The document introduces PyMC3 for Bayesian statistical modeling in Python, explaining the basics of Bayesian statistics including prior, likelihood, and posterior concepts. It discusses the application of Bayesian inference in various fields such as finance, music streaming, e-commerce, astronomy, life sciences, and medicine. Furthermore, it covers Markov Chain Monte Carlo (MCMC) methods, hierarchical models, and case studies including A/B testing, highlighting the advantages of Bayesian approaches over traditional frequentist methods.

1. PyMC3 – Bayesian Statistical Modelling in Python
Max Kochurov
22 June, 2019
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 1 / 29

2. About me
Max Kochurov tg,slack:@ferres / github:ferrine
Geoopt
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 2 / 29

3. Bayesian Statistics
Figure: Updating prior p(λ)
p(λ | D) =
p(D | λ)p(λ)
p(D)
• p(λ) – Prior, base knowledge
• p(D | λ) – Likelihood, new information
• p(λ | D) – Posterior, updated knowledge
• p(D) – Evidence, surprise in data
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 3 / 29

4. Bayesian Statistics
p(λ | D) =
p(D | λ)p(λ)
p(D)
• p(λ) – Prior, base knowledge
• p(D | λ) – Likelihood, new information
• p(λ | D) – Posterior, updated knowledge
• p(D) – Evidence, surprise in data
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 3 / 29

5. Bayesian Statistics
Compared to Frequentist
• Elegant way to put assumptions in the
model
• Can work with few Data!
• No need for p-values,
all you need is p(λ | D)
• . . .
p(λ | D) =
p(D | λ)p(λ)
p(D)
• p(λ) – Prior, base knowledge
• p(D | λ) – Likelihood, new information
• p(λ | D) – Posterior, updated knowledge
• p(D) – Evidence, surprise in data
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 3 / 29

6. Where do people use Bayesian inference?
• Finance (Quantopian): estimating the performance of trading algorithms
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 4 / 29

7. Where do people use Bayesian inference?
• Finance (Quantopian): estimating the performance of trading algorithms
• Music Streaming (Sounds): A/B testing, churn prediction, lifetime value, defining user
session
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 4 / 29

8. Where do people use Bayesian inference?
• Finance (Quantopian): estimating the performance of trading algorithms
• Music Streaming (Sounds): A/B testing, churn prediction, lifetime value, defining user
session
• E-Commerce (Salesforce): A/B tests, combining disparate sources of information,
hierarchical models
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 4 / 29

9. Where do people use Bayesian inference?
• Finance (Quantopian): estimating the performance of trading algorithms
• Music Streaming (Sounds): A/B testing, churn prediction, lifetime value, defining user
session
• E-Commerce (Salesforce): A/B tests, combining disparate sources of information,
hierarchical models
• Astronomy: estimating orbits of space objects, getting an image of a black hole
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 4 / 29

10. Where do people use Bayesian inference?
• Finance (Quantopian): estimating the performance of trading algorithms
• Music Streaming (Sounds): A/B testing, churn prediction, lifetime value, defining user
session
• E-Commerce (Salesforce): A/B tests, combining disparate sources of information,
hierarchical models
• Astronomy: estimating orbits of space objects, getting an image of a black hole
• Life science: epidemic analysis
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 4 / 29

11. Where do people use Bayesian inference?
• Finance (Quantopian): estimating the performance of trading algorithms
• Music Streaming (Sounds): A/B testing, churn prediction, lifetime value, defining user
session
• E-Commerce (Salesforce): A/B tests, combining disparate sources of information,
hierarchical models
• Astronomy: estimating orbits of space objects, getting an image of a black hole
• Life science: epidemic analysis
• Medicine: calculating effect sizes for drugs
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 4 / 29

12. Where do people use Bayesian inference?
• Finance (Quantopian): estimating the performance of trading algorithms
• Music Streaming (Sounds): A/B testing, churn prediction, lifetime value, defining user
session
• E-Commerce (Salesforce): A/B tests, combining disparate sources of information,
hierarchical models
• Astronomy: estimating orbits of space objects, getting an image of a black hole
• Life science: epidemic analysis
• Medicine: calculating effect sizes for drugs
• . . .
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 4 / 29

13. Where do people use Bayesian inference?
• Finance (Quantopian): estimating the performance of trading algorithms
• Music Streaming (Sounds): A/B testing, churn prediction, lifetime value, defining user
session
• E-Commerce (Salesforce): A/B tests, combining disparate sources of information,
hierarchical models
• Astronomy: estimating orbits of space objects, getting an image of a black hole
• Life science: epidemic analysis
• Medicine: calculating effect sizes for drugs
• . . .
Some will be covered later
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 4 / 29

