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A plausible model of recognition and postdiction in dynamic environment
Kevin Li, Maneesh Sahani
Gatsby Computational Neuroscience Unit, University College London
January 6, 2020
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 1 / 20
Introduction
1. Introduction
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 2 / 20
Introduction
Inference using an internal model (Helmholtz machine)
static world
p(z,x) −→ r(·)
z
x
r
xDayan, Hinton, Neal & Zemel, 1995
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 3 / 20
Introduction
Inference using an internal model (Helmholtz machine)
static world
p(z,x) −→ r(·)
z
x
r
x
dynamic world
p(z1:t,x1:t) −→ rt(·)
zt−1 zt
xtxt−1
rt−1 rt
xtxt−1Dayan, Hinton, Neal & Zemel, 1995
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 3 / 20
Introduction
Illusion 1
0 1 2 3time (sec)
500
1000
1500
frequ
ency
(Hz)
0 1 2 3time (sec)
500
1000
1500
frequ
ency
(Hz)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 4 / 20
Introduction
Illusion 1
0 1 2 3time (sec)
500
1000
1500
frequ
ency
(Hz)
0 1 2 3time (sec)
500
1000
1500
frequ
ency
(Hz)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 4 / 20
Introduction
Illusion 1
0 1 2 3time (sec)
500
1000
1500
frequ
ency
(Hz)
0 1 2 3time (sec)
500
1000
1500
frequ
ency
(Hz)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 4 / 20
Introduction
Illusion 1
0 1 2 3time (sec)
500
1000
1500
frequ
ency
(Hz)
0 1 2 3time (sec)
500
1000
1500
frequ
ency
(Hz)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 4 / 20
Introduction
Illusion 2: cutaneous rabbit
In the course of designing some experiments on thecutaneous perception of mechanical pulses deliveredto the back of the forearm, it was discovered that,under some conditions of timing, the taps producedseemed not to be properly localized under thecontactors. [...] They will seem to be distributed,with more or less uniform spacing, from the regionof the �rst contactor to that of the third. There is
a smooth progression of jumps up the arm, as
if a tiny rabbit were hopping from elbow to wrist.
Geldard & Sherrick, 1972, Science
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 5 / 20
Introduction
Illusion 2: cutaneous rabbit
In the course of designing some experiments on thecutaneous perception of mechanical pulses deliveredto the back of the forearm, it was discovered that,under some conditions of timing, the taps producedseemed not to be properly localized under thecontactors. [...] They will seem to be distributed,with more or less uniform spacing, from the regionof the �rst contactor to that of the third. There is
a smooth progression of jumps up the arm, as
if a tiny rabbit were hopping from elbow to wrist.
Geldard & Sherrick, 1972, Science
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 5 / 20
Introduction
In those examples:
Percept of the past is changed by new, future observation
�Law of Continuity�
What could be the neural basis for these statistical computation?
representing beliefs as distributional objects
updating beliefs of the past based on new evidence in real time?
learning to do all the above
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 6 / 20
Introduction
In those examples:
Percept of the past is changed by new, future observation
�Law of Continuity�
What could be the neural basis for these statistical computation?
representing beliefs as distributional objects
updating beliefs of the past based on new evidence in real time?
learning to do all the above
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 6 / 20
Introduction
In those examples:
Percept of the past is changed by new, future observation
�Law of Continuity�
What could be the neural basis for these statistical computation?
representing beliefs as distributional objects
updating beliefs of the past based on new evidence in real time?
learning to do all the above
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 6 / 20
Introduction
In those examples:
Percept of the past is changed by new, future observation
�Law of Continuity�
What could be the neural basis for these statistical computation?
representing beliefs as distributional objects
updating beliefs of the past based on new evidence in real time?
learning to do all the above
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 6 / 20
Introduction
In those examples:
Percept of the past is changed by new, future observation
�Law of Continuity�
What could be the neural basis for these statistical computation?
representing beliefs as distributional objects
updating beliefs of the past based on new evidence in real time?
learning to do all the above
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 6 / 20
Introduction
In those examples:
Percept of the past is changed by new, future observation
�Law of Continuity�
What could be the neural basis for these statistical computation?
representing beliefs as distributional objects
updating beliefs of the past based on new evidence in real time?
learning to do all the above
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 6 / 20
Introduction
In those examples:
Percept of the past is changed by new, future observation
�Law of Continuity�
What could be the neural basis for these statistical computation?
representing beliefs as distributional objects
updating beliefs of the past based on new evidence in real time?
