Bayesian Statistics conjugate prior distributions
Project description
Conjugate Prior
Python implementation of the conjugate prior table for Bayesian Statistics
See wikipedia page:
https://en.wikipedia.org/wiki/Conjugate_prior#Table_of_conjugate_distributions
Installation:
pip install conjugate-prior
Supported Models:
BetaBinomial
- Useful for independent trials such as click-trough-rate (ctr), web visitor conversion.BetaBernoulli
- Same as above.GammaExponential
- Useful for churn-rate analysis, cost, dwell-time.GammaPoisson
- Useful for time passed until event, as above.NormalNormalKnownVar
- Useful for modeling a centralized distribution with constant noise.NormalLogNormalKnownVar
- Useful for modeling a Length of a support phone call.InvGammaNormalKnownMean
- Useful for modeling the effect of a noise.InvGammaWeibullKnownShape
- Useful for reasoning about particle sizes over time.DirichletMultinomial
- Extension of BetaBinomial to more than 2 types of events (Limited support).
Basic API
model = GammaExponential(a, b)
- A Bayesian model with anExponential
likelihood, and aGamma
prior. Wherea
andb
are the prior parameters.model.pdf(x)
- Returns the probability-density-function of the prior function atx
.model.cdf(x)
- Returns the cumulative-density-function of the prior function atx
.model.mean()
- Returns the prior mean.model.plot(l, u)
- Plots the prior distribution betweenl
andu
.model.posterior(l, u)
- Returns the credible interval on(l,u)
(equivalent tocdf(u)-cdf(l)
).model.update(data)
- Returns a new model after observingdata
.model.predict(x)
- Predicts the likelihood of observingx
(if a posterior predictive exists).model.sample()
- Draw a single sample from the posterior distribution.
Coin flip example:
from conjugate_prior import BetaBinomial
heads = 95
tails = 105
prior_model = BetaBinomial() # Uninformative prior
updated_model = prior_model.update(heads, tails)
credible_interval = updated_model.posterior(0.45, 0.55)
print ("There's {p:.2f}% chance that the coin is fair".format(p=credible_interval*100))
predictive = updated_model.predict(50, 50)
print ("The chance of flipping 50 Heads and 50 Tails in 100 trials is {p:.2f}%".format(p=predictive*100))
Variant selection with Multi-armed-bandit
Assume we have 10
creatives (variants) we can choose for our ad campaign, at first we start with the uninformative prior.
After getting feedback (i.e. clicks) from displaying the ads, we update our model.
Then we sample the DirrechletMultinomial
model for the updated distribution.
from conjugate_prior import DirichletMultinomial
from collections import Counter
# Assuming we have 10 creatives
model = DirichletMultinomial(10)
mle = lambda M:[int(r.argmax()) for r in M]
selections = [v for k,v in sorted(Counter(mle(model.sample(100))).most_common())]
print("Percentage before 1000 clicks: ",selections)
# after a period of time, we got this array of clicks
clicks = [400,200,100,50,20,20,10,0,0,200]
model = model.update(clicks)
selections = [v for k,v in sorted(Counter(mle(model.sample(100))).most_common())]
print("Percentage after 1000 clicks: ",selections)
Naive Recommendation System with UCB
from conjugate_prior import BetaBinomialRanker ranker = BetaBinomialRanker(prior=0.1) # 10% click-through-rate ranker["cmpgn1"]+=(1,9) # 1 click, 9 skips ranker["cmpgn2"]+=(10,90) # 10 click, 90 skips ranker["cmpgn3"]+=(1,2) # 1 click, 3 skips
Balance exploration and exploitation w/UCB
print(ranker.rank_by_ucb())
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