, and gives a dynamical interpretation of Bayesian updating.
 Amari, S. and Nagaoka, H. (2000) Methods of Information Geometry. Translations of Mathematical Monographs, Vol. 191, American Mathematical Society, Providence, Oxford University Press, Oxford, Translated from the 1993 Japanese Original by Daishi Harada.