Bayesian updating of strategy performance

Skill positionModule 3 of 3 · Topic 3 of 4

3.3 Bayesian updating of strategy performance

—Lab score— / 100Total XP—Streak—

Yesterday's posterior is today's prior

A strategy comes with a back-test and then with live results. Bayesian updating handles both in one way. Start with a prior: what the back-test says about the strategy's yearly edge, with its uncertainty. When a new period of live results arrives, update. The posterior becomes the prior for the next period. Updating one period at a time gives exactly the same answer as updating once with all the data, and the order of the periods does not matter. That makes it natural for a live system that learns every day.

P_{k}=P_{k-1}+\frac{1}{s^{2}},\qquad m_{k}=\frac{P_{k-1}\,m_{k-1}+y_k/s^{2}}{P_{k}}

(1)

Here $P$ is the precision of the belief, $m$ its mean, $y_k$ the result of period $k$ and $s$ its standard error. One more idea matters for markets: they change. A strategy's edge today may differ from its edge five years ago, so very old results can mislead. A forgetting factor $\delta$ slightly below $1$ multiplies the precision by $\delta$ before each update, which widens the belief a little and lets old results fade. The price is that the belief never becomes extremely sure, which is honest when the world keeps moving.

A back-test meets two live yearsThe back-test says an edge of

6\%

a year with uncertainty

4\%

, so the prior precision is

1/16=0.0625

. Live year 1 returns

2\%

with standard error

5\%

, precision

0.04

. The posterior has precision

0.1025

, mean

(0.0625\times6+0.04\times2)/0.1025=4.44\%

and standard deviation

3.12\%

. Live year 2 returns

1\%

. Update again: precision

0.1425

, mean

3.47\%

, standard deviation

2.65\%

. All at once, with mean

1.5\%

and standard error

3.54\%

, gives the same

3.47\%

. The chance that the edge is above zero is now

0.905

Sequential updating: yesterday's posterior is today's prior. One period at a time equals all at once.
Back-test as a prior: it is a starting belief with an honest uncertainty, not a fact.
Forgetting factor $\delta<1$ : shrinks the precision before each update so old results fade.

Lab · what's graded

After the first live year15 pts
The spread after year one10 pts
After the second live year15 pts
The chance of a positive edge15 pts
Forgetting old results10 pts
Does the order matter?10 pts
Match the vocabulary10 pts
Why forget old results15 pts

0/ 8 answered

1Numeric answer15 ptsNot answered

The prior is $6\%$ with precision $0.0625$ . Live year 1 gives $2\%$ with precision $0.04$ . What is the posterior mean $(0.0625\times6+0.04\times2)/0.1025$ , in percent? Give two decimals.

m =

%accepted: ± 0.02

2Numeric answer10 ptsNot answered

The posterior precision is $0.1025$ . What is the posterior standard deviation $1/\sqrt{0.1025}$ , in percent? Give two decimals.

s =

%accepted: ± 0.02

3Numeric answer15 ptsNot answered

Now the prior is the last posterior (mean $4.44\%$ , precision $0.1025$ ) and live year 2 gives $1\%$ with precision $0.04$ . What is the new posterior mean $(0.1025\times4.44+0.04\times1)/0.1425$ , in percent? Give two decimals.

m =

%accepted: ± 0.02

4Numeric answer15 ptsNot answered

The posterior is $N(3.47\%,\ 2.65\%)$ . What is $P(m>0)=\Phi(3.47/2.65)=\Phi(1.31)$ ? Give three decimals.

p =

accepted: ± 0.004

5Numeric answer10 ptsNot answered

Before the next update, apply a forgetting factor $\delta=0.9$ to the precision $0.1025$ . What is the new precision $0.9\times0.1025$ ? Give five decimals.

d =

accepted: ± 0.00003

6Choice10 ptsNot answered

You update with year 1 then year 2, or with year 2 then year 1, with no forgetting. What is the difference in the final posterior?

7Match10 ptsNot answered

Match each term with its meaning.

