Seven scientists

In this post, I discuss one of my favorite problems from MacKay’s fantastic book on information theory . As the title suggests, the problem concerns seven scientists, and it illustrates the weakness of maximum likelihood estimation.

Maximum likelihood

The problem comes in two parts. In the first part, we use the maximum likelihood heuristic to solve the following problem:

N datapoints \{x_n\} are drawn from N distributions, all of which are Gaussian with mean \mu, but with different unknown standard deviations \sigma_n. What are the maximum likelihood parameters \mu, \{\sigma_n\}, given the data?

 

For example, seven scientists (A, B, C, D, E, F, G) with widely-differing experimental skills measure \mu. You expect some of them to do accurate work (i.e. to have small \sigma_n), and some of them to turn in wildly inaccurate answers (i.e. to have enormous \sigma_n). […] What is \mu and how reliable is each scientist?

 

MacKay, exercise 22.15.

In the book, the seven results by the seven scientists A to G are: -27.02, 3.57, 8.191, 9.898, 9.603, 9.945, and 10.056, respectively, as shown in this Figure:

FIGURE 1: Number line indicating the results that the seven scientists A to G obtained in their individual experiments. The true value seems to be around 10, assuming that the measurements of scientists A and B are wildly inaccurate.

To find the maximum likelihood values, we find the fixed point of the following rule:

\sigma_i \leftarrow \left|x_i - \frac{1}{N}\,\sum_{k = 1}^N \frac{x_k}{\sigma_k^2}\right|\;,

resulting in \mu = 0.1.

FIGURE 2: Results of the maximum likelihood estimate.

The dashed red line in Figure 2 indicates the maximum likelihood estimate of \mu, and error bars indicate the estimated standard deviation of each scientist.

Bayesian approach

The second part of the problem assesses the same situation with the full power of Bayesian statistics:

Give a Bayesian solution to exercise 22.15, where seven scientists of
varying capabilities have measured \mu with personal noise levels \sigma_n, and we are interested in inferring \mu. Let the prior on each \sigma_n be a broad prior, for example a gamma distribution with parameters (s, c)= (10, 0.1). Find the posterior distribution of \mu. Plot it, and explore its properties for a variety of data sets such as the one given, and the data set \{x_n\} = \{13.01, 7.39\}.

 

MacKay, exercise 24.3.

So, we assume that the \sigma_n come from a gamma distribution, i.e.

P(\sigma) = \frac{1}{\Gamma(c)\,s}\,\biggl(\frac{\sigma}{s}\biggr)^{c - 1}\,\exp\biggl(-\frac{\sigma}{s}\biggr)\;,

where \Gamma is the gamma function. Furthermore, if we’d know \mu and \sigma, then the probability to measure x is given by a Gaussian distribution

P(x|\mu, \sigma) = \frac{1}{\sqrt{2\,\pi}}\,\exp\biggl(-\frac{(x - \mu)^2}{2\,\sigma^2}\biggr)\;,

and we want to know P(\mu | \{x_n\}).

Estimating noise levels

As a first step, we seek the nth scientist’s noise level P(\sigma_n | \mu, x_n), which is given by Bayes’ theorem as

P(\sigma_n | \mu, x_n) = \frac{P(x_n | \mu, \sigma_n)\,P(\sigma_n)}{P(x_n | \mu)}\;,

where we’ve used that P(\sigma_n | \mu) = P(\sigma_n).

We already know the numerator of the right hand side. So we only need to find the normalizing constant P(x_n | \mu) by marginalizing P(x_n|\mu, \sigma_n) over \sigma_n, i.e. we integrate P(x|\mu, \sigma)\,P(\sigma) over \sigma from 0 to infinity:

P(x_n | \mu) = \int_0^\infty\;P(x_n | \mu, \sigma_n)\,P(\sigma_n)\,\rm{d}\sigma_n\;.

