Moment-Based Variational Inference for Markov Jump Processes - - PowerPoint PPT Presentation

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Moment-Based Variational Inference for Markov Jump Processes - - PowerPoint PPT Presentation

May 07, 2023 552 likes •852 views

Moment-Based Variational Inference for Markov Jump Processes Christian Wildner and Heinz Koeppl Department of Electrical Engineering and Information Technology Technische Universitt Darmstadt, Germany June 11, 2019 | Moment-Based

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June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 1

Christian Wildner and Heinz Koeppl Department of Electrical Engineering and Information Technology Technische Universität Darmstadt, Germany

Moment-Based Variational Inference for Markov Jump Processes

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SLIDE 2

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 2

Introduction

Model Class: Markov jump process / continuous time Markov chain

  • Applications in many domains (finance, social networks,

healthcare, systems biology, etc.)

  • Data-driven modelling requires latent state estimation
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SLIDE 3

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 3

Introduction

Model Class: Markov jump process / continuous time Markov chain

  • Applications in many domains (finance, social networks,

healthcare, systems biology, etc.)

  • Data-driven modelling requires latent state estimation

Problem: Hard/intractable for large state spaces

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SLIDE 4

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 4

Introduction

Model Class: Markov jump process / continuous time Markov chain

  • Applications in many domains (finance, social networks,

healthcare, systems biology, etc.)

  • Data-driven modelling requires latent state estimation

Problem: Hard/intractable for large state spaces Proposed solution: new variational inference approach based on

  • transition space partitioning
  • gradient-based optimization
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SLIDE 5

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 5

Markov Jump Processes

An MJP is fully defined by

  • an initial distribution
  • a transition function

with

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SLIDE 6

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 6

Markov Jump Processes

An MJP is fully defined by

  • an initial distribution
  • a transition function

with

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SLIDE 7

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 7

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X(t1 + h)

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X(t1 + 2h)

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X(t2)

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X(t2 + h)

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X(tN)

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Markov Jump Processes

An MJP is fully defined by

  • an initial distribution
  • a transition function

with

discretized representation of hidden MJP

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June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 8

Exact Inference

Goal: Compute posterior path distribution

· · · · · ·

Y(t1)

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X(t1)

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X(t1 + h)

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X(t1 + 2h)

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X(t2)

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Y(t2)

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X(t2 + h)

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X(tN)

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Y(tN)

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discretized representation of hidden MJP

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SLIDE 9

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 9

Exact Inference

Goal: Compute posterior path distribution Posterior paths are realized by smoothing process with modified transition function

· · · · · ·

Y(t1)

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X(t1)

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X(t1 + h)

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X(t1 + 2h)

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X(t2)

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Y(t2)

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X(t2 + h)

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X(tN)

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Y(tN)

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discretized representation of hidden MJP

slide-10
SLIDE 10

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 10

Exact Inference

Goal: Compute posterior path distribution Posterior paths are realized by smoothing process with modified transition function

· · · · · ·

Y(t1)

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X(t1)

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X(t1 + h)

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X(t1 + 2h)

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X(t2)

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Y(t2)

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X(t2 + h)

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X(tN)

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Y(tN)

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discretized representation of hidden MJP

hard to compute

slide-11
SLIDE 11

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 11

Variational Inference

Minimize path level KL divergence

exact smoothing process

slide-12
SLIDE 12

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 12

Variational Inference

Minimize path level KL divergence The usual decomposition applies

negative ELBO variational to prior marginal likelihood (evidence) exact smoothing process

slide-13
SLIDE 13

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 13

Variational Inference

Minimize path level KL divergence The usual decomposition applies Hard to construct suitable variational process class

negative ELBO variational to prior marginal likelihood (evidence) exact smoothing process

slide-14
SLIDE 14

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 14

Transition Space Partitioning

Smoothing process is in the class of controlled MJP with

prior transition function time and state dependent control factor

slide-15
SLIDE 15

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 15

Transition Space Partitioning

Smoothing process is in the class of controlled MJP with Partition transitions into groups and set

prior transition function time and state dependent control factor

slide-16
SLIDE 16

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 16

Transition Space Partitioning

Smoothing process is in the class of controlled MJP with Partition transitions into groups and set Example: random walk

