The Most Likely Distribution

Jaynes’ Principle of Uncertainty

In making inferences on the basis of partial information we must use the probability distribution which has maximum uncertainty subject to whatever is known

– after E.T. Jaynes (1957)

The central idea behind Jaynes’ Principle is that one accepts whatever information is available then interprets it in the widest sense consistent with the evidence at hand. The consequences to the play of a bridge hand are easily stated. Whether we think of maximum uncertainty or ratios of card combinations, it comes to the same thing: assume the most even splits possible under the current set of circumstances. Of course, calls and plays provide additional amounts of partial information, so the most likely distribution of the sides may change. One starts with a candidate distribution consisting of even splits, but one must not be over-reliant of just one distribution. That having been said, one must start somewhere, and the most likely distribution of sides is the preferred choice. This focuses the mind in the correct manner, at least as far as probability is concerned.

We illustrate below how this works in the simple situation where the bidding has gone 1NT – 3NT and a spade is led. This is the classical ‘blind lead’ situation where the 4th highest lead in a major suit is the common choice. We have discussed previously how an opening lead places restrictions on distributions, the more informative the lead, the more severe the restrictions and the fewer the distributions that remain to be considered. Now we consider how the most likely distribution of sides relates to declarer’s inferences.

The 8=7=6=5 Division of Sides

The opening lead is chosen the dummy is tabled, and declarer attempts to form a plan based on the information that has been made available to him. With regard to the division of sides, declarer knows immediately how many cards the opponents hold in each suit. In the absence of interference bidding he may obtain from the opening lead a good idea of how the cards in each suit are distributed between the defenders. The distribution of the cards encompasses the greatest number of card combinations is the most probable single distribution. We call this distribution the maximum likelihood estimate. As an example we shall consider the sides of 8=7=6=5. The most even splits are: Spades: 4-4, Hearts 4-3 or 3-4, Diamonds 3-3, and clubs 3-2 or 2-3. These even splits have to be assembled into combinations that make up 13 cards to a side, so some restrictions apply. Here are some of the more likely splits:

I	II	III	IV	V
♠ 4 – 4	♠ 4 – 4	♠ 5 – 3	♠ 4 – 4	♠ 5 – 3
♥ 3 – 4	♠ 4 – 3	♥ 3 – 4	♥ 3 – 4	♥ 4 – 3
♦ 3 – 3	♦ 3 – 3	♦ 3 – 3	♦ 4 – 2	♦ 2 – 4
♣ 3 – 2	♣ 2 – 3	♣ 2 – 3	♣ 2 – 3	♣ 2 – 3
100	100	80	75	60

 

The number below a distribution is the relative number on a scale of 100 of the card combinations encompassed by that distribution. Condition I is the most likely single distribution after a spade is led from length. Although Condition II encompasses the same number of card combinations, the hearts are of equal length so there is a good chance that a heart might be led instead of a spade. In the absence of clues from the bidding, it is reasonable to assume that a heart would be chosen for half of the combinations encompassed, so the weight after a spade is led should be reduced by half to 50, making it less likely that Conditions III – V.

The splits in the minors are of interest as well. The 5 conditions shown encompass the most even split in diamonds, 3-3, but also the 2-4 and 4-2 splits. The fact that a spade was led from length alters the odds in favor of the even split. The club splits divided between 3-2 and 2-3 with the latter more favored on the limited selection shown.

The Spot Card Effect

For the purposes of our illustration, the assumption is that a spade will be led whenever spades is the longest suit, the spades are equal in length to a minor, and half the time spades are of equal length with hearts. When one reads in an analysis of a deal that ‘a low spade was led’, one has been poorly informed. Which spade makes a difference to the odds, because sometimes declarer can make a pretty good guess as to whether or not the lead was from a 4-card or a 5-card suit. First we assume the lead is the ♠ 2 which looks very much like a lead from a 4-card suit. Given this is a long-suit lead, the possible distributions are reduced in number to 10, 4 of which contain a 4-card heart suit. For these we reduce the number of combinations by half. We can calculate the number of combinations for the relevant splits in hearts, diamonds, and clubs and express them as percentages of the total available for a given suit, as follows.

