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1. Ishanu Chattopadhyay

2. Disambiguation Of Stochasic Processes Competence Response Of Transgenic B. subtilis Bacillus subtilis Strains Comptence Circuit (Wild Type) Comptence Response Experimental Data: G. Suel (U. Of Texas, Southwestern Med. School)

22. Motivating Questions On Stochastic Processes

23. Basic Idea: Negate & Subtract! Just Like Real Numbers, · · · Well Almost x + (−y) = 0 ⇔ x = y Example: 0.375 + (−0.375) = 0

24. Basic Idea: Negate & Subtract! Just Like Real Numbers, · · · Well Almost x + (−y) = 0 ⇔ x = y Example: 0.375 + (−0.375) = 0

25. Symbols Can Be Anything Individual Symbols Need Not Have Algebraic Properties Manipulate the information content in symbolic streams. . . + − No matter what the symbols are. . .

26. Quantization Of Observed Sequences How Do We Get Symbols? σ6 σ5 σ8 σ7 σ1 σ1 σ3 · · · · · · σ5 σ3 σ3 · · · σ2 σ1 (a) Phase Space Quantization σ3 σ4 σ5 Binary Quantization σ4 σ3 σ2 (b) Data Range Discretization σ1 Continuous 8 symbol Signal Quantization

27. Probabilistic Finite State Automata As Stochastic Models Syntactic Deﬁnition: The Anatomy σ1 |0.9 σ2 |0.2 States q2 q1 Transition (Events From Alphabet) σ1 |0.8 σ1 |0.3 σ2 |0.1 q3 Alphabet Symbol σ2 |0.7 Probability of generating σ2 at state q3 PFSA Can act as symbolic generators

28. Probabilistic Finite State Automata As Stochastic Models Syntactic Deﬁnition: The Anatomy σ1 |0.9 σ2 |0.2 States q2 q1 Transition (Events From Alphabet) σ1 |0.8 σ1 |0.3 σ2 |0.1 q3 Alphabet Symbol σ2 |0.7 Probability of generating σ2 at state q3 PFSA Can act as symbolic generators

43. Causal States Of Dynamical Processes Equivalence Classes Of The Nerode Relation Future-equivalent Strings Go To The Same ”State“ Equality Of Distributions over... future symbols future substrings of length 2 future substrings What Does of length 3 ”Equivalence Of Futures“ future substrings of length 4 Mean ? all lengths of future substrings

44. Causal States Of Dynamical Processes Equivalence Classes Of The Nerode Relation Future-equivalent Strings Go To The Same ”State“ Equality Of Distributions over... future symbols future substrings of length 2 future substrings What Does of length 3 ”Equivalence Of Futures“ future substrings of length 4 Mean ? all lengths of future substrings

45. Stochastic Processes As Probabilistic Dynamical Systems Measures On Infinite Strings Present Dynamics completely specified by the probabilites:   p(τ1 Σω )    p(τ2 Σω )      Past p(τ3 Σ   ω ) Observation       . .   τ . All Possible Infinite Futures (Σω ) Quantized Stochastic Process −− − − − − − − − − −→ Probability space (Σω , B, P) Finite Encoding Probability space (Σω , B, P) −− − − − − − − − − − − − − − − −→ PFSA

49. Relationship Between Space of Automata & Space Of Measures Multiple PFSA May Map To Same Measure Any two PFSA can be Dynamics-equivalent made structure equivalent PFSA Space Of PFSA (A ) Non-minimal realizations exist Structue is not that important measure 2 measure 1 Space Of measures on Each measure deﬁnes inﬁnite strings (P) a probability space: (Σω , B, p1 )

105. PFSAs Are Encodings Of Measure Spaces · · · And Not Just Graphs! Any string 1(0.1) terminating in 10 0(0.9) q2 q1 1(0.2) 0(0.8) Any string terminating in 1 1(0.7) q3 0(0.3) Any string terminating in 00 • Causality Structure How many causal states? Interdependence between causal states • Probability Morph Σ⋆ Immediate future from causal states (Set of all ﬁnite strings on Analogous to tangent vectors alphabet Σ)

106. PFSAs Are Encodings Of Measure Spaces · · · And Not Just Graphs! Any string 1(0.1) terminating in 10 0(0.9) q2 q1 1(0.2) 0(0.8) Any string terminating in 1 1(0.7) q3 0(0.3) Any string terminating in 00 • Causality Structure How many causal states? Interdependence between causal states string: ω′ 01 1 • Probability Morph string: ω00 Σ⋆ Immediate future from causal states (Set of all 0 string: ﬁnite strings on Analogous to tangent vectors ω′ 00 alphabet Σ) (A) Transition Structure induced by state deﬁnitions

107. PFSAs Are Encodings Of Measure Spaces · · · And Not Just Graphs! Any string 1(0.1) terminating in 10 0(0.9) q2 q1 1(0.2) 0(0.8) Any string terminating in 1 1(0.7) q3 0(0.3) Any string terminating in 00 0(0.9) 1(0.1) 1(0.1) • Causality Structure 0(0.9) 1(0.2) How many causal states? (B) Symbol Interdependence between causal states 0(0.8) probabilities identical from 1(0.2) same state 0(0.8) • Probability Morph Σ⋆ Immediate future from causal states (Set of all ﬁnite strings on Analogous to tangent vectors alphabet Σ)

108. PFSAs Are Encodings Of Measure Spaces · · · And Not Just Graphs! Any string 1(0.1) terminating in 10 0(0.9) q2 q1 1(0.2) 0(0.8) Any string terminating in 1 1(0.7) q3 0(0.3) Any string terminating in 00 0(0.9) 1(0.1) 1(0.1) • Causality Structure 0(0.9) 1(0.2) How many causal states? (B) Symbol Interdependence between causal states string: 0(0.8) probabilities identical from ω′ 01 1(0.2) same state 0(0.8) 1 • Probability Morph string: Immediate future from causal states ω00 Σ⋆ (Set of all 0 string: ﬁnite strings on Analogous to tangent vectors ω′ 00 alphabet Σ) (A) Transition Structure induced by state deﬁnitions

109. PFSAs Are Encodings Of Measure Spaces · · · And Not Just Graphs! And the exact graphs are not even that important..... 1(0.1) 0(0.9) q1 q2 1(0.2) Any string 0(0.8) terminating in 1 1(0.7) q3 0(0.3) 0(0.9) 1(0.1) 1(0.1) 0(0.9) Σ⋆ (Set of all ﬁnite strings on alphabet Σ) State splitting produces

110. PFSAs Are Encodings Of Measure Spaces · · · And Not Just Graphs! And the exact graphs are not even that important..... 1(0.1) 0(0.9) q1 q2 1(0.2) Any string 0(0.8) terminating in 1 1(0.7) q3 0(0.3) 1(0.1) 0(0.9) 1(0.1) 0(0.9) 1(0.1) 0(0.9) 0(0.9) 1(0.1) Σ⋆ (Set of all ﬁnite strings on alphabet Σ) State splitting produces

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