prior transition function time and state dependent control factor

q

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slide-17
SLIDE 17

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 17

Partitioning Example

Example: random walk

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slide-18
SLIDE 18

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 18

Partitioning Example

Example: random walk

q

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p

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4

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slide-19
SLIDE 19

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 19

Partitioning Example

Example: random walk

q

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2

<latexit sha1_base64="fYpBIfYkVgQNWtDfbt/Ry7SmP6Q=">ACc3icfVFNSwMxE3X7/Vbj15Ca8GLZbcW9Fjw4rEFq0K7lGw6a4PJZklmhbL0F3jVH+cP8W76gbgVHAg83pvJvJmJMyksBsFnxVtb39jc2t7xd/f2Dw6Pjk8erM4Nhx7XUpunmFmQIoUeCpTwlBlgKpbwGL/czvTHVzBW6PQeJxlEij2nIhGcoaO6zeFRLWgE86B/QbgEtTZRGd4XBkMRprnClLklnbD4Mo4IZFzC1B/kFjLGX9gz9B1MmQIbFXOnU1p3zIgm2riXIp2zvysKpqydqNhlKoZju6rNyB+tXmqFyU1UiDTLEVK+6JTkqKms7npSBjgKCcOMG6EM0v5mBnG0W3H9+uU5xa1+v/TklrMXKDW0paHoN8BbcRm8zdljTLmYRVCxySloRx2rqu3uEq9v/Cx6ajfCq0ey2au3W8jLb5IxUyQUJyTVpkzvSIT3CZA38k4+Kl/emVf1zhepXmVZc0pK4V1+A+wmwCk=</latexit>

3

<latexit sha1_base64="XgeC2IRbpdeyNVzqw7vBy8FrXI=">ACc3icfVFNSwMxE3Xr7p+tXr0EloLXiy7Kuix4MVjBauCXUo2nW1Dk82SzApl6S/wqj/OH+Ld9ANxFRwIPN6bybyZiTMpLAbBR8VbW9/Y3Kpu+zu7e/sHtfrhg9W54dDjWmrzFDMLUqTQ4ESnjIDTMUSHuPJzVx/fAFjhU7vcZpBpNgoFYngDB1dzGoNYN2sAj6F4Qr0OyQZXQH9Uq/P9Q8V5Ail8za5zDIMCqYQcElzPx+biFjfMJG8OxgyhTYqFg4ndGWY4Y0ca9FOmC/VlRMGXtVMUuUzEc29/anPzWqVWmFxHhUizHCHly05JLilqOp+bDoUBjnLqAONGOLOUj5lhHN12fL9FeW5Rq/8/LanF3AVqLW156DHIF3AbscnCbUmznEkYRsUyp6QVcaxmvrtH+Hv7f8HDeTu8aJ/fXTY7l6vLVMkxaZBTEpIr0iG3pEt6hBMgr+SNvFc+vWOv4Z0sU73KquaIlMI7+wLuNcAq</latexit>

4

<latexit sha1_base64="SDL6jN7vAGo+5BbkP0rACN6+qE=">ACc3icfVFNSwMxE3Xr7p+Vo9eQmvBi2W3FvRY8OJRwarQLiWbztpgslmSWaEs/QVe9cf5Q7ybfiBuBQcCj/dmMm9m4kwKi0HwWfHW1jc2t6rb/s7u3v7BYe3owercOhxLbV5ipkFKVLoUAJT5kBpmIJj/HL9Ux/fAVjhU7vcZJBpNhzKhLBGTrqrjM8bAStYB70LwiXoNEli7gd1iqDwUjzXEGKXDJr+2GQYVQwg4JLmPqD3ELG+At7hr6DKVNgo2LudEqbjhnRBv3UqRz9ndFwZS1ExW7TMVwbFe1GfmjNUutMLmKCpFmOULKF52SXFLUdDY3HQkDHOXEAcaNcGYpHzPDOLrt+H6T8tyiVv9/WlKLmQvUWtry0GOQr+A2YpO525JmOZMwiopFTkr4lhNfXePcHX7f8FDuxVetNp3nUa3s7xMlZyQOjkjIbkXJDbkmPcALkjbyTj8qXd+LVvdNFqldZ1hyTUnjn3/BEwCs=</latexit>