14. Coin Example
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 5 / 29

15. Fair Coin Flips – expect fair
p ∼ Beta(3, 3)
flips ∼ Binomial(N, p)
Data: 12 out of 20 flips
What is p?
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 6 / 29

16. Fair Coin Flips – expect fair
p ∼ Beta(3, 3)
flips ∼ Binomial(N, p)
Data: 12 out of 20 flips
What is p?
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 6 / 29

17. Fair Coin Flips – expect fair
p ∼ Beta(3, 3)
flips ∼ Binomial(N, p)
Data: 12 out of 20 flips
What is p?
We expect an angel and see an angel, we are confident
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 6 / 29

18. Fair Coin Flips – expect unfair
What is p?
p ∼ Beta(1, 5)
flips ∼ Binomial(N, p)
Data: 12 out of 20 flips
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 7 / 29

19. Fair Coin Flips – expect unfair
What is p?
p ∼ Beta(1, 5)
flips ∼ Binomial(N, p)
Data: 12 out of 20 flips
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 7 / 29

20. Fair Coin Flips – expect unfair
What is p?
p ∼ Beta(1, 5)
flips ∼ Binomial(N, p)
Data: 12 out of 20 flips
We expect a devil but see an angel, we are less confident
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 7 / 29

21. Hierarchical Models
What if we have more diverse data? We can estimate an amount of devils + uncertainty!
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 8 / 29

22. Hierarchical Models
What if we have more diverse data? We can estimate an amount of devils + uncertainty!
Data: [8, 2, 2, 10, 0,
0, 2, 5, 6, 10] out of 20 flips
λDevil ∼ Exponential(1)
λAngel ∼ Exponential(1)
pi ∼ Beta(λAngel , λAngel + λDevil )
flipsi ∼ Binomial(pi , Ni )
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 8 / 29

23. Hierarchical Models
What if we have more diverse data? We can estimate an amount of devils + uncertainty!
Data: [8, 2, 2, 10, 0,
0, 2, 5, 6, 10] out of 20 flips
λDevil ∼ Exponential(1)
λAngel ∼ Exponential(1)
pi ∼ Beta(λAngel , λAngel + λDevil )
flipsi ∼ Binomial(pi , Ni )
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 8 / 29

24. Markov Chain Monte Carlo (MCMC) & Variational Inference
Few simple rules
• Try MCMC first
• If it is slow: wait a bit
• Got tired: try VI
• Didn’t work: feel sad
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 9 / 29

25. Markov Chain Monte Carlo (MCMC) & Variational Inference
Few simple rules
• Try MCMC first
• If it is slow: wait a bit
• Got tired: try VI
• Didn’t work: feel sad
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 9 / 29

26. Markov Chain Monte Carlo (MCMC) & Variational Inference
Few simple rules
• Try MCMC first
• If it is slow: wait a bit
• Got tired: try VI
• Didn’t work: feel sad
Did not work for you?
goto https://discourse.pymc.io
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 9 / 29

27. Inspecting Your Model
What is an amount of devils? (MCMC used) pm.traceplot(trace)
0 1 2 3 4 5 6 7 8
0.0
0.2
0.4
Frequency
devils
0 100 200 300 400 500
0.0
2.5
5.0
7.5
Samplevalue
devils
0.5 1.0 1.5 2.0 2.5
0.0
0.5
1.0
Frequency
angels
0 100 200 300 400 500
1
2
Samplevalue
angels
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
10
20
Frequency
p
0 100 200 300 400 500
0.00
0.25
0.50
0.75
Samplevalue
p
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 10 / 29

28. Inspecting Your Model
plt.hist(trace["angels"],
alpha=.8, label='Alpha')
plt.hist(trace["angels"]+trace["devils"],
alpha=.8, label='Beta')
plt.legend(fontsize=20)
0 1 2 3 4 5 6 7 8
0
50
100
150
200
250
300
Alpha
Beta
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 11 / 29

29. Inspecting Your Model
0.0 0.2 0.4 0.6 0.8 1.0
0
1
2
3
4
5
6
7
Prob Distribution
for a, b in zip(
trace["angels"],
trace["angels"]+trace["devils"]
):
plt.plot(
np.linspace(0, 1),
st.beta(a, b).pdf(np.linspace(0, 1)),
color="b", alpha=.025
)
a_mean = trace["angels"].mean()
b_mean = (trace["angels"]+trace["devils"]).mean()
plt.plot(
np.linspace(0, 1),
st.beta(a_mean, b_mean).pdf(np.linspace(0, 1)),
color="black", linewidth=4.0
)
plt.axvline(0.5)
plt.title("Prob Distribution", fontsize=20)
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 12 / 29

