the agent’s memory. \[logit\_mean\]](logit_vs_mean.eps){width="8cm"}

In Figure \[logit\_mean\] we see the results of this method, applied to the memory of a single randomly chosen agent from the simulation in Figure \[hopf\]. Also shown are the results of the primitive estimation method used by our agents. The primitive method appears considerably more effective at accurately predicting the current weights. However, a subtle phase shift may be perceived in the agent’s predictions relative to the true values. In order to investigate this shift, and to compare the two methods in greater detail we construct a fictitious sequence of Bernoulli random variables $\{X_k\}_{k=1}^{m}$ with success probability, $p(k)$, which changes over the course of the sequence: $p(k)=p_0 + (p_1-p_0)k/m$. We then use our two methods to find an estimate $\hat{p}_1$ of the value of $p_1$, for a very large number of example sequences, allowing us to estimate the distribution of $\hat{p}_1$.

![ Given a sequence $\{X_k\}_{k=1}^{m}$ where $m=100$, of Bernoulli random variables with changing success probability $p(k)=p_0 + (p_1-p_0)k/m$ where $p_0=0.2, p_1=0.25$. Black dots show the estimated probability density function of the logistic regression estimator $\hat{p}_1$, having mean 0.256 and root mean squared error $0.07$. Open circles show the estimated probability distribution of the number 