asaf cohen department of mathematics university of michigan financial mathematics seminar university...
TRANSCRIPT
PARAMETER ESTIMATION: THE PROPER WAY TO USE BAYESIAN
POSTERIOR PROCESSES WITH BROWNIAN NOISE
Asaf CohenDepartment of Mathematics
University of Michigan
Financial Mathematics SeminarUniversity of Michigan
September 10, 2014
1
Contents
Motivation: Brownian Motion with Unknown Drift
Introduction: Bayesian Parameter Estimation, Examples
Current & Future Work
Scaled Renewal Processes Brownian Motion (with unknown drift)
The Posterior Processes and their Limits
2
Contents
Motivation: Brownian Motion with Unknown Drift
Introduction: Bayesian Parameter Estimation, Examples
Current & Future Work
The Posterior Processes and their Limits
2
Scaled Renewal Processes Brownian Motion (with unknown drift)
A DM (Decision Maker) wants to estimate a parameter . His i.i.d. (given ) observations are
Introduction: Bayesian Parameter Estimation, Examples
Bayesian Parameter Estimation
• has a prior distribution
• The DM chooses an estimator that minimizes the expected loss function
Bayesian
• A fundamental tool is the posterior distribution
Parameter Estimation
3
e.g.,
Sampling is stopped accordance with a stopping rule
• The sample size is not fixed in advance .
4
Introduction: Bayesian Parameter Estimation, Examples
Bayesian Sequential Parameter Estimation
• A fundamental tool is the posterior process
• The DM chooses an estimator that minimizes the expected loss function
• has a prior distribution
Bayesian Parameter EstimationSequential
e.g.,
A DM wants to estimate a parameter
• The samples are observed continuously
5
Introduction: Bayesian Parameter Estimation, Examples
Bayesian Sequential Parameter Estimation with Renewal Process
• has a prior distribution
t
2
3
Time
1
• The posterior process
A DM wants to estimate a parameter
• The DM continuously observes
6
Introduction: Bayesian Parameter Estimation, Examples
Bayesian Sequential Parameter Estimation with Brownian Noise
• has a prior distribution
• The posterior process
A DM wants to estimate a parameter Standard Brownian Motion
noise coefficient
• The DM continuously observes
6
Introduction: Bayesian Parameter Estimation, Examples
Brownian Motion with Unknown Drift
• has a prior distribution
• The posterior process
A DM wants to estimate a parameter Standard Brownian Motion
noise coefficient
Contents
Motivation: Brownian Motion with Unknown Drift
Introduction: Bayesian Parameter Estimation, Examples
Current & Future Work
The Posterior Processes and their Limits
7
Scaled Renewal Processes Brownian Motion (with unknown drift)
• Observation:
8
The Model
• has a prior distribution
• The posterior process
A DM wants to estimate a parameter
Motivation: Brownian Motion with Unknown Drift
9
Motivation: Brownian Motion with Unknown Drift
The Model
• has a prior distribution
e.g.,
• A common solution: A First Exit Time of
A First Exit Time of the posterior process
A DM wants to estimate a parameter .
• Observation:
10
Literature
Motivation: Brownian Motion with Unknown Drift
• Kalman and Bucy (1961)• Zakai (1969) - Bayesian Posterior Process
Filtering Theory
• Shiryaev (1978) - formulated the problem (discrete and continuous time)• Gapeev and Peskir (2004) - Finite Horizon• Gapeev and Shiryaev (2011) - General Diffusion• Zhitlukhin and Shiryaev (2011) - Three hypotheses• Buonaguidi and Muliere (2013) - Levy Processes
Sequential 2-Hypothesis Testing
• Berry and Friestedt (1985) - formulated the problem• C. and Solan (2013) - Levy Processes
Bayesian Brownian Bandit
EconomicsBolton and Harris (1999), Felli and Harris (1996), Bergemann and Valimaki (1997), Keller and Rady (1999), Moscarini (2005) , Jovanovic (1979)...
11
Motivation: Brownian Motion with Unknown Drift
11
Motivation: Brownian Motion with Unknown Drift
12
Is This Model Useful?
• has a prior distribution
e.g.,
• A common solution: A First Exit Time of
• Observation:
A First Exit Time of the posterior process
A DM wants to estimate a parameter .
a “discrete process”
• What if ?
• Advantages of the Brownian model: easier for investigation
Motivation: Bayesian Parameter Estimation with Brownian Noise
Slide from before
13
Is This Model Useful?
