Introduction to Probability and Probabilistic Forecasting

Introduction to Probability and Probabilistic Forecasting

L i n k i n g S c i e n c e t o S o c i e t yL i n k i n g S c i e n c e t o S o c i e t y

Simon Mason

International Research Institute for Climate PredictionThe Earth Institute of Columbia University

AMS Short Course on Probabilistic Forecasting

San Diego, CA, January 9, 2005

Questions about the Future


Will it rain in San Diego next weekend?

Will it snow in San Diego next weekend?

Will the Red Sox win the 2005 World Championship?

Will Dick Cheney die a pauper?

Will this surfer live to 50?

We make forecasts to answer questions about the future:

Probability


For most situations the future is uncertain. In cases where the answer to a question about the future is uncertain, we tend to use probabilities to express this uncertainty in the outcome.

Probability and Events


We refer to a specific outcome, or a specific combination of outcomes, as an event, and refer to the probability of this event. What do these terms mean?

• event: a predefined outcome that forms the subject of a forecast (an outcome of interest) – examples:- rain in San Diego this afternoon;- tornado touch down anywhere in Iowa tomorrow;- average LAX January temperature of below 10°C;- NIÑO3 anomaly of more than +2°C by October;- global warming of more than +1°C by 2050.



Events can be:

- elementary (“hot”) or;

- compound (“hot and dry”; rain two days in a row).

Events either occur or do not occur – there are only these two possible outcomes (but there may be some uncertainty as to whether the outcome has occurred). If the event does not occur, its complement occurs.

Events need to be well-defined to avoid ambiguity. (What does “unfaithful” mean? What does “in San Diego” mean?)

Notation


An elementary event is often denoted by the letter E, and its complement by .

To distinguish an elementary event from a second elementary event, subscripts may be used:

first elementary event: E1

second elementary event: E2

A compound event occurs when the first AND the second elementary event occur (or, more generally, when all elementary events occur):

1 2E E

E



Uncertainty often is expressed using expressions such as “it is likely”, “the chances are”, etc.

Different degrees of uncertainty can be indicated: “possibly” indicates higher uncertainty than “probably”.

Compare:

- will it rain in San Diego next weekend?

- will it snow in San Diego next weekend?

• probability: a quantitative measure of the uncertainty in the event.

Probability and Uncertainty


Probabilities are used where there is uncertainty. Apart from ambiguity, there are two sources of uncertainty:

• our understanding is limited;

• there is some inherent randomness in the outcome.

• We do not know for certain what will happen.

• We cannot know for certain what will happen.

Probability and Uncertainty


But how can we quantify uncertainty?

• When probability is 1, the event will definitely occur. It is impossible for the event not to happen.

• When probability is 0, the event will definitely not occur. It is impossible for the event to happen.

• When the probability is between 0 and 1, the event may or may not happen.

Probability


• When probability is close to 1, the event is more likely to occur than not to occur.

• When probability is close to 0, the event is more likely not to occur than to occur.

• When the probability is 0.5, the event is just as likely to happen as not to happen.

Odds


• When the probability of an event, E, is 0.75, the probability that the event will not happen, the complement of the event, , is:

1 – 0.75 = 0.25

• When the probability is 0.75, the event is three times more likely to happen than not to happen:

0.75odds 30.25

P EP E

E

Probability


But how do we determine how likely the event is compared to its complement?

How do we obtain / calculate probabilities?

Interpretations of Probability: I



• What is the probability that it will rain in San Diego (at Lindberg Field) on January 31, 2005?

• How often has it rained on the same day in previous years (1927 – 2003)? Climatology.

Interpretations of Probability: I


• What is the probability that it will rain in San Diego (at Lindberg Field) on January 31, 2005?

Probability as Relative Frequency


What is the probability that it will rain in San Diego on January 31, 2005?

• Look for similar / identical situations.

• Repeat the experiment many times – only “unimportant” things are allowed to change.

Note that there may be sampling errors in the relative frequencies – uncertainty about the uncertainty! (The distribution of these sampling errors can be obtained using the binomial distribution).

Interpretations of Probability: II



• What is the probability that it will rain in San Diego (at Lindberg Field) tomorrow?

• How often has it rained on the same day in previous years with similar atmospheric conditions? Only unimportant things are allowed to change.

• This experiment has no precedent – today’s initial conditions are “important”, and they are unique.

Probability as Subjective Belief


The probability that it will rain in San Diego is best estimated by conditioning upon the current atmospheric state, a set of conditions that are unique.

• Make a forecast based on the physics of the atmosphere, and expert knowledge / experience.

• Produce an ensemble of forecasts based on sampling of known uncertainties in the physics of the atmosphere and / or in the initial conditions (Bright).

• The probability now represents the degree to which we believe that it will rain in San Diego tomorrow.

Interpretations of Probability


So two interpretations of probability are:

• relative frequency interpretation: how often the event has occurred in similar situations in the past;

• subjective interpretation: how confident we are the event will occur this time.