Pick the partner of each item on the left.

Sequential updating

Back-test prior

Forgetting factor

Live results

8Explain in words15 ptsNot answered

In one or two sentences, explain why a trader may want a forgetting factor when updating beliefs about a strategy.

Graded on:Markets change, so old data can misleadA forgetting factor gives older results less weight0 / 400 characters

Math modeKeyboard or LaTeX inputTask 1 of 8 · Ctrl ⏎ to check

Bayesian updating of strategy performance

Skill positionModule 3 of 3 · Topic 3 of 4

3.3 Bayesian updating of strategy performance

—Lab score— / 100Total XP—Streak—

Yesterday's posterior is today's prior

P_{k}=P_{k-1}+\frac{1}{s^{2}},\qquad m_{k}=\frac{P_{k-1}\,m_{k-1}+y_k/s^{2}}{P_{k}}

(1)

A back-test meets two live yearsThe back-test says an edge of

6\%

a year with uncertainty

4\%

, so the prior precision is

1/16=0.0625

. Live year 1 returns

2\%

with standard error

5\%

, precision

0.04

. The posterior has precision

0.1025

, mean

(0.0625\times6+0.04\times2)/0.1025=4.44\%

and standard deviation

3.12\%

. Live year 2 returns

1\%

. Update again: precision

0.1425

, mean

3.47\%

, standard deviation

2.65\%

. All at once, with mean

1.5\%

and standard error

3.54\%

, gives the same

3.47\%

. The chance that the edge is above zero is now

0.905

Sequential updating: yesterday's posterior is today's prior. One period at a time equals all at once.
Back-test as a prior: it is a starting belief with an honest uncertainty, not a fact.
Forgetting factor $\delta<1$ : shrinks the precision before each update so old results fade.

Lab · what's graded

After the first live year15 pts
The spread after year one10 pts
After the second live year15 pts
The chance of a positive edge15 pts
Forgetting old results10 pts
Does the order matter?10 pts
Match the vocabulary10 pts
Why forget old results15 pts

0/ 8 answered

1Numeric answer15 ptsNot answered

The prior is $6\%$ with precision $0.0625$ . Live year 1 gives $2\%$ with precision $0.04$ . What is the posterior mean $(0.0625\times6+0.04\times2)/0.1025$ , in percent? Give two decimals.

m =

%accepted: ± 0.02

2Numeric answer10 ptsNot answered

The posterior precision is $0.1025$ . What is the posterior standard deviation $1/\sqrt{0.1025}$ , in percent? Give two decimals.

s =

%accepted: ± 0.02

3Numeric answer15 ptsNot answered

m =

%accepted: ± 0.02

4Numeric answer15 ptsNot answered

The posterior is $N(3.47\%,\ 2.65\%)$ . What is $P(m>0)=\Phi(3.47/2.65)=\Phi(1.31)$ ? Give three decimals.

p =

accepted: ± 0.004

5Numeric answer10 ptsNot answered

Before the next update, apply a forgetting factor $\delta=0.9$ to the precision $0.1025$ . What is the new precision $0.9\times0.1025$ ? Give five decimals.

d =

accepted: ± 0.00003

6Choice10 ptsNot answered

You update with year 1 then year 2, or with year 2 then year 1, with no forgetting. What is the difference in the final posterior?

7Match10 ptsNot answered

Match each term with its meaning.

Pick the partner of each item on the left.

Sequential updating

Back-test prior

Forgetting factor

Live results

8Explain in words15 ptsNot answered

In one or two sentences, explain why a trader may want a forgetting factor when updating beliefs about a strategy.

Graded on:Markets change, so old data can misleadA forgetting factor gives older results less weight0 / 400 characters

Math modeKeyboard or LaTeX inputTask 1 of 8 · Ctrl ⏎ to check

Bayesian updating of strategy performance

Yesterday's posterior is today's prior

Lab · what's graded

Log in to run your code.

Bayesian updating of strategy performance

Yesterday's posterior is today's prior

Lab · what's graded

Log in to run your code.