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Hierarchical dynamic models and variational Bayesian inference

Hierarchical dynamic models describe how hidden states in the world generate sensory states. In Friston's treatment the inference rests on variational energy and variational action, usually under a Laplace assumption.12 The same scheme has since been carried from neuroscience into simulations of self-assembly and morphogenesis.7

Earliest held
1902, Bose, J. C.
Most discussed in
Hierarchical models in the brain, 2008
In the library
48 passages in 7 works
Rewritten
2026-10-12
18501875190019251950197520002025Friston K, 20081Friston K, 20082Friston K, Ao P, 20123Friston K, Ao P, 20124Sengupta B, Stemmler MB, Friston KJ, 20135Sengupta B, Stemmler MB, Friston KJ, 20136Friston K, Levin M, Sengupta B, Pezzulo…, 20157Pio-Lopez L, Kuchling F, Tung A…, 20228
01

Variational energy and action

Friston (2008) set out the conditions for inference in static and dynamic systems. In static systems the mode of the conditional density maximises variational energy. In dynamic systems the path of the mode must also be the mode of the path.2 He showed that the stationary solution in a moving frame of reference maximises variational action, and he compared this to Hamilton's principle of stationary action.2 Small perturbations to the path leave the variational energy unchanged. The path has the greatest variational action, the path-integral of variational energy, of all possible paths.2

02

The Laplace assumption

Friston (2008) noted that the Laplace assumption makes the conditional covariance a simple function of the modes.1 Sengupta, Stemmler and Friston (2013) gave the reasoning in terms of entropy. Minimising variational free energy entails maximising the entropy of the conditional density, which they likened to keeping one's options open.5 A Gaussian has the maximum entropy of any form specified by two moments. They called assuming a Gaussian form the Laplace assumption, and it lets the entropy be written in terms of the expectation.6 Pio-Lopez and colleagues (2022) state the same consequence: the variational density becomes a normal distribution.8

03

Generative models and action

Friston and Ao (2012) wrote the generative model as nonlinear functions of hidden states that generate sensory states, with random fluctuations acting as sensory noise. Hidden states are abstract quantities the agent uses to explain or predict sensations.3 Gaussian assumptions about the fluctuations give a probabilistic generative model of sensory states. They pointed to generalised filtering in hierarchical dynamic models for the full description. In their formulation the generative model does not include action, because action is not a hidden state.4

04

Use in morphogenesis

Friston, Levin, Sengupta and Pezzulo (2015) applied this machinery to pattern regulation. Their scheme used the Laplace assumption and was integrated numerically with a routine from the SPM software. They described it as a generalized Bayesian filter, in which the internal states become the expected values of the hidden states.7 They offered their simulations of self-assembly as a proof of principle. They argued the scheme is a sufficient explanation for simple self-assembly and bears on the role of genetics and epigenetics in morphogenesis.7 Pio-Lopez and colleagues (2022) again rely on the Laplace form of the variational density.8

SourcesEach quotation was checked word for word against the passage it opens.
  1. The advantage of the Laplace assumption is that the conditional covariance is a simple function of the modes.Friston K, 2008 · Hierarchical models in the brain · open at passage 30
  2. Intuitively, this means tiny perturbations to its path do not change the variational energy and it has the greatest variational action (i.e., path-integral of variational energy) of all possible paths.Friston K, 2008 · Hierarchical models in the brain · open at passage 34
  3. Hidden states are abstract quantities (like the motion of an object in the field of view) that the agent uses to explain or predict sensations.Friston K, Ao P, 2012 · Free energy, value, and attractors · open at passage 41
  4. The generative model does not include action, because action is not a hidden state.Friston K, Ao P, 2012 · Free energy, value, and attractors · open at passage 41
  5. minimising variational free energy entails maximising the entropy of the conditional densitySengupta B, Stemmler MB, Friston KJ, 2013 · Information and efficiency in the nervous system--a synthesis · open at passage 49
  6. Assuming a Gaussian form is known as the Laplace assumption and enables us to express the entropy of the conditional density in terms of its first moment or expectation.Sengupta B, Stemmler MB, Friston KJ, 2013 · Information and efficiency in the nervous system--a synthesis · open at passage 49
  7. The resulting scheme can be regarded as a generalized Bayesian filter, in which the internal states become the expected values of the external or hidden statesFriston K, Levin M, Sengupta B, Pezzulo G, 2015 · Knowing one's place: a free-energy approach to pattern regulation · open at passage 19
  8. Under a Laplace assumption, the variational density becomes a normal distribution:Pio-Lopez L, Kuchling F, Tung A, Pezzulo G, Levin M, 2022 · Active inference, morphogenesis, and computational psychiatry · open at passage 60
Linked ideas
Hierarchical dynamic models and variational energy supply the mathematical machinery in which the free-energy principle and active inference are expressed.
Both treat living systems as performing Bayesian inference, here with generative models over hidden states, there with biochemical circuits.
Predictive processing uses the same idea of internal states encoding expected values of hidden causes of sensation.
The separation of internal states from hidden external states in the generative model is usually framed through a Markov blanket.
The 2015 self-assembly simulations treat pattern regulation as inference, which bears on how a target form is specified and reached.
Precisions in these models set the weight of prediction errors at each level of a hierarchy.
Where it is discussedPassages matching hierarchical dynamic model, HDM, variational energy, variational action
2008Hierarchical models in the brain · Friston K39
1902Response in the Living and Non-Living · Bose, J. C.3
1907Comparative Electro-Physiology: A Physico-Physiological Study · Bose, J. C.2
2013Information and efficiency in the nervous system--a synthesis · Sengupta B, Stemmler MB, Friston KJ1
2022Active inference, morphogenesis, and computational psychiatry · Pio-Lopez L, Kuchling F, Tung A…1
2015Knowing one's place: a free-energy approach to pattern regulation · Friston K, Levin M, Sengupta B, Pezzulo…1
2012Free energy, value, and attractors · Friston K, Ao P1