common scaling for rightward transitions common scaling for leftward transitions

slide-20
SLIDE 20

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 20

Partitioning Example

Example: random walk

q

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p

<latexit sha1_base64="oxUfRjN90vSmO7sX2LN4M+VeLzs=">ACc3icfVFNSwMxE3X7/Vbj70Ea8GLZbcW9Fjw4lHBVsEuk1n2CyWZJZoSz9BV71x/lDvJtui7gWHAg83pvJvJmJMyksBsFnzVtZXVvf2Nzyt3d29/YPDo/6VueGQ49rqc1jzCxIkUIPBUp4zAwFUt4iF+uZ/rDKxgrdHqPkwixUapSARn6Ki7PmgEbSCMugyCBeg0SXzuH0+rA0GQ81zBSlyax9CoMo4IZFzC1B/kFjLGX9gInhxMmQIbFaXTKW06ZkgTbdxLkZbs74qCKWsnKnaZiuHY/tVm5I/WrLTC5CoqRJrlCmfd0pySVHT2dx0KAxwlBMHGDfCmaV8zAzj6Lbj+03Kc4ta/f9pRS1mLlBratDj0G+gtuITUq3Fc1yJmEYFfOcilbEsZr67h7h3+0vg367FV602nedRrezuMwmqZMTckZCckm65Ibckh7hBMgbeScftS+v7p14p/NUr7aoOSaV8M6/AWvXwGc=</latexit>

q

<latexit sha1_base64="pSTEOKv4dH9LMFMgPM2o3iCoaJQ=">ACc3icfVFNSwMxE3Xr7p+tXrsJVgLXiy7VdBjwYvHFqwKdpFsOmuDyWZNZoWy9Bd41R/nD/Fu+oG4Cg4EHu/NZN7MxJkUFoPgo+KtrK6tb1Q3/a3tnd29Wn3/xurcBhwLbW5i5kFKVIYoEAJd5kBpmIJt/HT5Uy/fQFjhU6vcZJBpNhjKhLBGTq/xQawbtYB70LwiXoNkli+g91CvD4UjzXEGKXDJr78Mgw6hgBgWXMPWHuYWM8Sf2CPcOpkyBjYq50yltOWZE23cS5HO2Z8VBVPWTlTsMhXDsf2tzchvrVqhclFVIg0yxFSvuiU5JKiprO56UgY4CgnDjBuhDNL+ZgZxtFtx/dblOcWtfr/05JazFyg1tKWhx6DfAG3EZvM3ZY0y5mEUVQsckpaEcdq6rt7hL+3/xfcdNrhabvTP2t2z5aXqZIGOSTHJCTnpEuSI8MCdAXskbea98eg3v0DtapHqVZc0BKYV38gVt5sBo</latexit>

q

<latexit sha1_base64="pSTEOKv4dH9LMFMgPM2o3iCoaJQ=">ACc3icfVFNSwMxE3Xr7p+tXrsJVgLXiy7VdBjwYvHFqwKdpFsOmuDyWZNZoWy9Bd41R/nD/Fu+oG4Cg4EHu/NZN7MxJkUFoPgo+KtrK6tb1Q3/a3tnd29Wn3/xurcBhwLbW5i5kFKVIYoEAJd5kBpmIJt/HT5Uy/fQFjhU6vcZJBpNhjKhLBGTq/xQawbtYB70LwiXoNkli+g91CvD4UjzXEGKXDJr78Mgw6hgBgWXMPWHuYWM8Sf2CPcOpkyBjYq50yltOWZE23cS5HO2Z8VBVPWTlTsMhXDsf2tzchvrVqhclFVIg0yxFSvuiU5JKiprO56UgY4CgnDjBuhDNL+ZgZxtFtx/dblOcWtfr/05JazFyg1tKWhx6DfAG3EZvM3ZY0y5mEUVQsckpaEcdq6rt7hL+3/xfcdNrhabvTP2t2z5aXqZIGOSTHJCTnpEuSI8MCdAXskbea98eg3v0DtapHqVZc0BKYV38gVt5sBo</latexit>

