In the maximum entropy inverse reinforcement learning paper, Ziebart et al. show that the state visitation frequency $\rho(s)$ of a state $s$ can be computed as $$ \rho_{\pi}(s) = \sum_{t}^{T} P(s_t=s|\pi), $$ which is the sum of the probability that the state being visited at each time step.

I just don't understand why is it the sum? From my perspective, a frequency should be the less than one, so that it should be the average value $$ \rho_{\pi}(s) = \frac{1}{T}\sum_{t}^{T} P(s_t=s|\pi). $$

  • $\begingroup$ My feeling is that they define the first equation and will then normalise it to make it a state distribution. $\endgroup$
    – David
    Apr 26, 2021 at 8:34
  • 1
    $\begingroup$ It should be the average and this is rarely mentioned by people except for a IRL summer camp at UCB. You can check this GithubIssue for details. $\endgroup$
    – skypitcher
    Apr 28, 2021 at 8:08

1 Answer 1


The equation you show does not appear in Ziebart et al (2008). They do provide a description of the computation in Algorithm 1.

It is the visitation frequency and it is not a probability distribution, so it does not need to be averaged.

If you look at Equation 2. in Arora & Doshi (2020), you a formulation that describes the Algorithm 1 quite well:

$\phi^\pi(s) = \phi^0(s) + \sum_{s'\in\mathcal{S}}P(s,\pi(s),s')\phi^\pi(s')$.

I am not very satisfied with this formulation, because, in my opinion, there should also be a summation over $a\in \mathcal{A}$, like $\eta(s)$ the expected number of visits, in Equation 9.2 in Sutton & Barto (2020).:


To summarize, in all three descriptions, you just calculate how often a state is visited by policy $\pi$.


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