learning to do all the above
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 6 / 20
Distributed distributional code
2. Distributed distributional code
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 7 / 20
Distributed distributional code
DDC: a framework for neural representation of uncertainty
A DDC encodes a probability distribution:
q(z)
by a set of tuning functions
γ(z) := [γ1 (z) , γ2 (z) , γ3 (z) , ..., γK (z)]
into a set of expectations
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
Zemel, Dayan & Pouget (1998); Sahani & Dayan (2003),Vértes & Sahani (2018)
, , , , , , , ,1 2 ... K[ ], , , , , , , ,
1 2 ... K[ ], , , , , , , ,
1 2 ... K[ ]
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 8 / 20
Distributed distributional code
DDC: a framework for neural representation of uncertainty
A DDC encodes a probability distribution:
q(z)
by a set of tuning functions
γ(z) := [γ1 (z) , γ2 (z) , γ3 (z) , ..., γK (z)]
into a set of expectations
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
Zemel, Dayan & Pouget (1998); Sahani & Dayan (2003),Vértes & Sahani (2018)
, , , , , , , ,1 2 ... K[ ]
, , , , , , , ,1 2 ... K[ ], , , , , , , ,
1 2 ... K[ ]
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 8 / 20
Distributed distributional code
DDC: a framework for neural representation of uncertainty
A DDC encodes a probability distribution:
q(z)
by a set of tuning functions
γ(z) := [γ1 (z) , γ2 (z) , γ3 (z) , ..., γK (z)]
into a set of expectations
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
Zemel, Dayan & Pouget (1998); Sahani & Dayan (2003),Vértes & Sahani (2018)
, , , , , , , ,1 2 ... K[ ], , , , , , , ,
1 2 ... K[ ]
, , , , , , , ,1 2 ... K[ ]
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 8 / 20
Distributed distributional code
DDC: a framework for neural representation of uncertainty
A DDC encodes a probability distribution:
q(z)
by a set of tuning functions
γ(z) := [γ1 (z) , γ2 (z) , γ3 (z) , ..., γK (z)]
into a set of expectations
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
Zemel, Dayan & Pouget (1998); Sahani & Dayan (2003),Vértes & Sahani (2018)
, , , , , , , ,1 2 ... K[ ], , , , , , , ,
1 2 ... K[ ], , , , , , , ,
1 2 ... K[ ]
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 8 / 20
Distributed distributional code
DDC: a framework for neural representation of uncertainty
A DDC encodes a probability distribution:
q(z)
by a set of tuning functions
γ(z) := [γ1 (z) , γ2 (z) , γ3 (z) , ..., γK (z)]
into a set of expectations
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
Zemel, Dayan & Pouget (1998); Sahani & Dayan (2003),Vértes & Sahani (2018)
, , , , , , , ,1 2 ... K[ ], , , , , , , ,
1 2 ... K[ ], , , , , , , ,
1 2 ... K[ ]
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 8 / 20
Distributed distributional code
Why DDC? It can encode a large family of distributions
given Eq [γk (z)] = rk, ∀k ∈ {1, 2, . . .K}
q(z) = arg maxH[q]
=1
Z(r)exp
[∑k
θk(r)γk(z)
]
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 9 / 20
Distributed distributional code
Why DDC? It can encode a large family of distributions
given Eq [γk (z)] = rk, ∀k ∈ {1, 2, . . .K}q(z) = arg maxH[q]
=1
Z(r)exp
[∑k
θk(r)γk(z)
]
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 9 / 20
Distributed distributional code
Why DDC? It can encode a large family of distributions
given Eq [γk (z)] = rk, ∀k ∈ {1, 2, . . .K}q(z) = arg maxH[q]
=1
Z(r)exp
[∑k
θk(r)γk(z)
]
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 9 / 20
Distributed distributional code
Why DDC? It can encode a large family of distributions
given Eq [γk (z)] = rk, ∀k ∈ {1, 2, . . .K}q(z) = arg maxH[q]
=1
Z(r)exp
[∑k
θk(r)γk(z)
]Normal q(z)
(z)
,[ ]rk
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 9 / 20
Distributed distributional code
Why DDC? It can encode a large family of distributions
given Eq [γk (z)] = rk, ∀k ∈ {1, 2, . . .K}q(z) = arg maxH[q]
=1
Z(r)exp
[∑k
θk(r)γk(z)
]
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 9 / 20
Distributed distributional code
Why DDC? It can encode a large family of distributions
given Eq [γk (z)] = rk, ∀k ∈ {1, 2, . . .K}q(z) = arg maxH[q]
=1
Z(r)exp
[∑k
θk(r)γk(z)
]Discrete q(z)
(z)
, , , , , , , , ,[ ]rk
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 9 / 20
Distributed distributional code
Why DDC? It can encode a large family of distributions
given Eq [γk (z)] = rk, ∀k ∈ {1, 2, . . .K}q(z) = arg maxH[q]
=1
Z(r)exp
[∑k
θk(r)γk(z)
]arbitrary q(z)
q(z)
k(z)
, , , , , , ,[ ]rk
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 9 / 20
Distributed distributional code
Why DDC? It can encode a large family of distributions
given Eq [γk (z)] = rk, ∀k ∈ {1, 2, . . .K}q(z) = arg maxH[q]
=1
Z(r)exp
[∑k
θk(r)γk(z)
]arbitrary q(z)
q(z)q(z)
k(z), K=8
, , , , , , ,[ ]rk
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 9 / 20
Distributed distributional code
Why DDC? It can encode a large family of distributions
given Eq [γk (z)] = rk, ∀k ∈ {1, 2, . . .K}q(z) = arg maxH[q]
=1
Z(r)exp
[∑k
θk(r)γk(z)
]arbitrary q(z)
q(z)q(z)
k(z), K=12
, , , , , , , , , , ,[ ]rk
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 9 / 20
Distributed distributional code
Why DDC? It can encode a large family of distributions
given Eq [γk (z)] = rk, ∀k ∈ {1, 2, . . .K}q(z) = arg maxH[q]
=1
Z(r)exp
[∑k
θk(r)γk(z)
]arbitrary q(z)
q(z)q(z)
k(z), K=16
, , , , , , , , , , , , , , ,[ ]rk
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 9 / 20
Distributed distributional code
Why DDC? It makes computation simple for neurons
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
Key computations involve expected values:
Message passing:q(z2) = Eq(z1) [p(z2|z1)]
Marginalization:
q(z2 ∈ (a, b)) = Eq(z1,z2) [1(a < z2 < b)]
Action evaluation:Q(a) = Eq(s) [R(s, a)]
How to compute E [h(z)]?
