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I have seen two $\chi^2$ test across the internet.

  1. [Common wiki Chi-suqare test] To check whether a prediction is a good fit or not (a classical null test):

    • Compute $\sum_{i=1}^{N} (x_i - \textbf{pred}_i)^2/\textbf{pred}_i $
  2. [Mackay 1991: Bayesian interpolant.] Consider $N$ Gaussian variable, and the corresponding mean $\mu_i$ and s.t.d. as $\sigma_i$.

    • The expected of $\chi^2$ misfit between the true interpolant and data is $N$.
    • Then the model parameters should be adjusted to match $\chi^2 = N$.

Question:

I have only seen the second one in Mackay's paper, in the original paper of Mackay 1991: Bayesian interpolant, it is said orthodox statistician would like any model parameter to match $\chi^2 = N$.

It seems to be a nice result but when I searched google, I can only find the first one. Does anyone see the second chi-square criterion before?

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  • $\begingroup$ The chi-square occurs in still more contexts than those two. There's no paper by Mackay called "Bayesian interpolant" that I can find. Do you mean MacKay, David J. C.. “Bayesian Interpolation.” Neural Computation 4(3) (1992): p415-447. ? $\endgroup$
    – Glen_b
    Jul 14, 2018 at 22:21
  • $\begingroup$ @Glen_b Yes! That is the paper! Mackay is a big guy in practical Bayesian. $\endgroup$ Jul 15, 2018 at 3:03

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Yes, this is pretty standard when the variance is specified (and the idea is widely used in sciences where suitable values of $\sigma$ are assumed to be known). However it's not as common as the chi-squared tests of goodness of fit and of independence / homogeneity of proportion.

(Setting the misfit value to $N$ is not something I've seen done before though; it would be more common to minimize it and adjust the DF, but I might be missing information about what he's trying to do there.)

If you have $X_1,X_2,\ldots,X_n$ independent and $\sim N(\mu_i,\sigma^2_i)$ then $Z_i=\frac{X_i-\mu_i}{\sigma_i}$, $i=1,2,\ldots,n$ are i.i.d standard normal, in which case $\sum_i Z_i^2$ is distributed as $\chi^2_n$.

Chi-squared tests occur in numerous other contexts; for an important example, see Wilks' theorem, which gives an asymptotic chi-square approximation for $-2\log \Lambda$ where $\Lambda$ is the likelihood ratio statistic.

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  • $\begingroup$ Thanks Glen_b. I had the similar feeling that chi-squared test with known mean and known std would definitely lead to a nice chi^2 distribution. But I would say chi-squared-test-of-goodness-of-fit is more practical to use. $\endgroup$ Jul 15, 2018 at 3:05
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    $\begingroup$ The chi-square goodness of fit test is actually a slightly modified version of the same thing. For example you imagine that the X's are Poisson-distributed counts then $\sigma_i^2=\mu_i$. If the expected counts are large enough to apply the normal approximation to the Poisson, then $Z_i=\frac{X_i-\mu_i}{\sqrt{\mu_i}}$ are approximately standard normal, so $\sum_i\frac{(X_i-\mu_i)^2}{\mu_i}$ will be approximately chi-squared. If you then condition on the marginal total you end up with the same statistic but lose a degree of freedom. So it's pretty much the same idea. $\endgroup$
    – Glen_b
    Jul 15, 2018 at 3:29

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