But all probabilities could be defined as subjective because of he subjectivity in defining which situations are “similar”.

Probability as Relative Frequency


The 77-year climatology does not provide a good estimate of the probability that it will rain in San Diego tomorrow because there are some “important” differences between the 77 instances of January 10 and January 10, 2005.

Sometimes we can improve upon climatological forecasts because of access to “important” information. But how do we know whether the information is important?

Is the probability of the event different when these conditions are present compared to when they are not?

Conditional Probabilities


The relative frequency of rainfall on January 10 could be obtained by considering only those January 10s on which January 9 rainfall occurrence was the same as on January 9, 2005.

If January 9 is wet: P(January 10 is wet)?

If January 9 is dry: P(January 10 is wet)?

1 2|P E E

1 2|P E E

This conditional probability is different from a compound event P(E1E2), because we know that E2 has (or has not) occurred already.



1 2|P E E

1P E

1 2|P E E



1 2dry Jan 10 and 9, E E

1 2

wet Jan 10 and dry Jan 9

E E 1 2

dry Jan 10 and wet Jan 9

E E

1 2

wet Jan 10 and 9E E

1wet Jan 10, E 2wet Jan 9, E

Venn diagram showing compound event:Jan 10 and 9

For conditional probabilities, the outcome of Jan 9 is known already, and so the sample space is reduced:


2dry Jan 9, E1wet Jan 10, E

1 2

wet Jan 10

|E E

1 2

dry Jan 10

E E

2wet Jan 9, E

1 2

wet Jan 10|E E

1wet Jan 10, E

1 2

dry Jan 10

E E

2wet Jan 9, E

1 2

1 2 2

|P E E

P E E P E

1 2

1 2 2

|P E E

P E E P E

Jan 10 and dry Jan 9

Jan 10 and wet Jan 9



What is the probability that it will rain in San Diego (at Lindberg Field) on January 10, 2005, given that it has (or has not) rained on January 9, 2005?

Jan 9 Wet (E2) Dry (Ē2)

Wet (E1) P(E1E2)=0.12 P(E1

Ē2)=0.10 P(E1)=0.22 Jan 10 Dry (Ē1) P(Ē1

E2)=0.10 P(Ē1Ē2)=0.68 P(Ē1)=0.78

P(E2)=0.22 P(Ē2)=0.78 1.00



What is the probability that it will rain in San Diego (at Lindberg Field) on January 10, 2005, given that it has has rained on January 9, 2005?


Wet (E1) P(E1E2)=0.12 P(E1

Ē2)=0.10 P(E1)=0.22 Jan 10 Dry (Ē1) P(Ē1

E2)=0.10 P(Ē1Ē2)=0.68 P(Ē1)=0.78

P(E2)=0.22 P(Ē2)=0.78 1.00

1 2 1 2 2|

0.12 0.220.54

P E E P E E P E



What is the probability that it will rain in San Diego (at Lindberg Field) on January 10, 2005, given that it has not rained on January 9, 2005?


Wet (E1) P(E1E2)=0.12 P(E1

Ē2)=0.10 P(E1)=0.22 Jan 10 Dry (Ē1) P(Ē1

E2)=0.10 P(Ē1Ē2)=0.68 P(Ē1)=0.78

P(E2)=0.22 P(Ē2)=0.78 1.00

1 2 1 2 2|

0.10 0.780.13

P E E P E E P E

Updating Probabilities


Based on the occurrence of January 9 rainfall, the probability of rainfall on January 10 has been updated from 0.22 to 0.54.

What if we now obtain a model forecast that states it will rain tomorrow, E3?

All we know about the model is that it has given a correct forecast 90% of the time over the last few days.

How can we update our probability for rain tomorrow?

Bayes’ Theorem


What is the probability that it will rain in San Diego (at Lindberg Field) on January 10, 2005, given that it has has rained on January 9, 2005 AND that the model forecasts rain? (To simplify, conditions on E2 are dropped.)

Model forecast Wet (E3) Dry (Ē3)

Wet (E1) P(E1E3)=? P(E1

Ē3)=? P(E1)=0.54 Jan 10 Dry (Ē1) P(Ē1

E3)=? P(Ē1Ē3)=? P(Ē1)=0.46

P(E3)=? P(Ē3)=? 1.00

1 3 1 3 3|P E E P E E P E


(E3) (Ē3) (E1) P(E1

E3) P(E1Ē3) P(E1)

(Ē1) P(Ē1E3) P(Ē1

Ē3) P(Ē1) P(E3) P(Ē3) 1.00

1 3

1 33

|P E E

P E EP E

1 3 1 3 3|P E E P E E P E


(E3) (Ē3) (E1) P(E1

E3) P(E1Ē3) P(E1)

(Ē1) P(Ē1E3) P(Ē1

Ē3) P(Ē1) P(E3) P(Ē3) 1.00

1 3

1 33

|P E E

P E EP E

1 3 1 3 3|P E E P E E P E

1 3 3 1 1|P E E P E E P E

3 1 11 3

3

||

P E E P EP E E

P E


(E3) (Ē3) (E1) P(E1

E3) P(E1Ē3) P(E1)