if h(z) ≈∑i
αiγi(z)⇒ E [h(z)] ≈∑i
αiri
, , , , , , , ,1 2 ... K[ ]
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 10 / 20
Distributed distributional code
Why DDC? It makes computation simple for neurons
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
Key computations involve expected values:
Message passing:q(z2) = Eq(z1) [p(z2|z1)]
Marginalization:
q(z2 ∈ (a, b)) = Eq(z1,z2) [1(a < z2 < b)]
Action evaluation:Q(a) = Eq(s) [R(s, a)]
How to compute E [h(z)]?
if h(z) ≈∑i
αiγi(z)⇒ E [h(z)] ≈∑i
αiri
q(z)
(z)
, , , , , , , ,1 2 ... K[ ]r
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 10 / 20
Distributed distributional code
Why DDC? It makes computation simple for neurons
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
Key computations involve expected values:
Message passing:q(z2) = Eq(z1) [p(z2|z1)]
Marginalization:
q(z2 ∈ (a, b)) = Eq(z1,z2) [1(a < z2 < b)]
Action evaluation:Q(a) = Eq(s) [R(s, a)]
How to compute E [h(z)]?
if h(z) ≈∑i
αiγi(z)⇒ E [h(z)] ≈∑i
αiri
q(z)
(z)
, , , , , , , ,1 2 ... K[ ]r
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 10 / 20
Distributed distributional code
Why DDC? It makes computation simple for neurons
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
Key computations involve expected values:
Message passing:q(z2) = Eq(z1) [p(z2|z1)]
Marginalization:
q(z2 ∈ (a, b)) = Eq(z1,z2) [1(a < z2 < b)]
Action evaluation:Q(a) = Eq(s) [R(s, a)]
How to compute E [h(z)]?
if h(z) ≈∑i
αiγi(z)⇒ E [h(z)] ≈∑i
αiri
q(z)
(z)
, , , , , , , ,1 2 ... K[ ]r
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 10 / 20
Distributed distributional code
Why DDC? It makes computation simple for neurons
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
Key computations involve expected values:
Message passing:q(z2) = Eq(z1) [p(z2|z1)]
Marginalization:
q(z2 ∈ (a, b)) = Eq(z1,z2) [1(a < z2 < b)]
Action evaluation:Q(a) = Eq(s) [R(s, a)]
How to compute E [h(z)]?
if h(z) ≈∑i
αiγi(z)⇒ E [h(z)] ≈∑i
αiri
q(z)
(z)
, , , , , , , ,1 2 ... K[ ]r
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 10 / 20
Distributed distributional code
Why DDC? It makes computation simple for neurons
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
Key computations involve expected values:
Message passing:q(z2) = Eq(z1) [p(z2|z1)]
Marginalization:
q(z2 ∈ (a, b)) = Eq(z1,z2) [1(a < z2 < b)]
Action evaluation:Q(a) = Eq(s) [R(s, a)]
How to compute E [h(z)]?
if h(z) ≈∑i
αiγi(z)⇒ E [h(z)] ≈∑i
αiri
q(z)
(z)
, , , , , , , ,1 2 ... K[ ]r
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 10 / 20
Distributed distributional code
Why DDC? It makes computation simple for neurons
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
Key computations involve expected values:
Message passing:q(z2) = Eq(z1) [p(z2|z1)]
Marginalization:
q(z2 ∈ (a, b)) = Eq(z1,z2) [1(a < z2 < b)]
Action evaluation:Q(a) = Eq(s) [R(s, a)]
How to compute E [h(z)]?
if h(z) ≈∑i
αiγi(z)⇒ E [h(z)] ≈∑i
αiri
q(z)h(z)
(z)
, , , , , , , ,1 2 ... K[ ]r
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 10 / 20
Distributed distributional code
Why DDC? It makes computation simple for neurons
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
Key computations involve expected values:
Message passing:q(z2) = Eq(z1) [p(z2|z1)]
Marginalization:
q(z2 ∈ (a, b)) = Eq(z1,z2) [1(a < z2 < b)]
Action evaluation:Q(a) = Eq(s) [R(s, a)]
How to compute E [h(z)]?
if h(z) ≈∑i
αiγi(z)
⇒ E [h(z)] ≈∑i
αiri
q(z)h(z)
(z)
, , , , , , , ,1 2 ... K[ ]r
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 10 / 20
Distributed distributional code
Why DDC? It makes computation simple for neurons
r := Eq(z) [γ1 (z) , γ2 (z) , ..., γK (z)]
Key computations involve expected values:
Message passing:q(z2) = Eq(z1) [p(z2|z1)]
Marginalization:
q(z2 ∈ (a, b)) = Eq(z1,z2) [1(a < z2 < b)]
Action evaluation:Q(a) = Eq(s) [R(s, a)]
How to compute E [h(z)]?
if h(z) ≈∑i
αiγi(z)⇒ E [h(z)] ≈∑i
αiri
q(z)h(z)
(z)
, , , , , , , ,1 2 ... K[ ]r
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 10 / 20
Distributed distributional code
Why DDC? It allows biologically plausible learning to infer
Previously, we saw marginal q(z)↔ Eq(z) [γ(z)]
For p(z,x) = p(z)p(x|z), p(z|x) is intractable
How to obtain approx. q(z|x)↔ Eq(z|x) [γ(z)]?
Clue: posterior mean...