Hearts	4 – 3	3 – 4	2 – 5	1 – 6		Heart Left	43%
	28%	52%	18%	2%		Heart Right	57%
Diamonds	4 – 2	3 – 3	2 – 4	1 – 5		Diamond Left	52%
	36%	44%	19%	2%		Diamond Right	48%
Clubs	4 – 1	3 – 2	2 – 3	1 – 4		Club Left	56%
	20%	45%	30%	5%		Club Right	44%

 

It is quite according to expectations that the hearts are most likely to split 3-4, as some combinations of 4-3 are partially eliminated by our assumption of parity. It is also expected that the diamonds would split evenly at 3-3, but there is an unexpected bias to the left in favor of the 4-2 split. Clubs also exhibit a left-hand bias with the 3-2 split half again as likely as a 2-3 split. The conclusion is that, although the spade lead appears at first glance to create a vacant place on the left, it cannot be concluded that a missing honor in a minor is more likely to be on the right. In fact, the contrary tendency applies.

How do we make sense of this conclusion? Easily, if one considers the maximum likelihood estimate expressed under Condition I shown above. Putting together the most frequent splits (modes) in each suit as shown above, and one obtains ♠ 4-4, ♥ 3-4, ♦ 3-3, and ♣ 3-2, which constitutes Condition I. It is good practice to consider the maximum likelihood distribution as a starting point as one is made conscious of the modes of the distributions of the suits taken individually. Remember this: the suit combinations can vary independently only to the degree that the sum of the cards in the suits must come to 13 in the end, and the degree of variation depends on the number of cards held in the suit.

Of interest is the probability that a given card in a suit, say a queen, will be dealt to the opening leader on the left or to his partner on the right. The ♥ Q is likely to be on the right roughly in the proportion of a 3-4 split. The ♦ Q is just slightly more likely to be on the left. The ♣ Q is actually more likely to be with the opening leader, roughly in the ratio of a 3-2 split. Thus Condition I provides a first approximation of the suit-dependent odds of finding a queen on the left or right.

The Lead from a Sparse Suit

The restrictions are more severe after a diamond is led, under the assumption that it is the longest suit or in a tie with clubs.

Spades	4 – 4	3 – 5	2 – 6			Spade Left	37%
	20%	63%	17%			Spade Right	62%
Hearts	4 – 3	3 – 4	2 – 5	1 – 6		Heart Left	40%
	12%	61%	22%	4%		Heart Right	60%
Clubs	4 – 1	3 – 2	2 – 3	1 – 4		Club Left	51%
	7%	49%	29%	15%		Club Right	49%

 

The maximum likelihood estimate of the distribution given a diamond is a long-suit lead is ♠ 3-5, ♥ 3-4, ♦ 4-2, ♣ 3-2 (weight =60) which is also the combination of the most likely splits taken individually in each suit. The information in the diamond lead indicates a major honor card is strongly favored to be on the right, that is, not in the hand of the opening leader. The most likely splits give a fairly good approximation of the probability of catching a major suit queen on the right (3:2).

The uncertainty is again a maximum for the club suit, as it is nearly 50-50 whether the ♣ Q would be on the right. The average number of clubs on the right or left is 2.5, an impossible number of cards to be dealt, which reflects the uncertainty. It goes against intuition, perhaps, that a 3-2 club split is more likely than a 2-3 split, that is, the longer club is more frequently with the opening leader. This apparent inconsistency can be resolved by considering the maximum likelihood estimate which is a result of taking into account all suits and how they interact.

The Effects of Uncertain Length

The effect of uncertainty is to increase the number of possible combinations that must be taken into consideration. If the spot card lead is unreadable, one must allow more suit combinations to enter the mix. We shall assume that spades can be 4, 5 or 6 cards in length. Adding these possibilities we find the following frequency of splits.

Hearts	5 – 2	4 – 3	3 – 4	2 – 5	1 – 6		Heart Left	43%
	2%	28%	44%	2%	2%		Heart Right	57%
Diamonds	5 – 1	4 – 2	3 – 3	2 – 4	1 – 5		Diamond Left	48%
	4%	29%	44%	22%	1%		Diamond Right	52%
Clubs	4 – 1	3 – 2	2 – 3	1 – 4			Club Left	49%
	12%	35%	38%	14%			Club Right	51%

 

The modes for the splits are: ♥ 3-4, ♦ 3-3, and ♣ 2-3 as before, so Condition III becomes the modal estimate. The 3-4 split in hearts is favored greatly, and overall any particular heart, say, the ♥ Q, is more likely to sit on the right with odds approximately those of a 3-4 split. The odds of particular diamond (♦ Q) or a particular club (♣ Q) sitting on the right is close to 50%, but with a slight bias towards the right. The margin is roughly that provided by a vacant place split of 12 on the left and 13 on the right in keeping with the a priori odds adjusted to the exposure of one card on the left.

