Transcript ntroduction to Bayesian Statistics

Bayesian vs. frequentist inference
frequentist:
1) Deductive hypothesis testing of Popper--ruling out
alternative explanations
Falsification: can prove that a theory is false by finding
contradicting evidence but, cannot confirm a theory
(may find a better theory or a falsification)
2) Statistical methods of Fisher – everything you have
learned this semester
Bayesian inference:
1) Requires explicit assignment of prior probabilities,
based on existing information, to the outcomes of
experiments.
2) Allows for assignment of probabilities of specific
outcomes
The frequentist assumption: there is a true, fixed value
for parameters of interest (eg. ave weight of yellow perch
in Lake Erie, ave height of college students) and the
expected value of the parameter equals the average
value obtained by random sampling (infinitely repeated).
Some problem:
-populations change constantly, so nothing really fixed
-truly random sample hard to obtain (convenience samples)
-experiments rarely repeated
Confidence in (frequentist) parameter estimates:
-K% confidence interval around Xbar (usually 95%)
-Tells us that  will lie within (Xbar1.96SE) in 95%
of the infinite number of possible samples we could
collect
-It does not state that there is a 95% probability that
the true mean does occur in the confidence interval
Frequentist hypothesis testing:
--The P-value of a statistical test is the probability of
observing the given result conditional on H0.
 = P(x|H0) = probability of x, given H0
-- The P-value does not say how probable the null is
given the data i.e., P(H0|x)
--”Proving” the null is false, does not prove the
alternative to be true.
What we really want to know!!
P (HA|x)
Probability of the alternative hypothesis
given the observed data
Likelihood that the treatment caused the
effect (for experiments)
Many researchers consider that a P-value <
0.05 means that P (HA|x) is high, in other
words they think it likely that the alternative
hypothesis is true …… when they really
tested P (x|H0).
Bayesian parameter estimation:
Begins with the joint probability of two events being
equal to the probability of the first event and the
conditional probability of the second event, given the
first event. It is usually expressed:
P(B|A) * P(A)
P (A|B) =
P(B)
Bayes’ Theorem
Rev Thomas Bayes
(b. 1702, London - d. 1761, Tunbridge Wells, Kent),
mathematician who first used probability inductively and
established a mathematical basis for probability inference (a
means of calculating, from the number of times an event has
not occurred, the probability that it will occur in future trials).
He set down his findings on probability in "Essay Towards
Solving a Problem in the Doctrine of Chances" (1763),
published posthumously in the Philosophical Transactions of
the Royal Society of London.
Consider 2 racehorses:
--Buttercup and Muffin
--Buttercup has won 7 of the last 12 races (53.8%)
between the two horses
--Muffin has won 5 of the last 12 (41.7%)
--So…… there is a greater likelihood of Buttercup
winning, right?
Unless, it is raining……….
--On 3 of Muffin’s previous 5 wins, it was raining
--So, if it is raining, you could estimate Muffin’s chance
of a win at 3/5 or 60%
--But then you ignore the fact that, overall Buttercup has
won more often.
--You must combine the two pieces of information
--Look at 4 possible situations
Raining
Not raining
3
2
Muffin looses 1
6
Muffin wins
--If it is raining on the day you place your bet, you want
to know the probability of Muffin wining in the rain.
-- # times X happened / # time X could have happened
--3 / 4 = 75% chance of Muffin winning
P(B|A) * P(A)
P (A|B) =
P(B)
P (Muffin wins|rain) = P (rain|Muffin wins) * P (Muffin wins)
P (rain)
It was raining on 3 of 5 days Muffin won, therefore,
P (rain|Muffin wins) = 3/5=0.6
P (Muffin wins) = 5/12 = 0.417
P (rain) = 4/12 = 0.333
P (Muffin wins|rain) = P (rain|Muffin wins) * P (Muffin wins)
P (rain)
P (Muffin wins|rain) = 0.6 * 0.417 / 0.333
= 0.758
P(B|A) * P(A)
P (A|B) =
P(B)
--In Bayesian terminology P(A) is the prior probability
of obtaining a specific parameter
--P(A) is the probability of observing A expected by the
investigator before the experiment is conducted
-- existing data
--objective statement without data
--subjective measure of belief
--Priors in ecology usually based on previous data,
information usually reported in the introduction of
papers.
Non-mathematical explanation of prior probabilities
--You want to estimate the whole season batting
average of a Player X based on the first 2 weeks of the
season
--Player X batted .550 during the first two weeks
--Any player is very unlikely to bat >.500 for the entire
season
-- The players, or the entire league’s batting average
from the last year could be used as a prior probability
-- Common sense dictates that Player X’s average will
tend down from .550 and toward last season’s total
average
--prior probabilities can be “non informative” in the
absence of any inforamtion
--in that case P(A) is a uniform distribution, with all
values equally likely
--Posterior probability is what you are trying to
figure out, the probability of the event in which you
are interested.
Bayesian Hypothesis testing:
--Can generalize; Bayes theorem can be extended to
assess the relative probabilities of alternative
hypotheses (alternative prior probability distributions)
--Can determine the likelihood of an hypothesis
posterior odds = prior odds * Bayes factor
-- different scales of Bayes factor proposed to say
whether the data favor HA
--More intuitive and satisfying than frequentist
statistics, but harder to do!
-- Math harder (integration) and little software
currently available
All students be prepared to answer these questions
?? As N increases, it becomes less likely to accept the
Null at a fixed alpha level. What guidelines should we use
to adjust alpha as sample size increases?
?? How can policy makers use a rejection of a null
hypothesis at the p=0.05 level to make a decision? What
does rejection at the 0.05 level mean anyway?
?? Why do researcher act as if they have tested P (HA|x)
when they really tested P (x|H0)? Do most researchers
realize what they are testing? Or do they realize and not
care….. They just want to support that ol’ alternative
hypothesis?
?? What will motivate statisticians to write easy to use
Bayesian software?