Ep(z|x) [γ(z)] = arg minφ
Ep(z|x)�‖γ(z)− φ‖22
�
�Amortize� using φW (x) := Wσ(x)
W ∗ = arg minW
Ep(z,x)�‖γ(z)−Wσ(x)‖22
�
r(x) := W ∗σ(x) = Eq(z|x) [γ(z)]
Find W ∗ by the delta rule:
∆W ∝ (γ(z)− φW (x))σ(x)ᵀ, {z,x} ∼ p
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 11 / 20
Distributed distributional code
Why DDC? It allows biologically plausible learning to infer
Previously, we saw marginal q(z)↔ Eq(z) [γ(z)]
For p(z,x) = p(z)p(x|z), p(z|x) is intractable
How to obtain approx. q(z|x)↔ Eq(z|x) [γ(z)]?
Clue: posterior mean...
Ep(z|x) [γ(z)] = arg minφ
Ep(z|x)�‖γ(z)− φ‖22
�
�Amortize� using φW (x) := Wσ(x)
W ∗ = arg minW
Ep(z,x)�‖γ(z)−Wσ(x)‖22
�
r(x) := W ∗σ(x) = Eq(z|x) [γ(z)]
Find W ∗ by the delta rule:
∆W ∝ (γ(z)− φW (x))σ(x)ᵀ, {z,x} ∼ p
z
x
γ(z)
simulation
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 11 / 20
Distributed distributional code
Why DDC? It allows biologically plausible learning to infer
Previously, we saw marginal q(z)↔ Eq(z) [γ(z)]
For p(z,x) = p(z)p(x|z), p(z|x) is intractable
How to obtain approx. q(z|x)↔ Eq(z|x) [γ(z)]?
Clue: posterior mean...
Ep(z|x) [γ(z)] = arg minφ
Ep(z|x)�‖γ(z)− φ‖22
�
�Amortize� using φW (x) := Wσ(x)
W ∗ = arg minW
Ep(z,x)�‖γ(z)−Wσ(x)‖22
�
r(x) := W ∗σ(x) = Eq(z|x) [γ(z)]
Find W ∗ by the delta rule:
∆W ∝ (γ(z)− φW (x))σ(x)ᵀ, {z,x} ∼ p
z
x
γ(z)
simulation
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 11 / 20
Distributed distributional code
Why DDC? It allows biologically plausible learning to infer
Previously, we saw marginal q(z)↔ Eq(z) [γ(z)]
For p(z,x) = p(z)p(x|z), p(z|x) is intractable
How to obtain approx. q(z|x)↔ Eq(z|x) [γ(z)]?
Clue: posterior mean...
Ep(z|x) [γ(z)] = arg minφ
Ep(z|x)�‖γ(z)− φ‖22
�
�Amortize� using φW (x) := Wσ(x)
W ∗ = arg minW
Ep(z,x)�‖γ(z)−Wσ(x)‖22
�
r(x) := W ∗σ(x) = Eq(z|x) [γ(z)]
Find W ∗ by the delta rule:
∆W ∝ (γ(z)− φW (x))σ(x)ᵀ, {z,x} ∼ p
z
x
γ(z)
simulation
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 11 / 20
Distributed distributional code
Why DDC? It allows biologically plausible learning to infer
Previously, we saw marginal q(z)↔ Eq(z) [γ(z)]
For p(z,x) = p(z)p(x|z), p(z|x) is intractable
How to obtain approx. q(z|x)↔ Eq(z|x) [γ(z)]?
Clue: posterior mean...
Ep(z|x) [γ(z)] = arg minφ
Ep(z|x)�‖γ(z)− φ‖22
�
�Amortize� using φW (x) := Wσ(x)
W ∗ = arg minW
Ep(z,x)�‖γ(z)−Wσ(x)‖22
�
r(x) := W ∗σ(x) = Eq(z|x) [γ(z)]
Find W ∗ by the delta rule:
∆W ∝ (γ(z)− φW (x))σ(x)ᵀ, {z,x} ∼ p
z
x
γ(z)
simulation
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 11 / 20
Distributed distributional code
Why DDC? It allows biologically plausible learning to infer
Previously, we saw marginal q(z)↔ Eq(z) [γ(z)]
For p(z,x) = p(z)p(x|z), p(z|x) is intractable
How to obtain approx. q(z|x)↔ Eq(z|x) [γ(z)]?
Clue: posterior mean...
Ep(z|x) [γ(z)] = arg minφ
Ep(z|x)�‖γ(z)− φ‖22
�
�Amortize� using φW (x) := Wσ(x)
W ∗ = arg minW
Ep(z,x)�‖γ(z)−Wσ(x)‖22
�
r(x) := W ∗σ(x) = Eq(z|x) [γ(z)]
Find W ∗ by the delta rule:
∆W ∝ (γ(z)− φW (x))σ(x)ᵀ, {z,x} ∼ p
z
x
γ(z)
simulation
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 11 / 20
Distributed distributional code
Why DDC? It allows biologically plausible learning to infer
Previously, we saw marginal q(z)↔ Eq(z) [γ(z)]
For p(z,x) = p(z)p(x|z), p(z|x) is intractable
How to obtain approx. q(z|x)↔ Eq(z|x) [γ(z)]?
Clue: posterior mean...
Ep(z|x) [γ(z)] = arg minφ
Ep(z|x)�‖γ(z)− φ‖22
�
�Amortize� using φW (x) := Wσ(x)
W ∗ = arg minW
Ep(z,x)�‖γ(z)−Wσ(x)‖22
�
r(x) := W ∗σ(x) = Eq(z|x) [γ(z)]
Find W ∗ by the delta rule:
∆W ∝ (γ(z)− φW (x))σ(x)ᵀ, {z,x} ∼ p
z
x
γ(z)
simulation
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 11 / 20
Distributed distributional code
Why DDC? It allows biologically plausible learning to infer
Previously, we saw marginal q(z)↔ Eq(z) [γ(z)]
For p(z,x) = p(z)p(x|z), p(z|x) is intractable
How to obtain approx. q(z|x)↔ Eq(z|x) [γ(z)]?