30. Case Studies
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 13 / 29

31. A/B testing
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 14 / 29

32. A/B testing, why Bayes
We would accept that
result
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
p-val=0.2398diff=0.1
diff=0.1
pval=.24
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 15 / 29

33. A/B testing, why Bayes
We would accept that
result
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
p-val=0.2398diff=0.1
diff=0.1
pval=.24
We wait too long to
check small diff
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
15
20
25
30
35
40
p-val=0.2398diff=0.01
diff=0.01
pval=.24
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 15 / 29

34. A/B testing, why Bayes
We would accept that
result
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
p-val=0.2398diff=0.1
diff=0.1
pval=.24
We wait too long to
check small diff
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
15
20
25
30
35
40
p-val=0.2398diff=0.01
diff=0.01
pval=.24
WTF?
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
15
20
25
30
35
40
p-val=0.0786diff=0.02
diff=0.02
pval=0.07
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 15 / 29

35. Expected Loss Framework
p-values are useless. What about calculating a loss of ignoring a better model?
• We now decide to use model A or B
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 16 / 29

36. Expected Loss Framework
p-values are useless. What about calculating a loss of ignoring a better model?
• We now decide to use model A or B
• A has some α ctr, B has some β ctr
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 16 / 29

37. Expected Loss Framework
p-values are useless. What about calculating a loss of ignoring a better model?
• We now decide to use model A or B
• A has some α ctr, B has some β ctr
• Say a loss (can be any) to ignore a better model given we use x
Loss(α, β, x) =
max(β − α, 0), x = A
max(α − β, 0), x = B
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 16 / 29

38. Expected Loss Framework
p-values are useless. What about calculating a loss of ignoring a better model?
• We now decide to use model A or B
• A has some α ctr, B has some β ctr
• Say a loss (can be any) to ignore a better model given we use x
Loss(α, β, x) =
max(β − α, 0), x = A
max(α − β, 0), x = B
• Loss depends on the magnitude of difference (p-values do not)
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 16 / 29

39. Expected Loss Framework
L(x) = Ep(α,β)Loss(α, β, x) → min
x
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 17 / 29

40. Expected Loss Framework
L(x) = Ep(α,β)Loss(α, β, x) → min
x
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
p-val=0.2398diff=0.1
L(A) = 0.122
L(B) = 0.019
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 17 / 29

41. Expected Loss Framework
L(x) = Ep(α,β)Loss(α, β, x) → min
x
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
p-val=0.2398diff=0.1
L(A) = 0.122
L(B) = 0.019
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
15
20
25
30
35
40
p-val=0.2398diff=0.01
L(A) = 0.012
L(B) = 0.002
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 17 / 29

42. Expected Loss Framework
L(x) = Ep(α,β)Loss(α, β, x) → min
x
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
p-val=0.2398diff=0.1
L(A) = 0.122
L(B) = 0.019
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
15
20
25
30
35
40
p-val=0.2398diff=0.01
L(A) = 0.012
L(B) = 0.002
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
15
20
25
30
35
40
p-val=0.0786diff=0.02
L(A) = 0.02
L(B) = 0.0005
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 17 / 29

43. Expected Loss Framework
L(x) = Ep(α,β)Loss(α, β, x) → min
x
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
p-val=0.2398diff=0.1
L(A) = 0.122
L(B) = 0.019
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
15
20
25
30
35
40
p-val=0.2398diff=0.01
L(A) = 0.012
L(B) = 0.002
0.0 0.2 0.4 0.6 0.8 1.0
0
5
10
15
20
25
30
35
40
p-val=0.0786diff=0.02
L(A) = 0.02
L(B) = 0.0005
Without p-values we stop first experiment earlier as B is a clear winner
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 17 / 29

44. Real(!) Impact of prior
Do you believe in ctr > 0.9 ? Or > 0.8? Or 0?
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 18 / 29

45. Real(!) Impact of prior
Do you believe in ctr > 0.9 ? Or > 0.8? Or 0?
You have your p(ctr) in mind, use it!
Informative prior
Non-Informative
prior
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 18 / 29

46. Real(!) Impact of prior
Do you believe in ctr > 0.9 ? Or > 0.8? Or 0?
You have your p(ctr) in mind, use it!
• You iterate quicker
• Accept small improvements
• Fewer problems of continuous monitoring
Informative prior
Non-Informative
prior
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 18 / 29

47. Real(!) Impact of prior
Do you believe in ctr > 0.9 ? Or > 0.8? Or 0?
You have your p(ctr) in mind, use it!
• You iterate quicker
• Accept small improvements
• Fewer problems of continuous monitoring
Do not use overconfident prior, choose slightly pessimistic one
Informative prior
Non-Informative
prior
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 18 / 29

