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The cost of acquiring information by natural selection
Carl T. Bergstrom
Department of Biology
University of Washington
with R. McGee, O. Kosterlitz, A.Kaznatcheev, and B. Kerr
Photo: Carl Bergstrom

Ben Kerr
Olivia Kosterlitz Artem Kaznatcheev
Ryan McGee

The information behind phenotype
Photos: Carl Bergstrom

Where does this information come from?

natural selection

In particular, organisms need to
match their phenotypes to their
environments, their niches, their
life-histories, etc.

The cost of acquiring information by natural selection

“The environment is light,
so grow light fur.”
— Mom Given that a light fur allele was
inherited,
it is likely that light fur has been favored
in the past, and thus it is likely
beneficial to develop light fur.

.
Every bit of adaptive
information in your
genome was paid for
in the blood of your
ancestors’ children.

But can we quantify it?
I think so.
Adaptive genetic information refers to the
inherited material that reduces an organism's
uncertainty about the current environment
so that its expected fitness is greater than it
would be due to chance alone.

Imagine the state of the environment as a random variable
E and the genome G as another random variable.
𝐼 𝐺; 𝐸 = 𝐻 𝐸 − 𝐻 𝐸 𝐺
Natural selection creates mutual
Information between E and G.

We can measure how much information has been
added looking at how much genotype frequencies have
changed due to selection.
𝐼 𝐺; 𝐸 = 𝐻 𝐸 − 𝐻 𝐸 𝐺
= 𝐷𝐾𝐿 (𝑃𝑡
(𝐺, 𝐸)||𝑃(𝐺)𝑃(𝐸))
= 𝐷𝐾𝐿(𝑃𝑡
(𝐺, 𝐸)||𝑃0
(𝐺, 𝐸))

As the frequency of the
best genotype increases
in the population, the
population gains
information about the
environment.

Nearly 70 years ��….

Substitution load
growth rate
of optimal type
⏞ ⏞
growth rate
of type i

“It is appropriate to
speak of a cost of
selection, since the
cost comes from the
fact that natural
selection is less
efficient than divine
intervention.”
- Joe Felsenstein
Substitution load

𝑤
𝑡
𝑤𝑚𝑎𝑥
𝑡
𝐿𝑡
𝐼𝑡
𝑤
(1)
𝑤
(2)
𝑤𝑚𝑎𝑥
(1)
𝑤𝑚𝑎𝑥
(2)

𝐿𝑡
𝐼𝑡
population growth rate population growth rate
optimal type growth rate optimal type growth rate
optimal growth rate

𝐿𝑠𝑢𝑏
𝑇
𝐼𝑇
population growth rate population growth rate
optimal type growth rate optimal type growth rate
optimal growth rate

time
growth rate
of optimal type
⏞ ⏞
growth rate
of type i
Substitution Load
growth rate
of optimal type
in condition j
⏞ ⏞
growth rate
of type i
in condition j
probability of
condition j
Mismatch Load

Regret
Loss and regret

In computational learning theory,
• loss is the payoff cost of mismatch between
strategy and environment, and
• regret is the cost of having to learn the best
strategy: the difference between the cumulative
loss as the learner updates its strategy over time,
and the loss it could have achieved had it played
an optimal fixed strategy from the beginning.

By thinking about
regret, we can broadly
extend our results
about substitution load.

Load of the evolving
population
Load of the best type
Information gain of
evolving population

environment
genotype “idiosyncratically” stochastic environments

environment
genotype Cycling environments

environment
genotype Frequency-dependent fitnesses
Rock-
Paper-
Scissors

Theorem: Information gain converges
to regret, in fixed or stochastic
environments, with or without
frequency-dependence, etc.

Theorem: Information gain is
bounded above by empirical regret,
a measure of regret taken relative to
the ex-post optimal strategy.

Teaser: Among learning
algorithms, natural selection is
never ”far” from optimal in the
sense of minimizing regret.

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The cost of acquiring information by natural selection

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