Clue: posterior mean...
Ep(z|x) [γ(z)] = arg minφ
Ep(z|x)�‖γ(z)− φ‖22
�
�Amortize� using φW (x) := Wσ(x)
W ∗ = arg minW
Ep(z,x)�‖γ(z)−Wσ(x)‖22
�
r(x) := W ∗σ(x) = Eq(z|x) [γ(z)]
Find W ∗ by the delta rule:
∆W ∝ (γ(z)− φW (x))σ(x)ᵀ, {z,x} ∼ p
z
x
γ(z)
simulation
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 11 / 20
Distributed distributional code
Why DDC? It allows biologically plausible learning to infer
Previously, we saw marginal q(z)↔ Eq(z) [γ(z)]
For p(z,x) = p(z)p(x|z), p(z|x) is intractable
How to obtain approx. q(z|x)↔ Eq(z|x) [γ(z)]?
Clue: posterior mean...
Ep(z|x) [γ(z)] = arg minφ
Ep(z|x)�‖γ(z)− φ‖22
�
�Amortize� using φW (x) := Wσ(x)
W ∗ = arg minW
Ep(z,x)�‖γ(z)−Wσ(x)‖22
�
r(x) := W ∗σ(x) = Eq(z|x) [γ(z)]
Find W ∗ by the delta rule:
∆W ∝ (γ(z)− φW (x))σ(x)ᵀ, {z,x} ∼ p
γ(z)
σ(x)
∆Wz
x
r
W
γ(z)
simulationlearning
to infer
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 11 / 20
Distributed distributional code
DDC Summary
De�nition
DDC of q(z) associated tuning functions γ(z) is r := Eq(z) [γ(z)]
MaxEnt interpretation
rE[γ(z)]
↼−−−−−−−−−−−−−−−−⇁γ(z),MaxEnt
q(z) ∝ exp [θ · γ(z)]
Expectation approximationh(z) ≈ α · γ(z) =⇒ E [h(z)] ≈ α · r
Learning to infer given p(z, x)
r(x) = Eq(z|x) [γ(z)] = W ∗σ(x), ∆W ∝ (γ − φW )σᵀ, {z,x} ∼ p(z,x)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 12 / 20
Distributed distributional code
DDC Summary
De�nition
DDC of q(z) associated tuning functions γ(z) is r := Eq(z) [γ(z)]
MaxEnt interpretation
rE[γ(z)]
↼−−−−−−−−−−−−−−−−⇁γ(z),MaxEnt
q(z) ∝ exp [θ · γ(z)]
Expectation approximationh(z) ≈ α · γ(z) =⇒ E [h(z)] ≈ α · r
Learning to infer given p(z, x)
r(x) = Eq(z|x) [γ(z)] = W ∗σ(x), ∆W ∝ (γ − φW )σᵀ, {z,x} ∼ p(z,x)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 12 / 20
Distributed distributional code
DDC Summary
De�nition
DDC of q(z) associated tuning functions γ(z) is r := Eq(z) [γ(z)]
MaxEnt interpretation
rE[γ(z)]
↼−−−−−−−−−−−−−−−−⇁γ(z),MaxEnt
q(z) ∝ exp [θ · γ(z)]
Expectation approximationh(z) ≈ α · γ(z) =⇒ E [h(z)] ≈ α · r
Learning to infer given p(z, x)
r(x) = Eq(z|x) [γ(z)] = W ∗σ(x), ∆W ∝ (γ − φW )σᵀ, {z,x} ∼ p(z,x)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 12 / 20
Distributed distributional code
DDC Summary
De�nition
DDC of q(z) associated tuning functions γ(z) is r := Eq(z) [γ(z)]
MaxEnt interpretation
rE[γ(z)]
↼−−−−−−−−−−−−−−−−⇁γ(z),MaxEnt
q(z) ∝ exp [θ · γ(z)]
Expectation approximationh(z) ≈ α · γ(z) =⇒ E [h(z)] ≈ α · r
Learning to infer given p(z, x)
r(x) = Eq(z|x) [γ(z)] = W ∗σ(x), ∆W ∝ (γ − φW )σᵀ, {z,x} ∼ p(z,x)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 12 / 20
Distributed distributional code
DDC Summary
De�nition
DDC of q(z) associated tuning functions γ(z) is r := Eq(z) [γ(z)]
MaxEnt interpretation
rE[γ(z)]
↼−−−−−−−−−−−−−−−−⇁γ(z),MaxEnt
q(z) ∝ exp [θ · γ(z)]
Expectation approximationh(z) ≈ α · γ(z) =⇒ E [h(z)] ≈ α · r
Learning to infer given p(z, x)
r(x) = Eq(z|x) [γ(z)] = W ∗σ(x), ∆W ∝ (γ − φW )σᵀ, {z,x} ∼ p(z,x)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 12 / 20
Online recognition and postdiction
3. Online recognition and postdiction
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 13 / 20
Online recognition and postdiction
A generic dynamic internal model
We assume a generic internal model
zt = f(zt−1, ξ(z))
xt = g(zt, ξ(x))
Assumptions
Discrete-time
Markov property
Stationarity
zt−1 zt
xtxt−1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 14 / 20
Online recognition and postdiction
A generic dynamic internal model
We assume a generic internal model
zt = f(zt−1, ξ(z))
xt = g(zt, ξ(x))
Assumptions
Discrete-time
Markov property
Stationarity
zt−1 zt
xtxt−1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 14 / 20
Online recognition and postdiction
Representing and computing beliefs of the whole history
Online recognition (�ltering): maintain q(zt|x1:t)
De�ne temporally extended encoding function ψt := ψ (z1:t) for DDC
A plausible ψt
ψ1 := γ (z1)
ψt := Uψt−1 + γ (zt)
