of successes in the sequence, serving as an alternative estimator for $p_1$, having mean 0.225 and root mean squared error 0.07. \[logit\_dist\_100\]](logit_vs_mean_dists.eps){width="8cm"}

![ Given a sequence $\{X_k\}_{k=1}^{m}$ where $m=20$, of Bernoulli random variables with changing success probability $p(k)=p_0 + (p_1-p_0)k/m$ where $p_0=0.2, p_1=0.3$. Light gray histogram gives estimated probability distribution of the logistic regression estimator $\hat{p}_1$, having mean 0.33 and root mean squared error $0.2$. Dark gray histogram gives estimated probability distribution of the number of successes in the sequence, having mean 0.25 and root mean squared error $0.1$. \[logit\_dist\_20\]](logit_vs_mean_m20_dists.eps){width="8cm"}

In Figures \[logit\_dist\_100\] and \[logit\_dist\_20\] we have constructed distributions for $\hat{p}_1$ in cases $m=100$ and $m=20$. It is clear that logistic regression produces estimates with much higher variance, but with smaller bias. Whilst the primitive method is the more efficient estimator, in the sense that the mean squared error in its predictions is smaller, its bias creates a systematic delay in its predictions. For larger memory values this bias will reduce, because the period of oscillation grows faster than the memory length, so the weights change by a smaller amount over the course of each agent’s memory.

Mixed memory and population dynamics
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![Populations of $m=10$ (open circles) and $m=100$ 