p

<latexit sha1_base64="oxUfRjN90vSmO7sX2LN4M+VeLzs=">ACc3icfVFNSwMxE3X7/Vbj70Ea8GLZbcW9Fjw4lHBVsEuk1n2CyWZJZoSz9BV71x/lDvJtui7gWHAg83pvJvJmJMyksBsFnzVtZXVvf2Nzyt3d29/YPDo/6VueGQ49rqc1jzCxIkUIPBUp4zAwFUt4iF+uZ/rDKxgrdHqPkwixUapSARn6Ki7PmgEbSCMugyCBeg0SXzuH0+rA0GQ81zBSlyax9CoMo4IZFzC1B/kFjLGX9gInhxMmQIbFaXTKW06ZkgTbdxLkZbs74qCKWsnKnaZiuHY/tVm5I/WrLTC5CoqRJrlCmfd0pySVHT2dx0KAxwlBMHGDfCmaV8zAzj6Lbj+03Kc4ta/f9pRS1mLlBratDj0G+gtuITUq3Fc1yJmEYFfOcilbEsZr67h7h3+0vg367FV602nedRrezuMwmqZMTckZCckm65Ibckh7hBMgbeScftS+v7p14p/NUr7aoOSaV8M6/AWvXwGc=</latexit>

p

<latexit sha1_base64="oxUfRjN90vSmO7sX2LN4M+VeLzs=">ACc3icfVFNSwMxE3X7/Vbj70Ea8GLZbcW9Fjw4lHBVsEuk1n2CyWZJZoSz9BV71x/lDvJtui7gWHAg83pvJvJmJMyksBsFnzVtZXVvf2Nzyt3d29/YPDo/6VueGQ49rqc1jzCxIkUIPBUp4zAwFUt4iF+uZ/rDKxgrdHqPkwixUapSARn6Ki7PmgEbSCMugyCBeg0SXzuH0+rA0GQ81zBSlyax9CoMo4IZFzC1B/kFjLGX9gInhxMmQIbFaXTKW06ZkgTbdxLkZbs74qCKWsnKnaZiuHY/tVm5I/WrLTC5CoqRJrlCmfd0pySVHT2dx0KAxwlBMHGDfCmaV8zAzj6Lbj+03Kc4ta/f9pRS1mLlBratDj0G+gtuITUq3Fc1yJmEYFfOcilbEsZr67h7h3+0vg367FV602nedRrezuMwmqZMTckZCckm65Ibckh7hBMgbeScftS+v7p14p/NUr7aoOSaV8M6/AWvXwGc=</latexit>

1

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2

<latexit sha1_base64="fYpBIfYkVgQNWtDfbt/Ry7SmP6Q=">ACc3icfVFNSwMxE3X7/Vbj15Ca8GLZbcW9Fjw4rEFq0K7lGw6a4PJZklmhbL0F3jVH+cP8W76gbgVHAg83pvJvJmJMyksBsFnxVtb39jc2t7xd/f2Dw6Pjk8erM4Nhx7XUpunmFmQIoUeCpTwlBlgKpbwGL/czvTHVzBW6PQeJxlEij2nIhGcoaO6zeFRLWgE86B/QbgEtTZRGd4XBkMRprnClLklnbD4Mo4IZFzC1B/kFjLGX9gz9B1MmQIbFXOnU1p3zIgm2riXIp2zvysKpqydqNhlKoZju6rNyB+tXmqFyU1UiDTLEVK+6JTkqKms7npSBjgKCcOMG6EM0v5mBnG0W3H9+uU5xa1+v/TklrMXKDW0paHoN8BbcRm8zdljTLmYRVCxySloRx2rqu3uEq9v/Cx6ajfCq0ey2au3W8jLb5IxUyQUJyTVpkzvSIT3CZA38k4+Kl/emVf1zhepXmVZc0pK4V1+A+wmwCk=</latexit>