Update beliefs about the past: compute
rt = Eq(z1:t|x1:t) [ψt]
from rt−1 and xtPostiction: readout statistics
h(zt−τ ) ≈ α ·ψ(z1:t) =⇒ Eq(zt−τ ) [h(zt−τ )] ≈ α · rt
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 15 / 20
Online recognition and postdiction
Representing and computing beliefs of the whole history
Online recognition (�ltering): maintain q(z1:t|x1:t) to allow postdiction
De�ne temporally extended encoding function ψt := ψ (z1:t) for DDC
A plausible ψt
ψ1 := γ (z1)
ψt := Uψt−1 + γ (zt)
Update beliefs about the past: compute
rt = Eq(z1:t|x1:t) [ψt]
from rt−1 and xtPostiction: readout statistics
h(zt−τ ) ≈ α ·ψ(z1:t) =⇒ Eq(zt−τ ) [h(zt−τ )] ≈ α · rt
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 15 / 20
Online recognition and postdiction
Representing and computing beliefs of the whole history
Online recognition (�ltering): maintain q(z1:t|x1:t) to allow postdiction
De�ne temporally extended encoding function ψt := ψ (z1:t) for DDC
A plausible ψt
ψ1 := γ (z1)
ψt := Uψt−1 + γ (zt)
Update beliefs about the past: compute
rt = Eq(z1:t|x1:t) [ψt]
from rt−1 and xtPostiction: readout statistics
h(zt−τ ) ≈ α ·ψ(z1:t) =⇒ Eq(zt−τ ) [h(zt−τ )] ≈ α · rt
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 15 / 20
Online recognition and postdiction
Representing and computing beliefs of the whole history
Online recognition (�ltering): maintain q(z1:t|x1:t) to allow postdiction
De�ne temporally extended encoding function ψt := ψ (z1:t) for DDC
A plausible ψt
ψ1 := γ (z1)
ψt := Uψt−1 + γ (zt)
Update beliefs about the past: compute
rt = Eq(z1:t|x1:t) [ψt]
from rt−1 and xtPostiction: readout statistics
h(zt−τ ) ≈ α ·ψ(z1:t) =⇒ Eq(zt−τ ) [h(zt−τ )] ≈ α · rt
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 15 / 20
Online recognition and postdiction
Representing and computing beliefs of the whole history
Online recognition (�ltering): maintain q(z1:t|x1:t) to allow postdiction
De�ne temporally extended encoding function ψt := ψ (z1:t) for DDC
A plausible ψt
ψ1 := γ (z1)
ψt := Uψt−1 + γ (zt)
Update beliefs about the past: compute
rt = Eq(z1:t|x1:t) [ψt]
from rt−1 and xt
Postiction: readout statistics
h(zt−τ ) ≈ α ·ψ(z1:t) =⇒ Eq(zt−τ ) [h(zt−τ )] ≈ α · rt
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 15 / 20
Online recognition and postdiction
Representing and computing beliefs of the whole history
Online recognition (�ltering): maintain q(z1:t|x1:t) to allow postdiction
De�ne temporally extended encoding function ψt := ψ (z1:t) for DDC
A plausible ψt
ψ1 := γ (z1)
ψt := Uψt−1 + γ (zt)
Update beliefs about the past: compute
rt = Eq(z1:t|x1:t) [ψt]
from rt−1 and xtPostiction: readout statistics
h(zt−τ ) ≈ α ·ψ(z1:t) =⇒ Eq(zt−τ ) [h(zt−τ )] ≈ α · rt
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 15 / 20
Online recognition and postdiction
Learning to postdict online
Want rt(x1:t) = Eqt [ψt (z1:t)]
Recall for p(z,x)r := φW ∗(x)
Likewise, for SSM p(z1:t,x1:t)
rt := φW t(x1:t)
Learning by the delta rule
∆W t ← (ψt − φt)(rt−1 ⊗ σt)ᵀ
{ψt,xt, rt−1} ∼ p(z1:t,x1:t), {hW i}t−1i=1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 16 / 20
Online recognition and postdiction
Learning to postdict online
Want rt(x1:t) = Eqt [ψt (z1:t)]
Recall for p(z,x)r := φW ∗(x)
Likewise, for SSM p(z1:t,x1:t)
rt := φW t(x1:t)
Learning by the delta rule
∆W t ← (ψt − φt)(rt−1 ⊗ σt)ᵀ
{ψt,xt, rt−1} ∼ p(z1:t,x1:t), {hW i}t−1i=1
γ(z)
σ(x)
∆Wz
x
r
W
γ(z)
simulationlearning
to infer
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 16 / 20
Online recognition and postdiction
Learning to postdict online
Want rt(x1:t) = Eqt [ψt (z1:t)]
Recall for p(z,x)r := φW ∗(x)
Likewise, for SSM p(z1:t,x1:t)
rt := φW t(x1:t)
Learning by the delta rule
∆W t ← (ψt − φt)(rt−1 ⊗ σt)ᵀ
{ψt,xt, rt−1} ∼ p(z1:t,x1:t), {hW i}t−1i=1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 16 / 20
Online recognition and postdiction
Learning to postdict online
Want rt(x1:t) = Eqt [ψt (z1:t)]
Recall for p(z,x)r := φW ∗(x)
Likewise, for SSM p(z1:t,x1:t)
rt := φW t(rt−1,xt)
