Bayesian updating

A curve says how much you believe each possible bias of a coin. Watch it reshape as the flips land, one by one.

Belief as a curve

The coin’s chance of heads, $p$, is unknown. Instead of one guess, the picture keeps a whole curve: a belief about every possible value. Before any flips that belief is the prior. After the data it is the posterior. The rule that turns one into the other is Bayes’ theorem, used again and again.

Before the clip you guessed what three heads would do. After three heads the curve is $4p^3$: it rises all the way to $p=1$, but it is spread wide. Its mean is $0.8$, not 1, and $p$ could easily be as low as 0.6: the curve still puts 13 % of its area below 0.6. Three flips are not much evidence.

How sure the curve is that the coin favours heads, $p>0.5$, is its area above 0.5: $1-0.5^4=0.9375$.

The clip’s coin then landed 17 more heads in 27 flips: 20 heads in 30 in all. The curve ends narrow, with mean $21/32=0.6563$, near the true chance 0.7.

Three coins in a bag

Start with a case small enough to do by hand. A bag holds three coins. One lands heads with chance 0.3, one with 0.5, one with 0.8. You draw a coin without looking, so each is equally likely: a prior of $\tfrac13$ each. You flip it once, and it lands heads. Which coin is it now likely to be?

Multiply each prior by the chance that coin gives heads, its likelihood from Maximum likelihood, and rescale so the three add to 1:

That is the whole method:

In words: posterior is prior times likelihood, rescaled. The bottom line is only the rescaling.

Now a tails. Use the posterior as the new prior, and the chances of tails, $0.7$, $0.5$ and $0.2$, as the likelihoods. The products are $0.13125$, $0.15625$ and $0.1$, adding to $0.3875$, so the beliefs become $0.3387$, $0.4032$ and $0.2581$.

After the clip, swap the order. Tails first and then heads gives the same three beliefs: either way each coin’s product is its chance of one heads times its chance of one tails, $0.21$, $0.25$ and $0.16$ (times $\tfrac13$). Order does not matter, only what was seen.

Many coins: a curve

Now let the bag hold coins of every bias from 0 to 1, all equally likely: a flat prior over $p$. After $h$ heads in $n$ flips, the likelihood of a coin with bias $p$ is $p^h(1-p)^{n-h}$, and with a flat prior the posterior is that curve, rescaled to area 1.

A curve of the shape $p^{a-1}(1-p)^{b-1}$ on $[0,1]$, rescaled to area 1, is called the beta law, written $\mathrm{Beta}(a,b)$. Here $a$ and $b$ are two positive shape numbers, not filter coefficients. So the posterior after $h$ heads in $n$ flips, from a flat prior, is

The flat prior is $\mathrm{Beta}(1,1)$, and three heads give $\mathrm{Beta}(4,1)$, the $4p^3$ above. A $\mathrm{Beta}(a,b)$ curve has mean $a/(a+b)$ and, for $a,b>1$, its peak at $(a-1)/(a+b-2)$. For the posterior that means

The peak is the maximum likelihood estimate; the mean is pulled slightly towards $\tfrac12$.

Predicting the next flip

What is the chance the next flip lands heads? Average each coin’s chance of heads, weighted by the belief in it. For the bag after one heads:

For the curve, the same average is the curve’s mean, $(h+1)/(n+2)$. After 7 heads in 10 flips it is $8/12=0.6667$. It is called the rule of succession: after no flips at all it gives $\tfrac12$.

Today’s posterior is tomorrow’s prior

Updating one piece of evidence at a time gives the same answer as updating on all of it at once, as the swap showed. That is why the method suits data that arrive in a stream: keep one belief, and fold in each new observation as it comes.

The repeated medical test of Bayes’ theorem was exactly this. A prior of 1 % became $16.7\%$ after one positive test, $79.8\%$ after two and $98.7\%$ after three, each posterior serving as the next prior.

Worked example

1. The bag, heads then tails. Priors $\tfrac13$ each. After heads: $0.1875$, $0.3125$, $0.5$, and the next flip is heads with chance $0.6125$. After tails as well: $0.3387$, $0.4032$, $0.2581$, and the next flip is heads with chance $0.5097$.

2. Seven heads in ten, from a flat prior. The posterior is $\mathrm{Beta}(8,4)$. Its peak is at $0.7$, the estimate of Maximum likelihood; its mean is $8/12=0.6667$, and its standard deviation $\sqrt{32/(144\times13)}=0.1307$.

This curve is the likelihood $p^7(1-p)^3$ divided by its area, $1/1320$. So its peak height is $1320\times0.002224=2.935$.

The curve’s mean sits left of its peak because the curve leans: it has a longer tail towards small $p$ than towards 1.

3. Two tails from a flat prior. No heads in two flips gives $\mathrm{Beta}(1,3)$, with mean $1/4$: not 0, because two flips cannot rule heads out.

Where you’ll meet this

Spam filters. A filter keeps a belief that a message is spam and updates it word by word, as the bag updated flip by flip.

Experiments with history. An A/B test can start from a prior built from earlier tests, so that a small new test is read in the light of the old ones.

Monitoring. A hospital monitor or a fault detector updates a belief about a patient’s or a machine’s state with every new reading.

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