We solve this integral analytically and find

\begin{aligned}P(x_n | \mu) =& \frac{1}{2\,\sqrt{\pi}\,s^2\,\Gamma(c)}\Biggl(\sqrt{2}\,\Gamma(c - 2)\,{}_0F_2\biggl(;\frac{3-c}{2}, \frac{4 - c}{2}; -\frac{(x_n - \mu)^2}{8\,s^2}\biggr) \\ &+ 2^{(1-c)/2}\,s^{2-c}\,|x_n - \mu|^{c-2}\,\Gamma\biggl(\frac{2 - c}{2}\biggr)\,{}_0F_2\biggl(;\frac{1}{2}, \frac{c}{2}; -\frac{(x_n - \mu)^2}{8\,s^2}\biggr) \\ &-\;\;\; 2^{-c/2}\,s^{1-c}\,|x_n - \mu|^{c-1}\,\Gamma\biggl(\frac{1-c}{2}\biggr)\,{}_0F_2\biggl(;\frac{3}{2}, \frac{c + 1}{2}; -\frac{(x_n - \mu)^2}{8\,s^2}\biggr)\Biggr)\;,\end{aligned}

where {}_0F_2 is the generalized hypergeometric function.

We now have everything we need to plot P(\sigma_n | x_n, \mu). If we would assume that \mu = 10, then we’d find that we’re quite uncertain about the noise level of scientist A, but it is generally large, while we are very certain about the noise levels of scientists D, F and G, which should be small, as you can see in Figure 3.

FIGURE 3: Bayesian estimate of the scientist’s individual noise levels \sigma_n, assuming the true value their experiments aim to measure is \mu = 10. Note the logarithmic scale of the \sigma-axis. The inset shows that \mu = 10 is close to measurement results, except the results of A and B, which is consistent with their large estimated \sigma values.

If we instead assumed the \mu that we found with the maximum likelihood estimate, \mu = 0.1, then we’d be relatively sure that B’s noise level is lower than all other noise levels (see Figure 4), which is consistent with Figure 2.

FIGURE 4: Same as Figure 3, but with \mu = 0.1, which we obtained with the maximum likelihood method.

The distribution of \mu

In the previous section we have found an expression for P(x_n | \mu). Since all measurements are independent from one another, the probability to find all the x_n is

P(\{x_n\} | \mu) = \prod_{n = 1}^N\,P(x_n | \mu)\;,

where N = 7 is the number of scientists.

Now we use Bayes’ theorem again to find

P(\mu | \{x_n\}) = \frac{P(\{x_n\} | \mu)\;P(\mu)}{P(\{x_n\})} \propto P(\{x_n\} | \mu)\;.

We don’t know anything about the prior P(\mu), but we could assume that it’s a uniform distribution between some large negative and positive values. In this case, P(\mu | \{x_n\}) is proportional to P(\{x_n\} | \mu), as indicated above.

Inserting the given values for the parameters c and s, we can now plot P(\mu | \{x_n\}) in Figure 5 (vertical scales are not given, since we have not computed the normalization).

FIGURE 5: Probability distribution (up to normalization) of the quantity \mu that the scientists have attempted to measure. The distribution has a pole around each measured x_n. The most likely true value is somewhere around \mu = 10.

The graph in Figure 5 diverges to +\infty for every \mu = x_n. The probability density around these singularities accumulates when multiple data are close to one another, resulting in an increased probability to find \mu around \mu = 10.

Another dataset

MacKay also asks what happens if we had just two scientists, measuring \{x_n\} = \{13.01,7.39\}. As we would expect, the probability distribution over \mu is exactly symmetric:

FIGURE 6: Same as Figure 5, but for the dataset \{x_n\} = \{13.01,7.39\}.

This symmetry reflects the fact that with just two measurements, we cannot decide which scientist is the more accurate one, and therefore, we cannot know if 13.01 or 7.39 is closer to the true \mu.

References

1.
MacKay, D. J. C. Information Theory, Inference, and Learning Algorithms. (Cambridge University Press, 2003).

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