Learning by the delta rule
∆W t ← (ψt − φt)(rt−1 ⊗ σt)ᵀ
{ψt,xt, rt−1} ∼ p(z1:t,x1:t), {hW i}t−1i=1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 16 / 20
Online recognition and postdiction
Learning to postdict online
Want rt(x1:t) = Eqt [ψt (z1:t)]
Recall for p(z,x)r := φW ∗(x)
Likewise, for SSM p(z1:t,x1:t)
rt := φW t(rt−1,xt)
= W t · (rt−1 ⊗ σ(xt))
Learning by the delta rule
∆W t ← (ψt − φt)(rt−1 ⊗ σt)ᵀ
{ψt,xt, rt−1} ∼ p(z1:t,x1:t), {hW i}t−1i=1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 16 / 20
Online recognition and postdiction
Learning to postdict online
Want rt(x1:t) = Eqt [ψt (z1:t)]
Recall for p(z,x)r := φW ∗(x)
Likewise, for SSM p(z1:t,x1:t)
rt := φW t(rt−1,xt)
= W t · (rt−1 ⊗ σ(xt))
Learning by the delta rule
∆W t ← (ψt − φt)(rt−1 ⊗ σt)ᵀ
{ψt,xt, rt−1} ∼ p(z1:t,x1:t), {hW i}t−1i=1
ψt
σ(xt)
rt−1
∆W tzt−1 zt
xt
rt
W t
rt−1
W t+1
ψtψt−1
Simulation Learning to infer
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 16 / 20
Online recognition and postdiction
Learning to postdict online
Want rt(x1:t) = Eqt [ψt (z1:t)]
Recall for p(z,x)r := φW ∗(x)
Likewise, for SSM p(z1:t,x1:t)
rt := φW t(rt−1,xt)
= W t · (rt−1 ⊗ σ(xt))
Learning by the delta rule
∆W t ← (ψt − φt)(rt−1 ⊗ σt)ᵀ
{ψt,xt, rt−1} ∼ p(z1:t,x1:t), {hW i}t−1i=1
ψt
σ(xt)
rt−1
∆W tzt−1 zt
xt
rt
W t
rt−1
W t+1
ψtψt−1
Simulation Learning to infer
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 16 / 20
Online recognition and postdiction
Learning to postdict online
Want rt(x1:t) = Eqt [ψt (z1:t)]
Recall for p(z,x)r := φW ∗(x)
Likewise, for SSM p(z1:t,x1:t)
rt := φW t(rt−1,xt)
= W t · (rt−1 ⊗ σ(xt))
Learning by the delta rule
∆W t ← (ψt − φt)(rt−1 ⊗ σt)ᵀ
{ψt,xt, rt−1} ∼ p(z1:t,x1:t), {hW i}t−1i=1
ψt
σ(xt)
rt−1
∆W tzt−1 zt
xt
rt
W t
rt−1
W t+1
ψtψt−1
Simulation Learning to infer
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 16 / 20
Testing DDC �ltering on simulated experiments
4. Testing DDC �ltering on simulated experiments
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 17 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
0
5st
ate
true states of tone and noisetone noise
0 50time
freq
spectrogram observations
0 4
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Auditory continuity illusion
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
stat
e tone mask
freq
1 2 3 4 5 6 7 8 9 10stimulus time (t− τ)
10987654321
perc
eptio
n tim
e (t)
0 1
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 18 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1
0
1
loca
tion
true tap locations
0 20 40 60 80# taps
30
1
neur
on
spiking observations
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
posit
ion
true tap
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Cutaneous rabbit
1 6 11 15# taps
0
1
2
trueperceived
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 19 / 20
Testing DDC �ltering on simulated experiments
Summary of DDC �ltering and Questions
Representation: DDC rt := Eq(z1:t|x1:t) [ψt(z1:t)]
Computation: bilinear rt := W ∗ · (rt−1 ⊗ σ(xt))
, or linear Eq [h(zt−τ )] ≈ αᵀrt
Learning to infer: delta rule ∆W ∝ (ψt − φW )(rt−1 ⊗ σt), similar for readout αQuestions:
Is the encoding function ψt = Uψt−1 + γ(zt) the best?
Does the brain encode the joint q(z1:t|x1:t)?
Any theoretical argument for using the bilinear rule?
Is there a way to also adapt U in a plausible way?
Can we encode the internal model by DDC? (talk to Eszter Vértes)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 20 / 20
Testing DDC �ltering on simulated experiments
Summary of DDC �ltering and Questions
Representation: DDC rt := Eq(z1:t|x1:t) [ψt(z1:t)]Computation: bilinear rt := W ∗ · (rt−1 ⊗ σ(xt)) , or linear Eq [h(zt−τ )] ≈ αᵀrt
Learning to infer: delta rule ∆W ∝ (ψt − φW )(rt−1 ⊗ σt), similar for readout αQuestions:
Is the encoding function ψt = Uψt−1 + γ(zt) the best?
Does the brain encode the joint q(z1:t|x1:t)?
Any theoretical argument for using the bilinear rule?
Is there a way to also adapt U in a plausible way?