48. Real(!) Impact of prior
Do you believe in ctr > 0.9 ? Or > 0.8? Or 0?
You have your p(ctr) in mind, use it!
• You iterate quicker
• Accept small improvements
• Fewer problems of continuous monitoring
Do not use overconfident prior, choose slightly pessimistic one
Choose smarter loss (”accept improvement at least δ”):
Loss(α, β, δ, x) =
max((β − δ) − α, 0), x = A
max(α − (β − δ), 0), x = B
Informative prior
Non-Informative
prior
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 18 / 29

49. Real(!) Impact of prior
Do you believe in ctr > 0.9 ? Or > 0.8? Or 0?
You have your p(ctr) in mind, use it!
• You iterate quicker
• Accept small improvements
• Fewer problems of continuous monitoring
Do not use overconfident prior, choose slightly pessimistic one
Choose smarter loss (”accept improvement at least δ”):
Loss(α, β, δ, x) =
max((β − δ) − α, 0), x = A
max(α − (β − δ), 0), x = B
Convert loss into $$$
Informative prior
Non-Informative
prior
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 18 / 29

50. Portfolio
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 19 / 29

51. Finance (Quantopian) case
Hedge-Fund in Boston
• Crowd sourcing for trade strategies, more that 700k
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 20 / 29

52. Finance (Quantopian) case
Hedge-Fund in Boston
• Crowd sourcing for trade strategies, more that 700k
• People tend to overfit the leaderboard, that’s ok
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 20 / 29

53. Finance (Quantopian) case
Hedge-Fund in Boston
• Crowd sourcing for trade strategies, more that 700k
• People tend to overfit the leaderboard, that’s ok
• But they do not have out of sample data
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 20 / 29

54. Finance (Quantopian) case
Hedge-Fund in Boston
• Crowd sourcing for trade strategies, more that 700k
• People tend to overfit the leaderboard, that’s ok
• But they do not have out of sample data
• Data changes over time, we should take care of it
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 20 / 29

55. Finance (Quantopian) case
Hedge-Fund in Boston
• Crowd sourcing for trade strategies, more that 700k
• People tend to overfit the leaderboard, that’s ok
• But they do not have out of sample data
• Data changes over time, we should take care of it
How to link 2 periods of algorithm evaluation to use more data?
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 20 / 29

56. Finance (Quantopian) case
Core ideas
• Use Gaussian process to capture changing
volatility and return in time
Figure: Gaussian Process
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 21 / 29

57. Finance (Quantopian) case
Core ideas
• Use Gaussian process to capture changing
volatility and return in time
• Allow structural changes in the model
• mean returns
• but not volatility
Figure: Gaussian Process
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 21 / 29

58. Finance (Quantopian) case
Core ideas
• Use Gaussian process to capture changing
volatility and return in time
• Allow structural changes in the model
• mean returns
• but not volatility
• Use posterior model returns to optimize
expected risk-return objective
Figure: Risk vs Returns
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 21 / 29

59. Finance (Quantopian) case
Figure: Sharpe Ratio
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 22 / 29

60. Supply chains
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 23 / 29

61. year 3019, PyMC X
Supply PyMC X Demand Mars
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 24 / 29

62. Supply
Business Effect / Observed
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 25 / 29

63. Demand
∼100 /month
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 26 / 29

64. Costs and Profits
Profit ∼ Stock | Demand Profit ∼ Demand | Stock
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 27 / 29

65. Costs and Profits
Profit ∼ Stock | Demand Profit ∼ Demand | Stock
How to combine them?
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 27 / 29

66. Expected Loss Framework
Profits: Frequentist vs Bayesian
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 28 / 29

67. Take outs
Bayesian Framework allows:
• effectively use your prior knowledge
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 29 / 29

68. Take outs
Bayesian Framework allows:
• effectively use your prior knowledge
• take uncertainty in account
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 29 / 29

69. Take outs
Bayesian Framework allows:
• effectively use your prior knowledge
• take uncertainty in account
• understand your data / problem better
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 29 / 29

70. Take outs
Bayesian Framework allows:
• effectively use your prior knowledge
• take uncertainty in account
• understand your data / problem better
BUT, you should really understand your problem
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 29 / 29

71. Take outs
Bayesian Framework allows:
• effectively use your prior knowledge
• take uncertainty in account
• understand your data / problem better
BUT, you should really understand your problem
Tutorials/MISC:
pymc3 docs: https://docs.pymc.io/nb_tutorials/index.html
case studies: https://twiecki.io
Max Kochurov PyMC3 – Bayesian Statistical Modelling in Python PyData – Moscow 29 / 29