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I'm reading sequential estimation section 2.3.5 PRML where Bishop introduces Robbins-Monro algorithm to calculate the root of $f(\theta) = E z | \theta = \int zp(z|\theta) dz, (z, \theta) \sim p(z, \theta)$.

The Robbins-Monro procedure then defines a sequence of successive estimates of the root $\theta^\star$ given by $$θ^{(N)} = θ^{(N−1)} + a_{N−1}z(θ^{(N−1)}) \tag{2.129}$$ where $z(θ^{(N)})$ is an observed value of $z$ when $θ$ takes the value $θ^{(N)}$. The coefficients $\{a_N\}$ represent a sequence of positive numbers that satisfy the conditions. (I skip the conditios since it does not help)

However, after some deduction and oberving the asypmtotic property of the equation of log likelihood (Equation 2.134), Bishop gives the procudure to estimate the parameters of normal distribution,

We can therefore apply the Robbins-Monro procedure, which now takes the form $$ θ^{(N)} = θ^{(N−1)} + a_{N−1}\frac{\partial}{\partial \theta^{(N-1)}}\ln p(x_N|\theta^{(N-1)}) \tag{2.135} $$ where $p(x|theta)$ is the normal density.

I'm confused that why it is $x_N$ but $x_{N-1}$. Does it be a typo or I didnot catch the idea of Robbins-Monro algorithm?

UPDATE
Another confusion arises that why the equation(2.129) does not integrate the the new observed data $x_N$ since I think the algorithm is similar to the gradient descent mothod.

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  • $\begingroup$ The text I read can be find here, page 96, microsoft.com/en-us/research/uploads/prod/2006/01/… $\endgroup$
    – Chia
    Commented May 11, 2023 at 13:00
  • $\begingroup$ Do you plan on updating the parameter before or after making an observation? $\endgroup$
    – whuber ♦
    Commented May 11, 2023 at 14:40
  • $\begingroup$ @whuber, PRML assumes that we have observed $N-1$ data say $x_1, ..., x_{N-1}$. I guess that we need to update after observing a new data $x_{N}$ so that I think the second entry of the update equation should be related to $x_{N}$ i.e. $\theta^{(N)} = \theta^{(N-1)} + a_{N-1}\times \text{something related to}\ x_N$. $\endgroup$
    – Chia
    Commented May 12, 2023 at 2:32
  • $\begingroup$ So I'm confused that whether there is any typo in equation(2.129). I would appreciate it if you can help. $\endgroup$
    – Chia
    Commented May 12, 2023 at 2:39
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    $\begingroup$ Although the change from "$z$" to "$x$" from (2.129) to (2.135) could use some explanation, I see no inherent inconsistency, because wouldn't $z(\theta^{(N-1)})$ be the "observed value of $z$ when $\theta$ takes the value $\theta^{(N-1)},$" which is precisely the data value observed after time $N-1,$ indicating it is $x_N$? $\endgroup$
    – whuber ♦
    Commented May 12, 2023 at 13:45

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