Can we encode the internal model by DDC? (talk to Eszter Vértes)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 20 / 20
Testing DDC �ltering on simulated experiments
Summary of DDC �ltering and Questions
Representation: DDC rt := Eq(z1:t|x1:t) [ψt(z1:t)]Computation: bilinear rt := W ∗ · (rt−1 ⊗ σ(xt)) , or linear Eq [h(zt−τ )] ≈ αᵀrtLearning to infer: delta rule ∆W ∝ (ψt − φW )(rt−1 ⊗ σt), similar for readout α
Questions:
Is the encoding function ψt = Uψt−1 + γ(zt) the best?
Does the brain encode the joint q(z1:t|x1:t)?
Any theoretical argument for using the bilinear rule?
Is there a way to also adapt U in a plausible way?
Can we encode the internal model by DDC? (talk to Eszter Vértes)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 20 / 20
Testing DDC �ltering on simulated experiments
Summary of DDC �ltering and Questions
Representation: DDC rt := Eq(z1:t|x1:t) [ψt(z1:t)]Computation: bilinear rt := W ∗ · (rt−1 ⊗ σ(xt)) , or linear Eq [h(zt−τ )] ≈ αᵀrtLearning to infer: delta rule ∆W ∝ (ψt − φW )(rt−1 ⊗ σt), similar for readout αQuestions:
Is the encoding function ψt = Uψt−1 + γ(zt) the best?
Does the brain encode the joint q(z1:t|x1:t)?
Any theoretical argument for using the bilinear rule?
Is there a way to also adapt U in a plausible way?
Can we encode the internal model by DDC? (talk to Eszter Vértes)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 20 / 20
Testing DDC �ltering on simulated experiments
Summary of DDC �ltering and Questions
Representation: DDC rt := Eq(z1:t|x1:t) [ψt(z1:t)]Computation: bilinear rt := W ∗ · (rt−1 ⊗ σ(xt)) , or linear Eq [h(zt−τ )] ≈ αᵀrtLearning to infer: delta rule ∆W ∝ (ψt − φW )(rt−1 ⊗ σt), similar for readout αQuestions:
Is the encoding function ψt = Uψt−1 + γ(zt) the best?
Does the brain encode the joint q(z1:t|x1:t)?
Any theoretical argument for using the bilinear rule?
Is there a way to also adapt U in a plausible way?
Can we encode the internal model by DDC? (talk to Eszter Vértes)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 20 / 20
Testing DDC �ltering on simulated experiments
Summary of DDC �ltering and Questions
Representation: DDC rt := Eq(z1:t|x1:t) [ψt(z1:t)]Computation: bilinear rt := W ∗ · (rt−1 ⊗ σ(xt)) , or linear Eq [h(zt−τ )] ≈ αᵀrtLearning to infer: delta rule ∆W ∝ (ψt − φW )(rt−1 ⊗ σt), similar for readout αQuestions:
Is the encoding function ψt = Uψt−1 + γ(zt) the best?
Does the brain encode the joint q(z1:t|x1:t)?
Any theoretical argument for using the bilinear rule?
Is there a way to also adapt U in a plausible way?
Can we encode the internal model by DDC? (talk to Eszter Vértes)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 20 / 20
Testing DDC �ltering on simulated experiments
Summary of DDC �ltering and Questions
Representation: DDC rt := Eq(z1:t|x1:t) [ψt(z1:t)]Computation: bilinear rt := W ∗ · (rt−1 ⊗ σ(xt)) , or linear Eq [h(zt−τ )] ≈ αᵀrtLearning to infer: delta rule ∆W ∝ (ψt − φW )(rt−1 ⊗ σt), similar for readout αQuestions:
Is the encoding function ψt = Uψt−1 + γ(zt) the best?
Does the brain encode the joint q(z1:t|x1:t)?
Any theoretical argument for using the bilinear rule?
Is there a way to also adapt U in a plausible way?
Can we encode the internal model by DDC? (talk to Eszter Vértes)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 20 / 20
Testing DDC �ltering on simulated experiments
Summary of DDC �ltering and Questions
Representation: DDC rt := Eq(z1:t|x1:t) [ψt(z1:t)]Computation: bilinear rt := W ∗ · (rt−1 ⊗ σ(xt)) , or linear Eq [h(zt−τ )] ≈ αᵀrtLearning to infer: delta rule ∆W ∝ (ψt − φW )(rt−1 ⊗ σt), similar for readout αQuestions:
Is the encoding function ψt = Uψt−1 + γ(zt) the best?
Does the brain encode the joint q(z1:t|x1:t)?
Any theoretical argument for using the bilinear rule?
Is there a way to also adapt U in a plausible way?
Can we encode the internal model by DDC? (talk to Eszter Vértes)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 20 / 20
Testing DDC �ltering on simulated experiments
Summary of DDC �ltering and Questions
Representation: DDC rt := Eq(z1:t|x1:t) [ψt(z1:t)]Computation: bilinear rt := W ∗ · (rt−1 ⊗ σ(xt)) , or linear Eq [h(zt−τ )] ≈ αᵀrtLearning to infer: delta rule ∆W ∝ (ψt − φW )(rt−1 ⊗ σt), similar for readout αQuestions:
Is the encoding function ψt = Uψt−1 + γ(zt) the best?
Does the brain encode the joint q(z1:t|x1:t)?
Any theoretical argument for using the bilinear rule?
Is there a way to also adapt U in a plausible way?
Can we encode the internal model by DDC? (talk to Eszter Vértes)
Kevin Li, Maneesh Sahani (Gatsby Unit, UCL) Model of recognition and postdiction January 6, 2020 20 / 20
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