Why has Condition I lost the status of the modal distribution? The reason is obvious: the inclusion of the possibilities of 5-card and 6-card spade suits, but not of 3-card suits, results in the average length of spades being 4.6, even though spades will be only 4-cards in length nearly half of the time (48%). The pressure of ‘virtual’ vacant places on the right is overcome by a flip from a 3-2 club split to a 2-3 club split, as one finds in Condition III. The heart and diamond splits are the same for both conditions, so the flip in the shortest odd-numbered suit, clubs, by itself accommodates the additional spade length. Common sense tells us that we shouldn’t put all our money on Condition I, which requires a lead from a 4-card suit, but keep in mind Condition III just in case the lead was from a 5-card suit. (Frivolous overcalls help.)

Opening Leads and Honors

Books have been written on how to choose an opening lead. The defender must take into account the opposition bidding and the location of his high cards. It is considered dangerous to choose a suit with gaps in the honors, because even a lead of Q from QJ98 can come to grief. Declarer knows the division of sides and which high cards are missing, but he doesn’t know on which side of the table they sit unless the opening lead is from an honor sequence. Generally in the absence of bidding the honors are divided evenly between the defenders, and consideration of the most likely distribution of sides remains a valid approach. If the lead is an honor card, that occurrence constitutes additional information that isn’t dictated by length alone, but it is more likely to have been made from a sequence in a longer suit rather than a shorter one. The most difficult situation to read when a long-suit lead is avoided because of gaps in the honors held.

Symmetry and the A Priori Argument

Some bridge analysts are reluctant to use the evidence of the opening lead as justification for an adjustment of the odds on the location of a particular card of interest, say the ♥ Q. A characteristic of the a priori conditions is symmetry, which is destroyed on the opening lead, but this feature remains central in the minds of some. To see how the opening lead has affected the odds, we can look at the most likely candidates involving the 4-3 and 3-4 splits in hearts which initially are equally probable.

I	II	III	IV	V	VI
♠ 4 – 4	♠ 4 – 4	♠ 5 – 3	♠ 3 – 5	♠ 5 – 3	♠ 3 – 5
♥ 3 – 4	♠ 4 – 3	♥ 3 – 4	♥ 4 – 3	♥ 4 – 3	♥ 3 – 4
♦ 3 – 3	♦ 3 – 3	♦ 3 – 3	♦ 3 – 3	♦ 2 – 4	♦ 4 – 2
♣ 3 – 2	♣ 2 – 3	♣ 2 – 3	♣ 3 – 2	♣ 2 – 3	♣ 3 – 2
100	100	80	80	60	60

 

The distributions form symmetric pairs with equal probabilities before a lead is made. The opening lead destroys the symmetry in probabilities. Under Condition II a heart is as likely to have been led as a spade, hence the number of combinations represented must be reduced by half. Under Conditions IV and VI a spade would not be led. So for these 6 conditions a spade lead adjusts the probabilities as follows:

4-3 split	50 + 60 = 110	38%	Heart Left	48%
3-4 split	100 + 80 = 180	62%	Heart Right	52%

 

Of course, if the spade lead were to come from the right instead of the left, the odds would be reversed with the 4-3 split the more probable. As the opening lead can be made from either side with equal probability, it is correct to say that the 4-3 and 3-4 splits are equally probable before a lead is made. But such is not the case after a spade is led from the left. Hypothetically, the lead could have been a spade from the right, not the left, but there is no evidence to support that assumption on this particular deal.

We note that on average the probability of the location of the ♥ Q corresponds to 12 vacant places on the left and 13 on the right. This average involves all remaining possible heart splits, but because the number of hearts is an odd number, there is no single split that closely reflects those odds, as there is with a 3-3 split in diamonds. The hearts can be split 4-3 or 3-4, but not both at the same time. The odds on the location of the ♥ Q with the hearts taken in isolation will be either 4:3 or 3:4. Rather than look at averages over several possible splits, one should consider the mode of the splits, and initially a 3-4 split clearly represents the greatest frequency of occurrence. As play progresses the general trend is to retain the more even splits. If a split in another suit gets established, that information will affect the odds of the heart splits. It may turn out eventually that a stage is reached where the 4-3 and 3-4 splits again become equally likely, but we can’t assume that will happen. An example: with ♠ 5-3 and ♦ 2-4, the distribution ♠ 5-3; ♥ 4-3; ♦ 2-4; ♣ 2-3 has the same weight as the non-symmetric ♠ 5-3; ♥ 3-4; ♦ 2-4; ♣ 3-2. This stage will be reached rarely, as it is normal for declarer to play on clubs before diamonds.