How cold is it? A Bayesian attempt to measure temperature

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It is getting colder in London, yet it is still quite mild considering that it is late November. Well, indoors it still feels like 20°C (68°F) to me, but I have been told last week that I should switch on the heating.

Luckily I found an old thermometer to check. The thermometer showed 18°C. Is it really below 20°C?

The thermometer is quite old and I’m not sure that is works properly anymore. So, what shall I do now? Perhaps I should consider that both measurements are uncertain and try to combine them.

I believe that I can sense the temperature within ±3°C, while I think that the thermometer still works within ±2°C. Assuming that both measurements follow a Gaussian (Normal) distribution, with the uncertainties given as standard deviations, I can use Bayes’ theorem to combine my hypothesis with the data. The posterior distribution will be Gaussian again with conjugated hyper-parameters:
[
mu=left.left(frac{mu_0}{sigma_0^2} + frac{sum_{i=1}^n x_i}{s^2}right)right/left(frac{1}{sigma_0^2} + frac{n}{s^2}right) \
sigma^2=left(frac{1}{sigma_0^2} + frac{n}{s^2}right)^{-1}
]With (K := frac{nsigma_0^2}{s^2+nsigma_0^2} ) this simplifies to:
[
mu = K, bar{x} + (1 – K), mu_0 mbox{, with } bar{x}=frac{1}{n}sum_{i=1}^n x_i\
sigma = s ,sqrt{K/n}
]In my case I have: (n=1,; x_1=18^{circ}C,; s=2^{circ}C,; mu_0=20^{circ}C,; sigma_0=3^{circ}C).

Hence, the posterior distribution has parameters (mu=18.6^{circ}C) and (sigma=1.7^{circ}C). Thus, my best guess would be that is actually a little colder than I thought. One could argue that the probability that is below 20° is 80%.

Over the last five days my perception of the temperature didn’t change, neither did the weather forecast, but the measurements showed: 18°C, 19°C, 17.5°C, 18°C, 18.5°C.

With that information the parameters update to (mu=18.3^{circ}C) and (sigma=0.9^{circ}C). I can’t deny it any longer it has got colder. The probability that is below 20°C is now at 97% and the heating is on.

Without any prior knowledge I may have used a t-test to check the measurements. But here I believe that I have information about the thermometer and my own temperature sensing abilities which I don’t want to ignore.

R code

Session Info

R version 3.1.2 (2014-10-31)<br />Platform: x86_64-apple-darwin13.4.0 (64-bit)<br /><br />locale:<br />[1] en_GB.UTF-8/en_GB.UTF-8/en_GB.UTF-8/C/en_GB.UTF-8/en_GB.UTF-8<br /><br />attached base packages:<br />[1] stats     graphics  grDevices utils     datasets  methods  <br />[7] base     <br /><br />other attached packages:<br />[1] BayesianFirstAid_0.1 rjags_3-14           coda_0.16-1         <br />[4] lattice_0.20-29     <br /><br />loaded via a namespace (and not attached):<br />[1] grid_3.1.2    MASS_7.3-35   mnormt_1.5-1  stringr_0.6.2

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