Motivation: Bayesian Parameter Estimation with Brownian Noise
• Kalman and Bucy (1961)• Zakai (1969) - Bayesian Posterior Process
Filtering Theory
• Shiryaev (1978) - formulated the problem (discrete and continuous time)• Gapeev and Peskir (2004) - Finite Horizon• Gapeev and Shiryaev (2011) - General Diffusion• Zhitlukhin and Shiryaev (2011) - Three hypotheses• Buonaguidi and Muliere (2013) - Levy Processes
Sequential 2-Hypothesis Testing
• Berry and Friestedt (1985) - formulated the problem• C. and Solan (2013) - Levy Processes
Bayesian Brownian Bandit
EconomicsBolton and Harris (1999), Felli and Harris (1996), Bergemann and Valimaki (1997), Keller and Rady (1999), Moscarini (2005) , Jovanovic (1979)...No Justification
Slide from before
14
Is This Model Useful?
Suppose that
1. For a given rule of strategy does the losses satisfy
2. Are the optimal rules of strategies for the observed processes and relatively close?
3. Are the optimal losses relatively close?
Usually not!
Usually not!
Usually not!So… Why to study Bayesian parameter estimation with Brownian noise?
Motivation: Brownian Motion with Unknown Drift
Questions:
14
Is This Model Useful?
Suppose that
1. For a given rule of strategy does the losses satisfy
2. Are the optimal rules of strategies for the observed processes and relatively close?
3. Are the optimal losses relatively close?
Usually not!
Usually not!
Usually not!So… Why to study Bayesian parameter estimation with Brownian noise?
Motivation: Brownian Motion with Unknown Drift
Questions:
Weird phenomena: a different approximation works….
How to calculate?Fundamental tool: posterior processes
Why different?
Contents
Motivation: Brownian Motion with Unknown Drift
Introduction: Bayesian Parameter Estimation, Examples
Current & Future Work
The Posterior Processes and their Limits
15
Scaled Renewal Processes Brownian Motion (with unknown drift)
are i.i.d. distributed as the RV
16
Renewal Process with Rate 1
Scaled Renewal Processes Brownian Motion (unknown drift)
2
3
Time
1
•
•
Renewal Process with Rate
2
3
1
tTime
17
High Rate
Take . How behaves?
By proper scaling:Similar to Brownian Motion
FCLT
Scaled Renewal Processes Brownian Motion (unknown drift)
18
High and Unknown Rate - The Proper Scaling
Assumption
• is chosen at time
• The DM continuously observes
FCLT
Why ?
Scaled Renewal Processes Brownian Motion (unknown drift)
Contents
Motivation: Brownian Motion with Unknown Drift
Introduction: Bayesian Parameter Estimation, Examples
Current & Future Work
The Posterior Processes and their Limits
19
Scaled Renewal Processes Brownian Motion (with unknown drift)
20
The Posterior Processes and their Limits
The Posterior Processes
21
The Posterior Processes and their Limits
Diffusion Limits
Assumption
has a density with the support
2
3
Time
1 •
•
•
22
The Posterior Processes and their Limits
Diffusion Limits
Assumption
has a density with the support
Theorem 1
Why ?
23
The Posterior Processes and their Limits
1.
2.
?
Using the Taylor’s Series3.
4.
Origins of
24
The Posterior Processes and their Limits
Heuristics for
Why ?
Recall that
The smaller : “easier” to estimate larger
When is a “large” R.V.?
• The interarrival time
for the discussion, or
The Posterior Processes and their Limits
Why ?
recall that
the smaller : “easier” to estimate larger
When is a “large” R.V.?
• The interarrival time
for the discussion, or
“easier” to estimate ? ?
x25
Heuristics for
26
The Posterior Processes and their Limits
The New
Theorem 2
, and equality holds iff .
What is the relation between and ?
In fact can be arbitrary large
So, can be very different from
27
The Posterior Processes and their Limits
Intuition (Theorem 2)
• is a sufficient statistic for :
• Usually is not a sufficient statistic for :
• However, if , it is sufficient!
and since, then .
28
The Posterior Processes and their Limits
Proof (Theorem 2)
Cauchy–Schwarz
Equality iff and are linearly dependent
iff
29
The Posterior Processes and their Limits
Back to the Motivation
Theorem 3
Since can be arbitrary large
can be very different from
Contents
Motivation: Brownian Motion with Unknown Drift
Introduction: Bayesian Parameter Estimation, Examples
The Posterior Processes and their Limits
Current & Future Work
30
Scaled Renewal Processes Brownian Motion (with unknown drift)
2. - replace FCLT with stable FCLT.
31
Current & Future Work
A) Behavior of posteriors in more general frameworks:
B) The Disorder Problem
1. General diffusions (e.g., Zakai Equation type).
Assume that at some unobservable random time the drift of a Brownian motion (rate of an arrival process) changes.
similar structure.Find the posterior belief that the change has already occurred?
32
Current & Future Work
C) Apply the results in context of queues with uncertainty about the service/arrival rates, asymptotic solution using Bayesian bandits.
D) Apply the results in context of risk processes.
Arrival processGeneral CDF • customers: Bandit (index solution)
• customers: non-classical Bandit (empty periods)
Asymptotic solution under heavy traffic using BM with unknown driftrouter
THANK YOU!
33