(Ē1) P(Ē1E3) P(Ē1

Ē3) P(Ē1) P(E3) P(Ē3) 1.00

3 1 11 3

3

||

P E E P EP E E

P E


(E3) (Ē3) (E1) P(E1

E3) P(E1Ē3) P(E1)

(Ē1) P(Ē1E3) P(Ē1

Ē3) P(Ē1) P(E3) P(Ē3) 1.00

3 1 11 3

3

||

P E E P EP E E

P E

3 1 11 3

1 3 1 3

||

P E E P EP E E

P E E P E E

3 1 11 3

3 1 1 3 1 1

||

| |P E E P E

P E EP E E P E P E E P E

Bayes’ Theorem



Wet (E1) P(E1E3)=? P(E1

Ē3)=? P(E1)=0.54 Jan 10 Dry (Ē1) P(Ē1

E3)=? P(Ē1Ē3)=? P(Ē1)=0.46

P(E3)=? P(Ē3)=? 1.00

1 3 1 3 3|P E E P E E P E

All terms on the right are unknown

3 1 11 3

3 1 1 3 1 1

||

| |P E E P E


The priors, at least, are known …

Bayes’ Theorem



Wet (E1) P(E1E3)=? P(E1

Ē3)=? P(E1)=0.54 Jan 10 Dry (Ē1) P(Ē1

E3)=? P(Ē1Ē3)=? P(Ē1)=0.46

P(E3)=? P(Ē3)=? 1.00

3 1 3 1| |P E E P E E

are likelihoods: they tell us how likely it is that rain was forecasted, assuming that it will / will not rain, respectively. Or: how often are rain days successfully forecasted / dry days unsuccessfully forecasted?

Bayes’ Theorem



Wet (E1) P(E1E3)=? P(E1

Ē3)=? P(E1)=0.54 Jan 10 Dry (Ē1) P(Ē1

E3)=? P(Ē1Ē3)=? P(Ē1)=0.46

P(E3)=? P(Ē3)=? 1.00

3 1 11 3

3 1 1 3 1 1

||

| |P E E P E


We do not have exact values for the likelihoods on the right side of the equation, but if we assume that the model has no bias, given that it has been correct 90% of the time, we can infer that 90% of the rain days have been forecasted.

Bayes’ Theorem



Wet (E1) P(E1E3)=? P(E1

Ē3)=? P(E1)=0.54 Jan 10 Dry (Ē1) P(Ē1

E3)=? P(Ē1Ē3)=? P(Ē1)=0.46

P(E3)=? P(Ē3)=? 1.00

3 1 11 3

3 1 1 3 1 1

||

| |

0.9 0.540.9 0.54 0.1 0.460.91

P E E P EP E E

P E E P E P E E P E

Bayes’ Theorem


Bayes’ theorem allows us to update probabilities (posterior probabilities):

The prior probabilities are imply the the best estimate of the probabilities before considering the new information. The may already have been previously updated.

The likelihoods indicate how likely the new information is, assuming a specific outcome. For example: how likely is it that the forecast would be for wet conditions assuming that it is going to be wet / dry. (I.e., the hit and false alarm rates of the ROC.)



A problem with conditional probabilities is that the sample space is reduced, and so the errors in estimating the relative frequencies increases.

These errors increase as the number of conditions is increased, and it is easy to reach the extreme case of having no previous cases with only “unimportant” differences from which to calculate the relative frequencies. (number of possible states = 2n).

In numerical weather prediction the infinite dimensions of the current atmospheric state are important, and so the current initial conditions are unique.



What if the probability of rainfall tomorrow depends on how much rainfall there is today rather than just its occurrence?

In this case the outcome of the current event is not dependent upon another event measured on a binary scale.

Jolliffe – some statistical models for calculating probabilities of events that are functions of continuous variables.



Similarly, many forecast verification procedures are based on conditional probabilities:

• reliability: given a forecast of 90% chance of rain, how often does rain occur? P(E|F=f)

• NB – notice that this involves a subjective interpretation of probability – we are verifying forecasts with similar levels of confidence, not forecasts with similar boundary / initial conditions.



• resolution: can we expect a different outcome given a different forecast?

Note reliability and resolution are often confused.

Resolution: is P(E) conditional upon the forecast?

Reliability: if F=f does P(E|F=f)=f?


REL = (BC)2

REL = (EC)2

RES = (AC)2

Note the y-axis gives the conditional probability of the event given the forecast.


RELIABILITY:Are the forecast probabilities correct?Do the forecast probabilities reflect an appropriate level of confidence?

RESOLUTION:Does the outcome depend on the forecast?Do different forecast probabilities imply actual differences in the probability of an event?

Reliability and Resolution

Documents

Introduction to Probability and Probabilistic Forecasting