1. Overview of our studies
Ryo Yamada
April, 2017

2. Data set Records are discrete Values {0,1},{0,1,2,…},R. {A,B,C,…}
Dimensions
No. samples

3. Not homogeneous values
We care “values” heterogeneous

4. N=1 One value A value set Space
Voxel type
Point set type
Essentially same : Discrete observation of “distribution”

5. N = 1, Discrete observation of distribution
Spectral Decomposition
Distribution -> Point
Dimension reduction from ∞ -> k
Moments
Fourrier, Wavelets, Deep learning
Partition

6. N > 1 We care non-homogeneity among N 2^N –(N+1) combinations
We separate information into 2 parts
Common part
Difference part
N(N-1)/2 pairs in particular
Distance/ divergence cares “Different part” only
MDS; dimension reduction to ~ (N-1) dimension
N distributions -> N points = 1 sample of N points distribution
Go back to Slide 4

7. Partition Whole can be separated into multiple parts
The way of partition mean something from heterogeneity standpoint
A value : Integer partition, Real value partition …
Space : Subsets
Classification, Clustering, Segregation
How classify
Parts
Single part
Go to slide 5
A set of parts
Go to slide 6

8. Parts in the whole Subsets are embedded in the whole set
Submanifold
Subsets/submanifolds themselves are something in lower dimension but recorded with full dimension
Treat subsets/submanifolds in their own dimension
Curve as a curve
When they are parameterized in the original space, go to slide3 (Spectral decomposition)
When not, new parameterization is necessary

9. Subsets/submanifolds
Parameterization is complex
Simplification/standardization
A unit segment, A unit circle, A unit sphere surface, A unit disc,….
Once standardized space with parameterization
N = 1 -> go to slide 5
N > 1 -> go to slide 6

10. Subsets/submanifolds
Subsets/submanifolds themselves are something in lower dimension but recorded with full dimension
This dimension reduction is “geometric”.

11. Submanifolds
Information geometry treat all-the-all distributions as points in the infinite-dimensional space
Particular types of distributions, such as normal distribution is a subset/submanifold of the whole space
Submanifolds have expressions parameterized
Then, subsets/submanifolds, incl. cell shapes and FACS distributions, corresponds to “distributions parameterized with finite number of paramters??”

12. Time series analysis Where does time come in???
Time is (usually) a special dimensional axis; independent from others and unidirectional.
Therefore, parameterization with time is almost always possible and “manifold-like” complex parameterization does not come in.
Points, distributions, subsets/submanifolds in space can be traced along time.

13. Spaces in general 1-d n-d Simplex : {A,B,C,…}, categorical
{0,1}
R -> Discrete records: 1-dimensional lattice
n-d
Discrete records: n-dimensional lattice
Simplex : {A,B,C,…}, categorical
Network: A substructure of simplex
Space structures are in the shape of “graph”, that determines “adjacency/neighbor” and “weight” of edge give information of “distance/diversity”.

14. STAN group P(theta|x) = P(x|theta)P(theta)/P(x)
P(theta|x1,x2) = P(x1,x2|theta)P(theta)/P(x1,x2)
P(x1,x2) : joint distribution
When P(x1,x2) = P(x1)P(x2) and P(x1,x2|theta) = P(x1|theta)P(x2|theta),
P(theta|x1,x2) = P(x1|theta)P(x2|theta)P(theta)/P(x1)P(x2)
= P(x1|theta)/P(x1) P(x2|theta)/P(x2) P(theta)
No need to care STAN with joint distributions

15. When P(x1,x2) != P(x1)P(x2) or P(x1,x2|theta) != P(x1|theta)P(x2|theta)
STAN cares joint distributions that is the target of slide 5
Model of STAN determines “Joint distribution”
Conditioned sampled data records are available and STAN estimates posterior distribution(s)
Meta-analysis project
Same sample set and multiple observations
One joint distribution with multiple projections to different planes.
Different sample sets and multiple observations
One joint distribution with multiple projections to similar planes in “some-independent” occasions

16. Decision theory Parameterized stochastic phenomena
Each stochastic event depends on the past with heavy memory
This should be submanifolds in information theory space but not easily defined without numeric labor
The difficulty is similar to “self-avoiding” path simulation.
Some stochastic rules are known to automatically generate self- avoiding path… that is related the “curve” … therefore… ????