3

<latexit sha1_base64="XgeC2IRbpdeyNVzqw7vBy8FrXI=">ACc3icfVFNSwMxE3Xr7p+tXr0EloLXiy7Kuix4MVjBauCXUo2nW1Dk82SzApl6S/wqj/OH+Ld9ANxFRwIPN6bybyZiTMpLAbBR8VbW9/Y3Kpu+zu7e/sHtfrhg9W54dDjWmrzFDMLUqTQ4ESnjIDTMUSHuPJzVx/fAFjhU7vcZpBpNgoFYngDB1dzGoNYN2sAj6F4Qr0OyQZXQH9Uq/P9Q8V5Ail8za5zDIMCqYQcElzPx+biFjfMJG8OxgyhTYqFg4ndGWY4Y0ca9FOmC/VlRMGXtVMUuUzEc29/anPzWqVWmFxHhUizHCHly05JLilqOp+bDoUBjnLqAONGOLOUj5lhHN12fL9FeW5Rq/8/LanF3AVqLW156DHIF3AbscnCbUmznEkYRsUyp6QVcaxmvrtH+Hv7f8HDeTu8aJ/fXTY7l6vLVMkxaZBTEpIr0iG3pEt6hBMgr+SNvFc+vWOv4Z0sU73KquaIlMI7+wLuNcAq</latexit>

4

<latexit sha1_base64="SDL6jN7vAGo+5BbkP0rACN6+qE=">ACc3icfVFNSwMxE3Xr7p+Vo9eQmvBi2W3FvRY8OJRwarQLiWbztpgslmSWaEs/QVe9cf5Q7ybfiBuBQcCj/dmMm9m4kwKi0HwWfHW1jc2t6rb/s7u3v7BYe3owercOhxLbV5ipkFKVLoUAJT5kBpmIJj/HL9Ux/fAVjhU7vcZJBpNhzKhLBGTrqrjM8bAStYB70LwiXoNEli7gd1iqDwUjzXEGKXDJr+2GQYVQwg4JLmPqD3ELG+At7hr6DKVNgo2LudEqbjhnRBv3UqRz9ndFwZS1ExW7TMVwbFe1GfmjNUutMLmKCpFmOULKF52SXFLUdDY3HQkDHOXEAcaNcGYpHzPDOLrt+H6T8tyiVv9/WlKLmQvUWtry0GOQr+A2YpO525JmOZMwiopFTkr4lhNfXePcHX7f8FDuxVetNp3nUa3s7xMlZyQOjkjIbkXJDbkmPcALkjbyTj8qXd+LVvdNFqldZ1hyTUnjn3/BEwCs=</latexit>

common scaling for rightward transitions common scaling for leftward transitions

1

<latexit sha1_base64="54GkNeQzaHP/Kdwz7Xg0OBe1OFQ=">ACc3icfVFNSwMxE3Xr7p+Vo9egrXgxbJbC3osePHYglXBLpJNZ20w2SzJrFCW/gKv+uP8Id5Nt0VcBQcCj/dmMm9m4kwKi0HwUfNWVtfWN+qb/tb2zu7efuPg1urcBhyLbW5j5kFKVIYokAJ95kBpmIJd/Hz1Vy/ewFjhU5vcJpBpNhTKhLBGTpqED7uN4N2UAb9C8IlaPbIvqPjdpoNY8V5Ail8zahzDIMCqYQcElzPxRbiFj/Jk9wYODKVNgo6J0OqMtx4xpo17KdKS/VlRMGXtVMUuUzGc2N/anPzWpVWmFxGhUizHCHli05JLilqOp+bjoUBjnLqAONGOLOUT5hHN12fL9FeW5Rq/8/rajF3AVqLW16AnIF3AbsUnptqJZziSMo2KRU9GKOFYz390j/L39v+C20w7P251Bt9nrLi9TJ0fkmJySkFyQHrkmfTIknAB5JW/kvfbpHXnH3ski1astaw5JbyzL+oXwCg=</latexit>

2

<latexit sha1_base64="fYpBIfYkVgQNWtDfbt/Ry7SmP6Q=">ACc3icfVFNSwMxE3X7/Vbj15Ca8GLZbcW9Fjw4rEFq0K7lGw6a4PJZklmhbL0F3jVH+cP8W76gbgVHAg83pvJvJmJMyksBsFnxVtb39jc2t7xd/f2Dw6Pjk8erM4Nhx7XUpunmFmQIoUeCpTwlBlgKpbwGL/czvTHVzBW6PQeJxlEij2nIhGcoaO6zeFRLWgE86B/QbgEtTZRGd4XBkMRprnClLklnbD4Mo4IZFzC1B/kFjLGX9gz9B1MmQIbFXOnU1p3zIgm2riXIp2zvysKpqydqNhlKoZju6rNyB+tXmqFyU1UiDTLEVK+6JTkqKms7npSBjgKCcOMG6EM0v5mBnG0W3H9+uU5xa1+v/TklrMXKDW0paHoN8BbcRm8zdljTLmYRVCxySloRx2rqu3uEq9v/Cx6ajfCq0ey2au3W8jLb5IxUyQUJyTVpkzvSIT3CZA38k4+Kl/emVf1zhepXmVZc0pK4V1+A+wmwCk=</latexit>

