Estimating the Peak Magnification of a Microlensing Event

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Estimating the Peak Magnification of a
Microlensing Event

• Michael Albrow
• Department of Physics and Astronomy
• University of Canterbury

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Background

• Fits based on early data often predict very large
  peak magnifications (usually with large
  uncertainties).
• In the majority of cases these events turn out to
  be low magnification.

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OGLE-2003-BUL-331
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The problem

• Methods based on ? 2 minimisation do not produce
  the most probable parameters.
• Maximising the likelihood function (minimising ?
  2 ) gives the parameter values that make the data
  most probable.
• We are interested in obtaining the most probable
  values of the parameters given the data.

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Bayes theorem

• P(?D,O) a P(D?,O) P(?O)
• We have prior statistical knowledge about
• the underlying distribution of ?.

Underlying probability of ? given an alert
Probability of ? given an alert and with data D
likelihood function exp(-? 2/2)
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2002 OGLE III
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OGLE-2003-BUL-331
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2003 OGLE Distribution of Brightness Increases
All data
? 2 at alert
Bayesian at alert
Ibase Ipeak
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2003 OGLE Distribution of u0
All data
? 2 at alert
Bayesian at alert
U0
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2003 OGLE Distribution of Observed Predicted
Peak Magnitudes
? 2
Bayesian
I0-I0,predicted at alert
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Typical low magnification eventOGLE-2003-BUL-171
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Typical high magnification eventOGLE-2003-BUL-208
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High magnification eventOGLE-2003-BUL-262
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Suggested follow up strategy

• Only follow events where the Bayesian
• prediction A0 gt 4.
• For all high magnification cases examined,
• this occurs when A lt 3.

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Generic Behaviour

• ? 2 minimisation
• Rapid parameter changes with new data
• Biased estimate of parameters
• Tends to overpredict peak magnification
• Probability maximisation
• Smooth parameter changes with new data
• Unbiased estimate of parameters
• Usually converges to the correct solution earlier
• Tends to initially underpredict peak
  magnification for high magnification events
• High magnification nature is identified early
  enough for observation

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2002 OGLE III
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