3

<latexit sha1_base64="XgeC2IRbpdeyNVzqw7vBy8FrXI=">ACc3icfVFNSwMxE3Xr7p+tXr0EloLXiy7Kuix4MVjBauCXUo2nW1Dk82SzApl6S/wqj/OH+Ld9ANxFRwIPN6bybyZiTMpLAbBR8VbW9/Y3Kpu+zu7e/sHtfrhg9W54dDjWmrzFDMLUqTQ4ESnjIDTMUSHuPJzVx/fAFjhU7vcZpBpNgoFYngDB1dzGoNYN2sAj6F4Qr0OyQZXQH9Uq/P9Q8V5Ail8za5zDIMCqYQcElzPx+biFjfMJG8OxgyhTYqFg4ndGWY4Y0ca9FOmC/VlRMGXtVMUuUzEc29/anPzWqVWmFxHhUizHCHly05JLilqOp+bDoUBjnLqAONGOLOUj5lhHN12fL9FeW5Rq/8/LanF3AVqLW156DHIF3AbscnCbUmznEkYRsUyp6QVcaxmvrtH+Hv7f8HDeTu8aJ/fXTY7l6vLVMkxaZBTEpIr0iG3pEt6hBMgr+SNvFc+vWOv4Z0sU73KquaIlMI7+wLuNcAq</latexit>

4

<latexit sha1_base64="SDL6jN7vAGo+5BbkP0rACN6+qE=">ACc3icfVFNSwMxE3Xr7p+Vo9eQmvBi2W3FvRY8OJRwarQLiWbztpgslmSWaEs/QVe9cf5Q7ybfiBuBQcCj/dmMm9m4kwKi0HwWfHW1jc2t6rb/s7u3v7BYe3owercOhxLbV5ipkFKVLoUAJT5kBpmIJj/HL9Ux/fAVjhU7vcZJBpNhzKhLBGTrqrjM8bAStYB70LwiXoNEli7gd1iqDwUjzXEGKXDJr+2GQYVQwg4JLmPqD3ELG+At7hr6DKVNgo2LudEqbjhnRBv3UqRz9ndFwZS1ExW7TMVwbFe1GfmjNUutMLmKCpFmOULKF52SXFLUdDY3HQkDHOXEAcaNcGYpHzPDOLrt+H6T8tyiVv9/WlKLmQvUWtry0GOQr+A2YpO525JmOZMwiopFTkr4lhNfXePcHX7f8FDuxVetNp3nUa3s7xMlZyQOjkjIbkXJDbkmPcALkjbyTj8qXd+LVvdNFqldZ1hyTUnjn3/BEwCs=</latexit>

λ1(t)p

<latexit sha1_base64="x5IMz86wHVBj0M7Y964uj9IajY=">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</latexit>

λ1(t)p

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λ1(t)p

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λ2(t)q

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λ2(t)q

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λ2(t)q

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SLIDE 21

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 21

Complexity Reduction

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SLIDE 22

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 22

Complexity Reduction

transition space partitioning into r classes

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SLIDE 23

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 23

Complexity Reduction

transition space partitioning into r classes

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SLIDE 24

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 24

Complexity Reduction

transition space partitioning into r classes expected summary statistic

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SLIDE 25

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 25

Control Problem

Using the Markov property, derive moment equations for

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SLIDE 26

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 26

Control Problem

Using the Markov property, derive moment equations for Obtain non-linear, deterministic optimal control problem

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SLIDE 27

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 27

Control Problem

Using the Markov property, derive moment equations for Obtain non-linear, deterministic optimal control problem Solve via natural gradient descent in the controls

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SLIDE 28

June 11, 2019 | Moment-Based Variational Inference for Markov Jump Processes | Christian Wildner, Heinz Koeppl | 28

Example Application

Scenario: Stochastic gene expression Studied by fluorescence microscopy Goals: Parameter estimation Model comparison Optimal experiment design

t)

t y(t)